Post on 31-Jan-2016
description
Nikos SKANTZOS
2010
Stochastic methods in Finance
Fair price What is the fair price of an
option Consider a call option on an
asset
Delta hedging For every time interval BuySell the asset to
make the positionCall ndash Spot nbr Assetsinsensitive to variations of the Spot
Fair price is the amount spent during delta-hedgingOption Price ==Δ1+ Δ2+ Δ3 + Δ4+ Δ5
It is fair because that is how much we spent
Δ1 Δ2 Δ3 Δ4 Δ5
Black-Scholes the mother model Black-Scholes based option-pricing on
no-arbitrage amp delta hedging
Previously pricing was based mainly on intuition and risk-based calculations
Fair value of securities was unknown
Black-Scholes main ideasAssume a Spot Dynamics
The rule for updating the spot has two terms
Drift the spot follows a main trend
Vol the spot fluctuates around the main trend
Black-Scholes assume as update rule
This process is a ldquolognormalrdquo process
ldquolognormalrdquo means that drift and fluctuations are proportional to S t
nsfluctuatiomain trend
tttttt WStSSS
tttt dWSdtSdS
Tttt SSSSS 10
σ size of fluctuations
μ steepness of main trend
ΔWt random variable (posneg)
Black-Scholes assumptions No-arbitrage drift = risk-free rate
Impose no-arbitrage by requiring thatexpected spot = market forward
Calculations simplify if fluctuations are normal
is Gaussian normal of zero mean variance ~ T
Volatility (size of fluctuations) is assumed constant
Risk-free rate is assumed constant
No-transaction costs underlying is liquid etc
tW
Black-Scholes formula Call option = endashrT ∙ E[ max(S(T)-K0)]
= discounted average of the call-payoff over various realizations of final spot
Solution
)()( 2121 dNKedNSeC TrTr
T
TrrKS
d
212
1
21
ln
T
TrrKS
d
212
2
21
ln
Interpretation of BS formula
money in the finishes
spot y that probabilit
2
long go toshares ofnumber option theof Delta
1 )()( 21 dNKedNeSC TrTr
Price = value of position at maturity ndash value of cash-flow at maturity
How much does the portfolio value change when spot changes
Δ=0 Delta-neutral value
if S S+dS then portfolio value does not change
Vega=0 Vega-neutral value
if σ σ+dσ then portfolio value does not change
ldquoGreeksrdquo measure sensitivity of portfolio value
portfolio value
S
Delta neutral position S
S+dSpartPortfoliopartS=0
Black-Scholes vs market Comparison with market
BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility
smile Inverse calculation ldquoimplied volrdquo Call on EURUSD
0
10000
20000
30000
40000
50000
60000
70000
80000
12000 12500 13000 13500 14000 14500 15000 15500
strike
USD
cas
h
Black-Scholes
Market
Smile
1300
1350
1400
1450
1500
12000 12500 13000 13500 14000 14500 15000 15500 16000
Strike
Vola
tility
Black-Scholes
Market
Spot probability density Distribution of terminal spot
(given initial spot) obtained from
Fat tails
Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes
Main causes
bullSpot dynamics is not lognormal
bullSpot fluctuations (vol) are not constant
2
mkt2
0
Call2
KeSSP Tr
T
Market observable
What information does the smile give
It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price
It is not the volatility of the spot dynamics
It does not give any information about the spot dynamics even if we combine smiles of various tenors
Therefore it cannot be used (directly) to price path-dependent options
The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)
If there was an instantaneous volatility σ(t) the BS could be interpreted as
dtT
T
t
22BS
1 the accumulated vol
Types of smile quotes The smile is a static representation of the
implied volatilities at a given moment of time
What if the spot changes
Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change
Sticky strike if spot changes implied vol of a given strike doesnrsquot change
Moneyness Δ=DF1N(d1)
Spotladders price delta amp gamma
Vanilla
Knock-out spot=128 strike=125 barrier=15
0051
152
253
35
08 1 12 14 16 18
spot
gamma
06y
1y
0
01
02
03
04
05
06
08 1 12 14 16 18
spot
price
06y
1y
0
02
04
06
08
1
12
08 1 12 14 16 18
spot
delta
06y
1y
Linear regime S-K
1 underlying is needed to hedge
Sensitivity of Delta to spot is maximum
0000050001
000150002
000250003
000350004
00045
08 09 1 11 12 13 14
spot
price
06y
1y
-006
-005
-004
-003
-002
-001
0
001
002
003
08 09 1 11 12 13spot
delta
06y
1y
-06
-04
-02
0
02
04
08 1 12spot
gamma
06y
1y
Spot is far from barrier and far from OTM risk is minimum price is maximum
Δlt0 price gets smaller if spot increases
Spotladders vega vanna amp volga Vanilla
Knock-out spot=128 strike=125 barrier=135
0
000001
000002
000003
000004
000005
000006
08 1 12 14 16 18
spot
vega
06y
1y
-2
-1
0
1
2
3
08 1 12 14 16 18
spotvanna
06y
1y
0051
152
253
35
08 1 12 14 16 18
spot
volga
06y
1y
-0000008
-0000006
-0000004
-0000002
0
0000002
0000004
08 1 12
spot
vega
06y
1y
-1
-05
0
05
1
15
08 09 1 11 12 13
spot
vanna
06y
1y
-1
-05
0
05
1
15
2
08 1 12spot
volga
06y
1y
Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol
Simple analytic techniques ldquomoment matchingrdquo
Average-rate option payoff with N fixing dates
Basket option with two underlyings
TV pricing can be achieved quickly via ldquomoment matchingrdquo
Mark-to-market requires correlated stochastic processes for spotsvols (more complex)
01
max Asian 1
KSN
N
ii
0max Basket
2
22
1
11 K
tS
TSa
tS
TSa
ldquoMoment matchingrdquo To price Asian (average option) in TV we consider
that The spot process is lognormal The sum of all spots is lognormal also
Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)
Central idea of moment matching Find first and second moment of sum of lognormals
E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known
moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Fair price What is the fair price of an
option Consider a call option on an
asset
Delta hedging For every time interval BuySell the asset to
make the positionCall ndash Spot nbr Assetsinsensitive to variations of the Spot
Fair price is the amount spent during delta-hedgingOption Price ==Δ1+ Δ2+ Δ3 + Δ4+ Δ5
It is fair because that is how much we spent
Δ1 Δ2 Δ3 Δ4 Δ5
Black-Scholes the mother model Black-Scholes based option-pricing on
no-arbitrage amp delta hedging
Previously pricing was based mainly on intuition and risk-based calculations
Fair value of securities was unknown
Black-Scholes main ideasAssume a Spot Dynamics
The rule for updating the spot has two terms
Drift the spot follows a main trend
Vol the spot fluctuates around the main trend
Black-Scholes assume as update rule
This process is a ldquolognormalrdquo process
ldquolognormalrdquo means that drift and fluctuations are proportional to S t
nsfluctuatiomain trend
tttttt WStSSS
tttt dWSdtSdS
Tttt SSSSS 10
σ size of fluctuations
μ steepness of main trend
ΔWt random variable (posneg)
Black-Scholes assumptions No-arbitrage drift = risk-free rate
Impose no-arbitrage by requiring thatexpected spot = market forward
Calculations simplify if fluctuations are normal
is Gaussian normal of zero mean variance ~ T
Volatility (size of fluctuations) is assumed constant
Risk-free rate is assumed constant
No-transaction costs underlying is liquid etc
tW
Black-Scholes formula Call option = endashrT ∙ E[ max(S(T)-K0)]
= discounted average of the call-payoff over various realizations of final spot
Solution
)()( 2121 dNKedNSeC TrTr
T
TrrKS
d
212
1
21
ln
T
TrrKS
d
212
2
21
ln
Interpretation of BS formula
money in the finishes
spot y that probabilit
2
long go toshares ofnumber option theof Delta
1 )()( 21 dNKedNeSC TrTr
Price = value of position at maturity ndash value of cash-flow at maturity
How much does the portfolio value change when spot changes
Δ=0 Delta-neutral value
if S S+dS then portfolio value does not change
Vega=0 Vega-neutral value
if σ σ+dσ then portfolio value does not change
ldquoGreeksrdquo measure sensitivity of portfolio value
portfolio value
S
Delta neutral position S
S+dSpartPortfoliopartS=0
Black-Scholes vs market Comparison with market
BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility
smile Inverse calculation ldquoimplied volrdquo Call on EURUSD
0
10000
20000
30000
40000
50000
60000
70000
80000
12000 12500 13000 13500 14000 14500 15000 15500
strike
USD
cas
h
Black-Scholes
Market
Smile
1300
1350
1400
1450
1500
12000 12500 13000 13500 14000 14500 15000 15500 16000
Strike
Vola
tility
Black-Scholes
Market
Spot probability density Distribution of terminal spot
(given initial spot) obtained from
Fat tails
Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes
Main causes
bullSpot dynamics is not lognormal
bullSpot fluctuations (vol) are not constant
2
mkt2
0
Call2
KeSSP Tr
T
Market observable
What information does the smile give
It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price
It is not the volatility of the spot dynamics
It does not give any information about the spot dynamics even if we combine smiles of various tenors
Therefore it cannot be used (directly) to price path-dependent options
The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)
If there was an instantaneous volatility σ(t) the BS could be interpreted as
dtT
T
t
22BS
1 the accumulated vol
Types of smile quotes The smile is a static representation of the
implied volatilities at a given moment of time
What if the spot changes
Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change
Sticky strike if spot changes implied vol of a given strike doesnrsquot change
Moneyness Δ=DF1N(d1)
Spotladders price delta amp gamma
Vanilla
Knock-out spot=128 strike=125 barrier=15
0051
152
253
35
08 1 12 14 16 18
spot
gamma
06y
1y
0
01
02
03
04
05
06
08 1 12 14 16 18
spot
price
06y
1y
0
02
04
06
08
1
12
08 1 12 14 16 18
spot
delta
06y
1y
Linear regime S-K
1 underlying is needed to hedge
Sensitivity of Delta to spot is maximum
0000050001
000150002
000250003
000350004
00045
08 09 1 11 12 13 14
spot
price
06y
1y
-006
-005
-004
-003
-002
-001
0
001
002
003
08 09 1 11 12 13spot
delta
06y
1y
-06
-04
-02
0
02
04
08 1 12spot
gamma
06y
1y
Spot is far from barrier and far from OTM risk is minimum price is maximum
Δlt0 price gets smaller if spot increases
Spotladders vega vanna amp volga Vanilla
Knock-out spot=128 strike=125 barrier=135
0
000001
000002
000003
000004
000005
000006
08 1 12 14 16 18
spot
vega
06y
1y
-2
-1
0
1
2
3
08 1 12 14 16 18
spotvanna
06y
1y
0051
152
253
35
08 1 12 14 16 18
spot
volga
06y
1y
-0000008
-0000006
-0000004
-0000002
0
0000002
0000004
08 1 12
spot
vega
06y
1y
-1
-05
0
05
1
15
08 09 1 11 12 13
spot
vanna
06y
1y
-1
-05
0
05
1
15
2
08 1 12spot
volga
06y
1y
Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol
Simple analytic techniques ldquomoment matchingrdquo
Average-rate option payoff with N fixing dates
Basket option with two underlyings
TV pricing can be achieved quickly via ldquomoment matchingrdquo
Mark-to-market requires correlated stochastic processes for spotsvols (more complex)
01
max Asian 1
KSN
N
ii
0max Basket
2
22
1
11 K
tS
TSa
tS
TSa
ldquoMoment matchingrdquo To price Asian (average option) in TV we consider
that The spot process is lognormal The sum of all spots is lognormal also
Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)
Central idea of moment matching Find first and second moment of sum of lognormals
E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known
moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Black-Scholes the mother model Black-Scholes based option-pricing on
no-arbitrage amp delta hedging
Previously pricing was based mainly on intuition and risk-based calculations
Fair value of securities was unknown
Black-Scholes main ideasAssume a Spot Dynamics
The rule for updating the spot has two terms
Drift the spot follows a main trend
Vol the spot fluctuates around the main trend
Black-Scholes assume as update rule
This process is a ldquolognormalrdquo process
ldquolognormalrdquo means that drift and fluctuations are proportional to S t
nsfluctuatiomain trend
tttttt WStSSS
tttt dWSdtSdS
Tttt SSSSS 10
σ size of fluctuations
μ steepness of main trend
ΔWt random variable (posneg)
Black-Scholes assumptions No-arbitrage drift = risk-free rate
Impose no-arbitrage by requiring thatexpected spot = market forward
Calculations simplify if fluctuations are normal
is Gaussian normal of zero mean variance ~ T
Volatility (size of fluctuations) is assumed constant
Risk-free rate is assumed constant
No-transaction costs underlying is liquid etc
tW
Black-Scholes formula Call option = endashrT ∙ E[ max(S(T)-K0)]
= discounted average of the call-payoff over various realizations of final spot
Solution
)()( 2121 dNKedNSeC TrTr
T
TrrKS
d
212
1
21
ln
T
TrrKS
d
212
2
21
ln
Interpretation of BS formula
money in the finishes
spot y that probabilit
2
long go toshares ofnumber option theof Delta
1 )()( 21 dNKedNeSC TrTr
Price = value of position at maturity ndash value of cash-flow at maturity
How much does the portfolio value change when spot changes
Δ=0 Delta-neutral value
if S S+dS then portfolio value does not change
Vega=0 Vega-neutral value
if σ σ+dσ then portfolio value does not change
ldquoGreeksrdquo measure sensitivity of portfolio value
portfolio value
S
Delta neutral position S
S+dSpartPortfoliopartS=0
Black-Scholes vs market Comparison with market
BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility
smile Inverse calculation ldquoimplied volrdquo Call on EURUSD
0
10000
20000
30000
40000
50000
60000
70000
80000
12000 12500 13000 13500 14000 14500 15000 15500
strike
USD
cas
h
Black-Scholes
Market
Smile
1300
1350
1400
1450
1500
12000 12500 13000 13500 14000 14500 15000 15500 16000
Strike
Vola
tility
Black-Scholes
Market
Spot probability density Distribution of terminal spot
(given initial spot) obtained from
Fat tails
Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes
Main causes
bullSpot dynamics is not lognormal
bullSpot fluctuations (vol) are not constant
2
mkt2
0
Call2
KeSSP Tr
T
Market observable
What information does the smile give
It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price
It is not the volatility of the spot dynamics
It does not give any information about the spot dynamics even if we combine smiles of various tenors
Therefore it cannot be used (directly) to price path-dependent options
The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)
If there was an instantaneous volatility σ(t) the BS could be interpreted as
dtT
T
t
22BS
1 the accumulated vol
Types of smile quotes The smile is a static representation of the
implied volatilities at a given moment of time
What if the spot changes
Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change
Sticky strike if spot changes implied vol of a given strike doesnrsquot change
Moneyness Δ=DF1N(d1)
Spotladders price delta amp gamma
Vanilla
Knock-out spot=128 strike=125 barrier=15
0051
152
253
35
08 1 12 14 16 18
spot
gamma
06y
1y
0
01
02
03
04
05
06
08 1 12 14 16 18
spot
price
06y
1y
0
02
04
06
08
1
12
08 1 12 14 16 18
spot
delta
06y
1y
Linear regime S-K
1 underlying is needed to hedge
Sensitivity of Delta to spot is maximum
0000050001
000150002
000250003
000350004
00045
08 09 1 11 12 13 14
spot
price
06y
1y
-006
-005
-004
-003
-002
-001
0
001
002
003
08 09 1 11 12 13spot
delta
06y
1y
-06
-04
-02
0
02
04
08 1 12spot
gamma
06y
1y
Spot is far from barrier and far from OTM risk is minimum price is maximum
Δlt0 price gets smaller if spot increases
Spotladders vega vanna amp volga Vanilla
Knock-out spot=128 strike=125 barrier=135
0
000001
000002
000003
000004
000005
000006
08 1 12 14 16 18
spot
vega
06y
1y
-2
-1
0
1
2
3
08 1 12 14 16 18
spotvanna
06y
1y
0051
152
253
35
08 1 12 14 16 18
spot
volga
06y
1y
-0000008
-0000006
-0000004
-0000002
0
0000002
0000004
08 1 12
spot
vega
06y
1y
-1
-05
0
05
1
15
08 09 1 11 12 13
spot
vanna
06y
1y
-1
-05
0
05
1
15
2
08 1 12spot
volga
06y
1y
Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol
Simple analytic techniques ldquomoment matchingrdquo
Average-rate option payoff with N fixing dates
Basket option with two underlyings
TV pricing can be achieved quickly via ldquomoment matchingrdquo
Mark-to-market requires correlated stochastic processes for spotsvols (more complex)
01
max Asian 1
KSN
N
ii
0max Basket
2
22
1
11 K
tS
TSa
tS
TSa
ldquoMoment matchingrdquo To price Asian (average option) in TV we consider
that The spot process is lognormal The sum of all spots is lognormal also
Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)
Central idea of moment matching Find first and second moment of sum of lognormals
E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known
moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Black-Scholes main ideasAssume a Spot Dynamics
The rule for updating the spot has two terms
Drift the spot follows a main trend
Vol the spot fluctuates around the main trend
Black-Scholes assume as update rule
This process is a ldquolognormalrdquo process
ldquolognormalrdquo means that drift and fluctuations are proportional to S t
nsfluctuatiomain trend
tttttt WStSSS
tttt dWSdtSdS
Tttt SSSSS 10
σ size of fluctuations
μ steepness of main trend
ΔWt random variable (posneg)
Black-Scholes assumptions No-arbitrage drift = risk-free rate
Impose no-arbitrage by requiring thatexpected spot = market forward
Calculations simplify if fluctuations are normal
is Gaussian normal of zero mean variance ~ T
Volatility (size of fluctuations) is assumed constant
Risk-free rate is assumed constant
No-transaction costs underlying is liquid etc
tW
Black-Scholes formula Call option = endashrT ∙ E[ max(S(T)-K0)]
= discounted average of the call-payoff over various realizations of final spot
Solution
)()( 2121 dNKedNSeC TrTr
T
TrrKS
d
212
1
21
ln
T
TrrKS
d
212
2
21
ln
Interpretation of BS formula
money in the finishes
spot y that probabilit
2
long go toshares ofnumber option theof Delta
1 )()( 21 dNKedNeSC TrTr
Price = value of position at maturity ndash value of cash-flow at maturity
How much does the portfolio value change when spot changes
Δ=0 Delta-neutral value
if S S+dS then portfolio value does not change
Vega=0 Vega-neutral value
if σ σ+dσ then portfolio value does not change
ldquoGreeksrdquo measure sensitivity of portfolio value
portfolio value
S
Delta neutral position S
S+dSpartPortfoliopartS=0
Black-Scholes vs market Comparison with market
BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility
smile Inverse calculation ldquoimplied volrdquo Call on EURUSD
0
10000
20000
30000
40000
50000
60000
70000
80000
12000 12500 13000 13500 14000 14500 15000 15500
strike
USD
cas
h
Black-Scholes
Market
Smile
1300
1350
1400
1450
1500
12000 12500 13000 13500 14000 14500 15000 15500 16000
Strike
Vola
tility
Black-Scholes
Market
Spot probability density Distribution of terminal spot
(given initial spot) obtained from
Fat tails
Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes
Main causes
bullSpot dynamics is not lognormal
bullSpot fluctuations (vol) are not constant
2
mkt2
0
Call2
KeSSP Tr
T
Market observable
What information does the smile give
It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price
It is not the volatility of the spot dynamics
It does not give any information about the spot dynamics even if we combine smiles of various tenors
Therefore it cannot be used (directly) to price path-dependent options
The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)
If there was an instantaneous volatility σ(t) the BS could be interpreted as
dtT
T
t
22BS
1 the accumulated vol
Types of smile quotes The smile is a static representation of the
implied volatilities at a given moment of time
What if the spot changes
Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change
Sticky strike if spot changes implied vol of a given strike doesnrsquot change
Moneyness Δ=DF1N(d1)
Spotladders price delta amp gamma
Vanilla
Knock-out spot=128 strike=125 barrier=15
0051
152
253
35
08 1 12 14 16 18
spot
gamma
06y
1y
0
01
02
03
04
05
06
08 1 12 14 16 18
spot
price
06y
1y
0
02
04
06
08
1
12
08 1 12 14 16 18
spot
delta
06y
1y
Linear regime S-K
1 underlying is needed to hedge
Sensitivity of Delta to spot is maximum
0000050001
000150002
000250003
000350004
00045
08 09 1 11 12 13 14
spot
price
06y
1y
-006
-005
-004
-003
-002
-001
0
001
002
003
08 09 1 11 12 13spot
delta
06y
1y
-06
-04
-02
0
02
04
08 1 12spot
gamma
06y
1y
Spot is far from barrier and far from OTM risk is minimum price is maximum
Δlt0 price gets smaller if spot increases
Spotladders vega vanna amp volga Vanilla
Knock-out spot=128 strike=125 barrier=135
0
000001
000002
000003
000004
000005
000006
08 1 12 14 16 18
spot
vega
06y
1y
-2
-1
0
1
2
3
08 1 12 14 16 18
spotvanna
06y
1y
0051
152
253
35
08 1 12 14 16 18
spot
volga
06y
1y
-0000008
-0000006
-0000004
-0000002
0
0000002
0000004
08 1 12
spot
vega
06y
1y
-1
-05
0
05
1
15
08 09 1 11 12 13
spot
vanna
06y
1y
-1
-05
0
05
1
15
2
08 1 12spot
volga
06y
1y
Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol
Simple analytic techniques ldquomoment matchingrdquo
Average-rate option payoff with N fixing dates
Basket option with two underlyings
TV pricing can be achieved quickly via ldquomoment matchingrdquo
Mark-to-market requires correlated stochastic processes for spotsvols (more complex)
01
max Asian 1
KSN
N
ii
0max Basket
2
22
1
11 K
tS
TSa
tS
TSa
ldquoMoment matchingrdquo To price Asian (average option) in TV we consider
that The spot process is lognormal The sum of all spots is lognormal also
Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)
Central idea of moment matching Find first and second moment of sum of lognormals
E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known
moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Black-Scholes assumptions No-arbitrage drift = risk-free rate
Impose no-arbitrage by requiring thatexpected spot = market forward
Calculations simplify if fluctuations are normal
is Gaussian normal of zero mean variance ~ T
Volatility (size of fluctuations) is assumed constant
Risk-free rate is assumed constant
No-transaction costs underlying is liquid etc
tW
Black-Scholes formula Call option = endashrT ∙ E[ max(S(T)-K0)]
= discounted average of the call-payoff over various realizations of final spot
Solution
)()( 2121 dNKedNSeC TrTr
T
TrrKS
d
212
1
21
ln
T
TrrKS
d
212
2
21
ln
Interpretation of BS formula
money in the finishes
spot y that probabilit
2
long go toshares ofnumber option theof Delta
1 )()( 21 dNKedNeSC TrTr
Price = value of position at maturity ndash value of cash-flow at maturity
How much does the portfolio value change when spot changes
Δ=0 Delta-neutral value
if S S+dS then portfolio value does not change
Vega=0 Vega-neutral value
if σ σ+dσ then portfolio value does not change
ldquoGreeksrdquo measure sensitivity of portfolio value
portfolio value
S
Delta neutral position S
S+dSpartPortfoliopartS=0
Black-Scholes vs market Comparison with market
BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility
smile Inverse calculation ldquoimplied volrdquo Call on EURUSD
0
10000
20000
30000
40000
50000
60000
70000
80000
12000 12500 13000 13500 14000 14500 15000 15500
strike
USD
cas
h
Black-Scholes
Market
Smile
1300
1350
1400
1450
1500
12000 12500 13000 13500 14000 14500 15000 15500 16000
Strike
Vola
tility
Black-Scholes
Market
Spot probability density Distribution of terminal spot
(given initial spot) obtained from
Fat tails
Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes
Main causes
bullSpot dynamics is not lognormal
bullSpot fluctuations (vol) are not constant
2
mkt2
0
Call2
KeSSP Tr
T
Market observable
What information does the smile give
It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price
It is not the volatility of the spot dynamics
It does not give any information about the spot dynamics even if we combine smiles of various tenors
Therefore it cannot be used (directly) to price path-dependent options
The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)
If there was an instantaneous volatility σ(t) the BS could be interpreted as
dtT
T
t
22BS
1 the accumulated vol
Types of smile quotes The smile is a static representation of the
implied volatilities at a given moment of time
What if the spot changes
Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change
Sticky strike if spot changes implied vol of a given strike doesnrsquot change
Moneyness Δ=DF1N(d1)
Spotladders price delta amp gamma
Vanilla
Knock-out spot=128 strike=125 barrier=15
0051
152
253
35
08 1 12 14 16 18
spot
gamma
06y
1y
0
01
02
03
04
05
06
08 1 12 14 16 18
spot
price
06y
1y
0
02
04
06
08
1
12
08 1 12 14 16 18
spot
delta
06y
1y
Linear regime S-K
1 underlying is needed to hedge
Sensitivity of Delta to spot is maximum
0000050001
000150002
000250003
000350004
00045
08 09 1 11 12 13 14
spot
price
06y
1y
-006
-005
-004
-003
-002
-001
0
001
002
003
08 09 1 11 12 13spot
delta
06y
1y
-06
-04
-02
0
02
04
08 1 12spot
gamma
06y
1y
Spot is far from barrier and far from OTM risk is minimum price is maximum
Δlt0 price gets smaller if spot increases
Spotladders vega vanna amp volga Vanilla
Knock-out spot=128 strike=125 barrier=135
0
000001
000002
000003
000004
000005
000006
08 1 12 14 16 18
spot
vega
06y
1y
-2
-1
0
1
2
3
08 1 12 14 16 18
spotvanna
06y
1y
0051
152
253
35
08 1 12 14 16 18
spot
volga
06y
1y
-0000008
-0000006
-0000004
-0000002
0
0000002
0000004
08 1 12
spot
vega
06y
1y
-1
-05
0
05
1
15
08 09 1 11 12 13
spot
vanna
06y
1y
-1
-05
0
05
1
15
2
08 1 12spot
volga
06y
1y
Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol
Simple analytic techniques ldquomoment matchingrdquo
Average-rate option payoff with N fixing dates
Basket option with two underlyings
TV pricing can be achieved quickly via ldquomoment matchingrdquo
Mark-to-market requires correlated stochastic processes for spotsvols (more complex)
01
max Asian 1
KSN
N
ii
0max Basket
2
22
1
11 K
tS
TSa
tS
TSa
ldquoMoment matchingrdquo To price Asian (average option) in TV we consider
that The spot process is lognormal The sum of all spots is lognormal also
Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)
Central idea of moment matching Find first and second moment of sum of lognormals
E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known
moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Black-Scholes formula Call option = endashrT ∙ E[ max(S(T)-K0)]
= discounted average of the call-payoff over various realizations of final spot
Solution
)()( 2121 dNKedNSeC TrTr
T
TrrKS
d
212
1
21
ln
T
TrrKS
d
212
2
21
ln
Interpretation of BS formula
money in the finishes
spot y that probabilit
2
long go toshares ofnumber option theof Delta
1 )()( 21 dNKedNeSC TrTr
Price = value of position at maturity ndash value of cash-flow at maturity
How much does the portfolio value change when spot changes
Δ=0 Delta-neutral value
if S S+dS then portfolio value does not change
Vega=0 Vega-neutral value
if σ σ+dσ then portfolio value does not change
ldquoGreeksrdquo measure sensitivity of portfolio value
portfolio value
S
Delta neutral position S
S+dSpartPortfoliopartS=0
Black-Scholes vs market Comparison with market
BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility
smile Inverse calculation ldquoimplied volrdquo Call on EURUSD
0
10000
20000
30000
40000
50000
60000
70000
80000
12000 12500 13000 13500 14000 14500 15000 15500
strike
USD
cas
h
Black-Scholes
Market
Smile
1300
1350
1400
1450
1500
12000 12500 13000 13500 14000 14500 15000 15500 16000
Strike
Vola
tility
Black-Scholes
Market
Spot probability density Distribution of terminal spot
(given initial spot) obtained from
Fat tails
Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes
Main causes
bullSpot dynamics is not lognormal
bullSpot fluctuations (vol) are not constant
2
mkt2
0
Call2
KeSSP Tr
T
Market observable
What information does the smile give
It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price
It is not the volatility of the spot dynamics
It does not give any information about the spot dynamics even if we combine smiles of various tenors
Therefore it cannot be used (directly) to price path-dependent options
The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)
If there was an instantaneous volatility σ(t) the BS could be interpreted as
dtT
T
t
22BS
1 the accumulated vol
Types of smile quotes The smile is a static representation of the
implied volatilities at a given moment of time
What if the spot changes
Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change
Sticky strike if spot changes implied vol of a given strike doesnrsquot change
Moneyness Δ=DF1N(d1)
Spotladders price delta amp gamma
Vanilla
Knock-out spot=128 strike=125 barrier=15
0051
152
253
35
08 1 12 14 16 18
spot
gamma
06y
1y
0
01
02
03
04
05
06
08 1 12 14 16 18
spot
price
06y
1y
0
02
04
06
08
1
12
08 1 12 14 16 18
spot
delta
06y
1y
Linear regime S-K
1 underlying is needed to hedge
Sensitivity of Delta to spot is maximum
0000050001
000150002
000250003
000350004
00045
08 09 1 11 12 13 14
spot
price
06y
1y
-006
-005
-004
-003
-002
-001
0
001
002
003
08 09 1 11 12 13spot
delta
06y
1y
-06
-04
-02
0
02
04
08 1 12spot
gamma
06y
1y
Spot is far from barrier and far from OTM risk is minimum price is maximum
Δlt0 price gets smaller if spot increases
Spotladders vega vanna amp volga Vanilla
Knock-out spot=128 strike=125 barrier=135
0
000001
000002
000003
000004
000005
000006
08 1 12 14 16 18
spot
vega
06y
1y
-2
-1
0
1
2
3
08 1 12 14 16 18
spotvanna
06y
1y
0051
152
253
35
08 1 12 14 16 18
spot
volga
06y
1y
-0000008
-0000006
-0000004
-0000002
0
0000002
0000004
08 1 12
spot
vega
06y
1y
-1
-05
0
05
1
15
08 09 1 11 12 13
spot
vanna
06y
1y
-1
-05
0
05
1
15
2
08 1 12spot
volga
06y
1y
Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol
Simple analytic techniques ldquomoment matchingrdquo
Average-rate option payoff with N fixing dates
Basket option with two underlyings
TV pricing can be achieved quickly via ldquomoment matchingrdquo
Mark-to-market requires correlated stochastic processes for spotsvols (more complex)
01
max Asian 1
KSN
N
ii
0max Basket
2
22
1
11 K
tS
TSa
tS
TSa
ldquoMoment matchingrdquo To price Asian (average option) in TV we consider
that The spot process is lognormal The sum of all spots is lognormal also
Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)
Central idea of moment matching Find first and second moment of sum of lognormals
E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known
moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Interpretation of BS formula
money in the finishes
spot y that probabilit
2
long go toshares ofnumber option theof Delta
1 )()( 21 dNKedNeSC TrTr
Price = value of position at maturity ndash value of cash-flow at maturity
How much does the portfolio value change when spot changes
Δ=0 Delta-neutral value
if S S+dS then portfolio value does not change
Vega=0 Vega-neutral value
if σ σ+dσ then portfolio value does not change
ldquoGreeksrdquo measure sensitivity of portfolio value
portfolio value
S
Delta neutral position S
S+dSpartPortfoliopartS=0
Black-Scholes vs market Comparison with market
BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility
smile Inverse calculation ldquoimplied volrdquo Call on EURUSD
0
10000
20000
30000
40000
50000
60000
70000
80000
12000 12500 13000 13500 14000 14500 15000 15500
strike
USD
cas
h
Black-Scholes
Market
Smile
1300
1350
1400
1450
1500
12000 12500 13000 13500 14000 14500 15000 15500 16000
Strike
Vola
tility
Black-Scholes
Market
Spot probability density Distribution of terminal spot
(given initial spot) obtained from
Fat tails
Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes
Main causes
bullSpot dynamics is not lognormal
bullSpot fluctuations (vol) are not constant
2
mkt2
0
Call2
KeSSP Tr
T
Market observable
What information does the smile give
It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price
It is not the volatility of the spot dynamics
It does not give any information about the spot dynamics even if we combine smiles of various tenors
Therefore it cannot be used (directly) to price path-dependent options
The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)
If there was an instantaneous volatility σ(t) the BS could be interpreted as
dtT
T
t
22BS
1 the accumulated vol
Types of smile quotes The smile is a static representation of the
implied volatilities at a given moment of time
What if the spot changes
Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change
Sticky strike if spot changes implied vol of a given strike doesnrsquot change
Moneyness Δ=DF1N(d1)
Spotladders price delta amp gamma
Vanilla
Knock-out spot=128 strike=125 barrier=15
0051
152
253
35
08 1 12 14 16 18
spot
gamma
06y
1y
0
01
02
03
04
05
06
08 1 12 14 16 18
spot
price
06y
1y
0
02
04
06
08
1
12
08 1 12 14 16 18
spot
delta
06y
1y
Linear regime S-K
1 underlying is needed to hedge
Sensitivity of Delta to spot is maximum
0000050001
000150002
000250003
000350004
00045
08 09 1 11 12 13 14
spot
price
06y
1y
-006
-005
-004
-003
-002
-001
0
001
002
003
08 09 1 11 12 13spot
delta
06y
1y
-06
-04
-02
0
02
04
08 1 12spot
gamma
06y
1y
Spot is far from barrier and far from OTM risk is minimum price is maximum
Δlt0 price gets smaller if spot increases
Spotladders vega vanna amp volga Vanilla
Knock-out spot=128 strike=125 barrier=135
0
000001
000002
000003
000004
000005
000006
08 1 12 14 16 18
spot
vega
06y
1y
-2
-1
0
1
2
3
08 1 12 14 16 18
spotvanna
06y
1y
0051
152
253
35
08 1 12 14 16 18
spot
volga
06y
1y
-0000008
-0000006
-0000004
-0000002
0
0000002
0000004
08 1 12
spot
vega
06y
1y
-1
-05
0
05
1
15
08 09 1 11 12 13
spot
vanna
06y
1y
-1
-05
0
05
1
15
2
08 1 12spot
volga
06y
1y
Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol
Simple analytic techniques ldquomoment matchingrdquo
Average-rate option payoff with N fixing dates
Basket option with two underlyings
TV pricing can be achieved quickly via ldquomoment matchingrdquo
Mark-to-market requires correlated stochastic processes for spotsvols (more complex)
01
max Asian 1
KSN
N
ii
0max Basket
2
22
1
11 K
tS
TSa
tS
TSa
ldquoMoment matchingrdquo To price Asian (average option) in TV we consider
that The spot process is lognormal The sum of all spots is lognormal also
Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)
Central idea of moment matching Find first and second moment of sum of lognormals
E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known
moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
How much does the portfolio value change when spot changes
Δ=0 Delta-neutral value
if S S+dS then portfolio value does not change
Vega=0 Vega-neutral value
if σ σ+dσ then portfolio value does not change
ldquoGreeksrdquo measure sensitivity of portfolio value
portfolio value
S
Delta neutral position S
S+dSpartPortfoliopartS=0
Black-Scholes vs market Comparison with market
BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility
smile Inverse calculation ldquoimplied volrdquo Call on EURUSD
0
10000
20000
30000
40000
50000
60000
70000
80000
12000 12500 13000 13500 14000 14500 15000 15500
strike
USD
cas
h
Black-Scholes
Market
Smile
1300
1350
1400
1450
1500
12000 12500 13000 13500 14000 14500 15000 15500 16000
Strike
Vola
tility
Black-Scholes
Market
Spot probability density Distribution of terminal spot
(given initial spot) obtained from
Fat tails
Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes
Main causes
bullSpot dynamics is not lognormal
bullSpot fluctuations (vol) are not constant
2
mkt2
0
Call2
KeSSP Tr
T
Market observable
What information does the smile give
It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price
It is not the volatility of the spot dynamics
It does not give any information about the spot dynamics even if we combine smiles of various tenors
Therefore it cannot be used (directly) to price path-dependent options
The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)
If there was an instantaneous volatility σ(t) the BS could be interpreted as
dtT
T
t
22BS
1 the accumulated vol
Types of smile quotes The smile is a static representation of the
implied volatilities at a given moment of time
What if the spot changes
Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change
Sticky strike if spot changes implied vol of a given strike doesnrsquot change
Moneyness Δ=DF1N(d1)
Spotladders price delta amp gamma
Vanilla
Knock-out spot=128 strike=125 barrier=15
0051
152
253
35
08 1 12 14 16 18
spot
gamma
06y
1y
0
01
02
03
04
05
06
08 1 12 14 16 18
spot
price
06y
1y
0
02
04
06
08
1
12
08 1 12 14 16 18
spot
delta
06y
1y
Linear regime S-K
1 underlying is needed to hedge
Sensitivity of Delta to spot is maximum
0000050001
000150002
000250003
000350004
00045
08 09 1 11 12 13 14
spot
price
06y
1y
-006
-005
-004
-003
-002
-001
0
001
002
003
08 09 1 11 12 13spot
delta
06y
1y
-06
-04
-02
0
02
04
08 1 12spot
gamma
06y
1y
Spot is far from barrier and far from OTM risk is minimum price is maximum
Δlt0 price gets smaller if spot increases
Spotladders vega vanna amp volga Vanilla
Knock-out spot=128 strike=125 barrier=135
0
000001
000002
000003
000004
000005
000006
08 1 12 14 16 18
spot
vega
06y
1y
-2
-1
0
1
2
3
08 1 12 14 16 18
spotvanna
06y
1y
0051
152
253
35
08 1 12 14 16 18
spot
volga
06y
1y
-0000008
-0000006
-0000004
-0000002
0
0000002
0000004
08 1 12
spot
vega
06y
1y
-1
-05
0
05
1
15
08 09 1 11 12 13
spot
vanna
06y
1y
-1
-05
0
05
1
15
2
08 1 12spot
volga
06y
1y
Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol
Simple analytic techniques ldquomoment matchingrdquo
Average-rate option payoff with N fixing dates
Basket option with two underlyings
TV pricing can be achieved quickly via ldquomoment matchingrdquo
Mark-to-market requires correlated stochastic processes for spotsvols (more complex)
01
max Asian 1
KSN
N
ii
0max Basket
2
22
1
11 K
tS
TSa
tS
TSa
ldquoMoment matchingrdquo To price Asian (average option) in TV we consider
that The spot process is lognormal The sum of all spots is lognormal also
Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)
Central idea of moment matching Find first and second moment of sum of lognormals
E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known
moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Black-Scholes vs market Comparison with market
BS lt MtM when inout of the money Plug MtM in BS formula to calculate volatility
smile Inverse calculation ldquoimplied volrdquo Call on EURUSD
0
10000
20000
30000
40000
50000
60000
70000
80000
12000 12500 13000 13500 14000 14500 15000 15500
strike
USD
cas
h
Black-Scholes
Market
Smile
1300
1350
1400
1450
1500
12000 12500 13000 13500 14000 14500 15000 15500 16000
Strike
Vola
tility
Black-Scholes
Market
Spot probability density Distribution of terminal spot
(given initial spot) obtained from
Fat tails
Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes
Main causes
bullSpot dynamics is not lognormal
bullSpot fluctuations (vol) are not constant
2
mkt2
0
Call2
KeSSP Tr
T
Market observable
What information does the smile give
It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price
It is not the volatility of the spot dynamics
It does not give any information about the spot dynamics even if we combine smiles of various tenors
Therefore it cannot be used (directly) to price path-dependent options
The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)
If there was an instantaneous volatility σ(t) the BS could be interpreted as
dtT
T
t
22BS
1 the accumulated vol
Types of smile quotes The smile is a static representation of the
implied volatilities at a given moment of time
What if the spot changes
Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change
Sticky strike if spot changes implied vol of a given strike doesnrsquot change
Moneyness Δ=DF1N(d1)
Spotladders price delta amp gamma
Vanilla
Knock-out spot=128 strike=125 barrier=15
0051
152
253
35
08 1 12 14 16 18
spot
gamma
06y
1y
0
01
02
03
04
05
06
08 1 12 14 16 18
spot
price
06y
1y
0
02
04
06
08
1
12
08 1 12 14 16 18
spot
delta
06y
1y
Linear regime S-K
1 underlying is needed to hedge
Sensitivity of Delta to spot is maximum
0000050001
000150002
000250003
000350004
00045
08 09 1 11 12 13 14
spot
price
06y
1y
-006
-005
-004
-003
-002
-001
0
001
002
003
08 09 1 11 12 13spot
delta
06y
1y
-06
-04
-02
0
02
04
08 1 12spot
gamma
06y
1y
Spot is far from barrier and far from OTM risk is minimum price is maximum
Δlt0 price gets smaller if spot increases
Spotladders vega vanna amp volga Vanilla
Knock-out spot=128 strike=125 barrier=135
0
000001
000002
000003
000004
000005
000006
08 1 12 14 16 18
spot
vega
06y
1y
-2
-1
0
1
2
3
08 1 12 14 16 18
spotvanna
06y
1y
0051
152
253
35
08 1 12 14 16 18
spot
volga
06y
1y
-0000008
-0000006
-0000004
-0000002
0
0000002
0000004
08 1 12
spot
vega
06y
1y
-1
-05
0
05
1
15
08 09 1 11 12 13
spot
vanna
06y
1y
-1
-05
0
05
1
15
2
08 1 12spot
volga
06y
1y
Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol
Simple analytic techniques ldquomoment matchingrdquo
Average-rate option payoff with N fixing dates
Basket option with two underlyings
TV pricing can be achieved quickly via ldquomoment matchingrdquo
Mark-to-market requires correlated stochastic processes for spotsvols (more complex)
01
max Asian 1
KSN
N
ii
0max Basket
2
22
1
11 K
tS
TSa
tS
TSa
ldquoMoment matchingrdquo To price Asian (average option) in TV we consider
that The spot process is lognormal The sum of all spots is lognormal also
Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)
Central idea of moment matching Find first and second moment of sum of lognormals
E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known
moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Spot probability density Distribution of terminal spot
(given initial spot) obtained from
Fat tails
Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes
Main causes
bullSpot dynamics is not lognormal
bullSpot fluctuations (vol) are not constant
2
mkt2
0
Call2
KeSSP Tr
T
Market observable
What information does the smile give
It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price
It is not the volatility of the spot dynamics
It does not give any information about the spot dynamics even if we combine smiles of various tenors
Therefore it cannot be used (directly) to price path-dependent options
The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)
If there was an instantaneous volatility σ(t) the BS could be interpreted as
dtT
T
t
22BS
1 the accumulated vol
Types of smile quotes The smile is a static representation of the
implied volatilities at a given moment of time
What if the spot changes
Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change
Sticky strike if spot changes implied vol of a given strike doesnrsquot change
Moneyness Δ=DF1N(d1)
Spotladders price delta amp gamma
Vanilla
Knock-out spot=128 strike=125 barrier=15
0051
152
253
35
08 1 12 14 16 18
spot
gamma
06y
1y
0
01
02
03
04
05
06
08 1 12 14 16 18
spot
price
06y
1y
0
02
04
06
08
1
12
08 1 12 14 16 18
spot
delta
06y
1y
Linear regime S-K
1 underlying is needed to hedge
Sensitivity of Delta to spot is maximum
0000050001
000150002
000250003
000350004
00045
08 09 1 11 12 13 14
spot
price
06y
1y
-006
-005
-004
-003
-002
-001
0
001
002
003
08 09 1 11 12 13spot
delta
06y
1y
-06
-04
-02
0
02
04
08 1 12spot
gamma
06y
1y
Spot is far from barrier and far from OTM risk is minimum price is maximum
Δlt0 price gets smaller if spot increases
Spotladders vega vanna amp volga Vanilla
Knock-out spot=128 strike=125 barrier=135
0
000001
000002
000003
000004
000005
000006
08 1 12 14 16 18
spot
vega
06y
1y
-2
-1
0
1
2
3
08 1 12 14 16 18
spotvanna
06y
1y
0051
152
253
35
08 1 12 14 16 18
spot
volga
06y
1y
-0000008
-0000006
-0000004
-0000002
0
0000002
0000004
08 1 12
spot
vega
06y
1y
-1
-05
0
05
1
15
08 09 1 11 12 13
spot
vanna
06y
1y
-1
-05
0
05
1
15
2
08 1 12spot
volga
06y
1y
Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol
Simple analytic techniques ldquomoment matchingrdquo
Average-rate option payoff with N fixing dates
Basket option with two underlyings
TV pricing can be achieved quickly via ldquomoment matchingrdquo
Mark-to-market requires correlated stochastic processes for spotsvols (more complex)
01
max Asian 1
KSN
N
ii
0max Basket
2
22
1
11 K
tS
TSa
tS
TSa
ldquoMoment matchingrdquo To price Asian (average option) in TV we consider
that The spot process is lognormal The sum of all spots is lognormal also
Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)
Central idea of moment matching Find first and second moment of sum of lognormals
E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known
moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
What information does the smile give
It represents the price of vanillas Take the vol at a given strike Insert it to Black-Scholes formula Obtain the vanilla market price
It is not the volatility of the spot dynamics
It does not give any information about the spot dynamics even if we combine smiles of various tenors
Therefore it cannot be used (directly) to price path-dependent options
The quoted BS implied-vol is an artificial volatility ldquowrong quote into the wrong formula to give the right pricerdquo (RRebonato)
If there was an instantaneous volatility σ(t) the BS could be interpreted as
dtT
T
t
22BS
1 the accumulated vol
Types of smile quotes The smile is a static representation of the
implied volatilities at a given moment of time
What if the spot changes
Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change
Sticky strike if spot changes implied vol of a given strike doesnrsquot change
Moneyness Δ=DF1N(d1)
Spotladders price delta amp gamma
Vanilla
Knock-out spot=128 strike=125 barrier=15
0051
152
253
35
08 1 12 14 16 18
spot
gamma
06y
1y
0
01
02
03
04
05
06
08 1 12 14 16 18
spot
price
06y
1y
0
02
04
06
08
1
12
08 1 12 14 16 18
spot
delta
06y
1y
Linear regime S-K
1 underlying is needed to hedge
Sensitivity of Delta to spot is maximum
0000050001
000150002
000250003
000350004
00045
08 09 1 11 12 13 14
spot
price
06y
1y
-006
-005
-004
-003
-002
-001
0
001
002
003
08 09 1 11 12 13spot
delta
06y
1y
-06
-04
-02
0
02
04
08 1 12spot
gamma
06y
1y
Spot is far from barrier and far from OTM risk is minimum price is maximum
Δlt0 price gets smaller if spot increases
Spotladders vega vanna amp volga Vanilla
Knock-out spot=128 strike=125 barrier=135
0
000001
000002
000003
000004
000005
000006
08 1 12 14 16 18
spot
vega
06y
1y
-2
-1
0
1
2
3
08 1 12 14 16 18
spotvanna
06y
1y
0051
152
253
35
08 1 12 14 16 18
spot
volga
06y
1y
-0000008
-0000006
-0000004
-0000002
0
0000002
0000004
08 1 12
spot
vega
06y
1y
-1
-05
0
05
1
15
08 09 1 11 12 13
spot
vanna
06y
1y
-1
-05
0
05
1
15
2
08 1 12spot
volga
06y
1y
Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol
Simple analytic techniques ldquomoment matchingrdquo
Average-rate option payoff with N fixing dates
Basket option with two underlyings
TV pricing can be achieved quickly via ldquomoment matchingrdquo
Mark-to-market requires correlated stochastic processes for spotsvols (more complex)
01
max Asian 1
KSN
N
ii
0max Basket
2
22
1
11 K
tS
TSa
tS
TSa
ldquoMoment matchingrdquo To price Asian (average option) in TV we consider
that The spot process is lognormal The sum of all spots is lognormal also
Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)
Central idea of moment matching Find first and second moment of sum of lognormals
E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known
moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Types of smile quotes The smile is a static representation of the
implied volatilities at a given moment of time
What if the spot changes
Sticky delta if spot changes implied vol of a given ldquomoneynessrdquo doesnrsquot change
Sticky strike if spot changes implied vol of a given strike doesnrsquot change
Moneyness Δ=DF1N(d1)
Spotladders price delta amp gamma
Vanilla
Knock-out spot=128 strike=125 barrier=15
0051
152
253
35
08 1 12 14 16 18
spot
gamma
06y
1y
0
01
02
03
04
05
06
08 1 12 14 16 18
spot
price
06y
1y
0
02
04
06
08
1
12
08 1 12 14 16 18
spot
delta
06y
1y
Linear regime S-K
1 underlying is needed to hedge
Sensitivity of Delta to spot is maximum
0000050001
000150002
000250003
000350004
00045
08 09 1 11 12 13 14
spot
price
06y
1y
-006
-005
-004
-003
-002
-001
0
001
002
003
08 09 1 11 12 13spot
delta
06y
1y
-06
-04
-02
0
02
04
08 1 12spot
gamma
06y
1y
Spot is far from barrier and far from OTM risk is minimum price is maximum
Δlt0 price gets smaller if spot increases
Spotladders vega vanna amp volga Vanilla
Knock-out spot=128 strike=125 barrier=135
0
000001
000002
000003
000004
000005
000006
08 1 12 14 16 18
spot
vega
06y
1y
-2
-1
0
1
2
3
08 1 12 14 16 18
spotvanna
06y
1y
0051
152
253
35
08 1 12 14 16 18
spot
volga
06y
1y
-0000008
-0000006
-0000004
-0000002
0
0000002
0000004
08 1 12
spot
vega
06y
1y
-1
-05
0
05
1
15
08 09 1 11 12 13
spot
vanna
06y
1y
-1
-05
0
05
1
15
2
08 1 12spot
volga
06y
1y
Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol
Simple analytic techniques ldquomoment matchingrdquo
Average-rate option payoff with N fixing dates
Basket option with two underlyings
TV pricing can be achieved quickly via ldquomoment matchingrdquo
Mark-to-market requires correlated stochastic processes for spotsvols (more complex)
01
max Asian 1
KSN
N
ii
0max Basket
2
22
1
11 K
tS
TSa
tS
TSa
ldquoMoment matchingrdquo To price Asian (average option) in TV we consider
that The spot process is lognormal The sum of all spots is lognormal also
Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)
Central idea of moment matching Find first and second moment of sum of lognormals
E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known
moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Spotladders price delta amp gamma
Vanilla
Knock-out spot=128 strike=125 barrier=15
0051
152
253
35
08 1 12 14 16 18
spot
gamma
06y
1y
0
01
02
03
04
05
06
08 1 12 14 16 18
spot
price
06y
1y
0
02
04
06
08
1
12
08 1 12 14 16 18
spot
delta
06y
1y
Linear regime S-K
1 underlying is needed to hedge
Sensitivity of Delta to spot is maximum
0000050001
000150002
000250003
000350004
00045
08 09 1 11 12 13 14
spot
price
06y
1y
-006
-005
-004
-003
-002
-001
0
001
002
003
08 09 1 11 12 13spot
delta
06y
1y
-06
-04
-02
0
02
04
08 1 12spot
gamma
06y
1y
Spot is far from barrier and far from OTM risk is minimum price is maximum
Δlt0 price gets smaller if spot increases
Spotladders vega vanna amp volga Vanilla
Knock-out spot=128 strike=125 barrier=135
0
000001
000002
000003
000004
000005
000006
08 1 12 14 16 18
spot
vega
06y
1y
-2
-1
0
1
2
3
08 1 12 14 16 18
spotvanna
06y
1y
0051
152
253
35
08 1 12 14 16 18
spot
volga
06y
1y
-0000008
-0000006
-0000004
-0000002
0
0000002
0000004
08 1 12
spot
vega
06y
1y
-1
-05
0
05
1
15
08 09 1 11 12 13
spot
vanna
06y
1y
-1
-05
0
05
1
15
2
08 1 12spot
volga
06y
1y
Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol
Simple analytic techniques ldquomoment matchingrdquo
Average-rate option payoff with N fixing dates
Basket option with two underlyings
TV pricing can be achieved quickly via ldquomoment matchingrdquo
Mark-to-market requires correlated stochastic processes for spotsvols (more complex)
01
max Asian 1
KSN
N
ii
0max Basket
2
22
1
11 K
tS
TSa
tS
TSa
ldquoMoment matchingrdquo To price Asian (average option) in TV we consider
that The spot process is lognormal The sum of all spots is lognormal also
Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)
Central idea of moment matching Find first and second moment of sum of lognormals
E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known
moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Spotladders vega vanna amp volga Vanilla
Knock-out spot=128 strike=125 barrier=135
0
000001
000002
000003
000004
000005
000006
08 1 12 14 16 18
spot
vega
06y
1y
-2
-1
0
1
2
3
08 1 12 14 16 18
spotvanna
06y
1y
0051
152
253
35
08 1 12 14 16 18
spot
volga
06y
1y
-0000008
-0000006
-0000004
-0000002
0
0000002
0000004
08 1 12
spot
vega
06y
1y
-1
-05
0
05
1
15
08 09 1 11 12 13
spot
vanna
06y
1y
-1
-05
0
05
1
15
2
08 1 12spot
volga
06y
1y
Vanna Sensitivity of Vega with respect to SpotVolga Sensitivity of Vega with respect to Vol
Simple analytic techniques ldquomoment matchingrdquo
Average-rate option payoff with N fixing dates
Basket option with two underlyings
TV pricing can be achieved quickly via ldquomoment matchingrdquo
Mark-to-market requires correlated stochastic processes for spotsvols (more complex)
01
max Asian 1
KSN
N
ii
0max Basket
2
22
1
11 K
tS
TSa
tS
TSa
ldquoMoment matchingrdquo To price Asian (average option) in TV we consider
that The spot process is lognormal The sum of all spots is lognormal also
Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)
Central idea of moment matching Find first and second moment of sum of lognormals
E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known
moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Simple analytic techniques ldquomoment matchingrdquo
Average-rate option payoff with N fixing dates
Basket option with two underlyings
TV pricing can be achieved quickly via ldquomoment matchingrdquo
Mark-to-market requires correlated stochastic processes for spotsvols (more complex)
01
max Asian 1
KSN
N
ii
0max Basket
2
22
1
11 K
tS
TSa
tS
TSa
ldquoMoment matchingrdquo To price Asian (average option) in TV we consider
that The spot process is lognormal The sum of all spots is lognormal also
Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)
Central idea of moment matching Find first and second moment of sum of lognormals
E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known
moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
ldquoMoment matchingrdquo To price Asian (average option) in TV we consider
that The spot process is lognormal The sum of all spots is lognormal also
Note a sum of lognormal variables is not lognormal Therefore this method is an approximation (but quite accurate for practical purposes)
Central idea of moment matching Find first and second moment of sum of lognormals
E[Σi Si] E[ (Σi Si)2 ] Assume sum of lognormals is lognormal (with known
moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Asian options analytics (1) Prerequisites for the analysis statistics of random increments Increments of spot process have 0 mean and variance T
(time to maturity)
E[Wt]=0 E[Wt2]=t
If t1ltt2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt1
2] = t1
(because Wt1 is independent of Wt2-Wt1)
More generally E[Wt1∙Wt2] = min(t1t2)
From this and with some algebra it follows that E[St1 ∙ St2] = S0
2 exp[r ∙(t1+t2) + σ2 ∙ min(t1t2)]
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Asian options analytics (2) Asian payoff contains sum of spots
What are its mean (first moment) and variance
Looks complex but on the right-hand side all quantities are known and can be easily calculated
Therefore the first and second moment of the sum of spots can be calculated
N
iiSN
X1
1
N
ji
ttttrN
iji
NN
jj
N
ii
N
i
trN
i
NttrN
ii
N
ii
jiji
iii
eSN
SSN
SSN
X
eSN
eSN
SN
SN
X
1
)(min202
1 1j2
112
2
10
1
)10(0
11
2
221
E1
E11
EE
1E
1E
11EE
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Asian options analytics (3) Now assume that X follows lognormal process with λ the (flat) vol μ
the drift
Has solution (as in standard Black-Scholes)
Take averages in above and obtain first and second moment in terms of μλ
Solving for drift and vol produces
tttt dWXdtXdX
TWTT eSX 2
21
0
TT
WTT
TT
eXeeSX
eSXT
2221 2222
02
0
EEE
E
0
Elog
1
S
X
TT
TT
X
X
T 2
2
E
Elog
1
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Asian options analytics (4) Since we wrote Asian payoff as max(XT-K0) We can quote the Black-Scholes formula
With
And μ λ are written in terms of E[X] E[X2] which we have calculated as sums over all the fixing dates
The ldquoaveragingrdquo reduces volatility we expect lower price than vanilla
Basket is based on similar ideas
)()(DFAsian 210 dNKdNSe T
T
TK
S
d
20
1
2
1ln
T
TK
S
d
20
2
2
1ln
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Smile-dynamics models Large number of alternative models
Volatility becomes itself stochastic Spot process is not lognormal Random variables are not Gaussian Random path has memory (ldquonon-markovianrdquo) The time increment is a random variable (Levy processes) And many many morehellip
A successful model must allow quick and exact pricing of vanillas to reproduce smile
Wilmott ldquomaths is like the equipment in mountain climbing too much of it and you will be pulled down by its weight too few and you wonrsquot make it to the toprdquo
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Dupire Local Vol Comes from a need to price path-dependent
options while reproducing the vanilla mkt prices
Underlying follows still lognormal process buthellip Vol depends on underlying at each time and time itself It is therefore indirectly stochastic
Local vol is a time- and spot-dependent vol(something the BS implied vol is not)
No-arbitrage fixes drift μ to risk-free rate
ttttt dWtSSdtSdS
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Local Vol
tTSK
KK
KTt tCK
CrrKCrCtS
2
21
1212
Technology invented independently by B Dupire Risk (1994) v7 pp18-20 E Derman and I Kani Fin Anal J (1996) v53 pp25-36
They expressed local vol in terms of market-quoted vanillasand its timestrike derivatives
Or equivalently in terms of BS implied-vols
tTSKt t
dd
KKtTK
d
KtTKK
KrrK
TtTtS
BS
21
2
BS2BS
2
0BS
1BS
02
BS
221
BS12
BS
0
BS21
2
21
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Dupire Local Vol
Contains derivatives of mkt quotes with respect to
Maturity Strike
The denominator can cause numerical problems CKKlt0 (smile is locally concave) σ2lt0 σ is imaginary
The Local-vol can be seen as an instantaneous volatility depends on where is the spot at each time step
Can be used to price path-dependent options
T
tStStS SSS TT 112211 2
1
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Local Vol rule of thumb Rule of thumb
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman amp Kani)
Sinitial
Sfinal
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Local-Vol and vanillas
Example Take smile quotes Build local-vol Use them in simulation
and price vanillas Compare resulting price
of vanillas vs market quotes(in smile terms)
By design the local-vol model reproduces automatically vanillas
No further calibration necessary only market quotes needed
EURUSD market
Lines market quotes
Markers LV pricer
Blue 3 years maturity
Green 5 years maturity
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Analytic Local-Vol (2)
Alternative assume a form for the local-vol σ(Stt)
Do that for example by
From historical market data calculate log-returns
These equal to the volatility
Make a scatter plot of all these Pass a regression The regression will give an idea of
the historically realised local-vol function
tSS
St
t
tt log
Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Analytic Local-Vol (2) A popular choice is
Ft the forward at time t Three calibration parameters
σ0 controlling ATM vol α controlling skew (RR) β controlling overall shift (BF)
Calibration is on vanilla prices Solve Dupire forward PDE with initial condition C=(S0-K)
+
SF
F
F
FtS tt
2
000 111
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Stochastic models Stochastic models introduce one extra source of
randomness for example Interest rate dynamics Vol dynamics Jumps in vol spot other underlying Combinations of the aboveDupire Local Vol is therefore not a real stochastic model
Main problem Calibration minimize
(model output ndash market observable)2
Example (model ATM vol ndash market ATM vol)2
Parameter space should not be too small model cannot reproduce all market-quotes
across tenors too large more than one solution exists to calibration
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Heston model Coupled dynamics of underlying and volatility
Interpretation of model parameters
μ drift of underlying κ speed of mean-reversion ρ correlation of Brownian motions ε volatility of variance
Analytic solution exists for vanillas S L Heston A Closed form solution for options with stochastic
volatility Rev Fin Stud (1993) v6 pp327-343
1dWSvdtSdS tttt
2dWvdtvvdv ttt
dtdWdWE 21
Processes Lognormal for spot Mean-reverting for
variance Correlated Brownian
motions
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Effect of Heston parameters on smile
Affecting overall shift in vol Speed of mean-reversion κ Long-run variance vinfin
Affecting skew Correlation ρ Vol of variance ε
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Local-vol vs Stochastic-vol Dupire and Heston reproduce vanillas perfectly But can differ dramatically when pricing exotics
Rule of thumb skewed smiles use Local Vol convex smiles use Heston
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Hull-White model It models mean-reverting underlyings such as
Interest rates Electricity oil gas etc
3 parameters to calibrate obtained from historical data
rmean (describes long-term mean) obtained from calibration
a speed of mean reversion σ volatility
Has analytic solution for the bond price P = E[ e-
intr(t)dt ]
ttt dWdtrardr mean
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Three-factor model in FOREX
Three factor model in FOREX spot + domesticforeign rates
To replicate FX volatilities match
FXmkt with FXmodel
Θ(s) is a function of all model parameters FXdfadaf
ffff
meanff
ddddmean
dd
FXfd
dWdtrardr
dWdtrardr
dWSdtSrrdS
T
t
dsstT
22modelFX
1
Hull-White is often coupled to another underlying
Common calibration issue Variance squeezeldquo
FX vol + IR vols up to a certain date have exceeded the FX-model vol
Solution (among other possibilities)
Time-dependent parameters (piecewise constant)
parameter
time
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Two-factor model in commodities
Commodity models introduce the ldquoconvenience yieldrdquo (termed δ)
δ = benefit of direct access ndash cost of carry
Not observable but related to physical ownership of asset
No-arbitrage implies Forward F(tT) = St ∙ E [ eint(r(t)-δ(t))dt ]
δt is taken as a correction to the drift of the spot price process
What is the process for St rt δt
Problem δt is unobserved Spot is not easy to observe
for electricity it does not exist For oil the future is taken as a proxy
Commodity models based on assumptions on δ
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Gibson-Scwartz model Classic commodities model
Spot is lognormal (as in Black-Scholes) Convenience yield is mean-reverting
Very similar to interest rate modeling (although δt can be posneg)
Fluctuation of δ is in practise an order of magnitude higher than that of r no need for stochastic interest rates
Analysis based on combining techniques Calculate implied convenience yield from observed
future prices
2
1
ttt
ttttttt
dWdtd
dWSdtrSdS
Miltersen extension
Time-dependent parameters
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Merton jump model This model adds a new element to the
stochastic models jumps in spot Motivated by real historic data
Disadvantages Risk cannot be
eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages Can produce smile Adds a realistic
element to dynamics Has exact solution
for vanillas
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Merton jump modelExtra term to the Black-Scholes process
If jump does not occur
If jump occurs Then
Therefore Y size of the jump
Model has two extra parameters size of the jump Y frequency of the jump λ
tt
t dWdtS
dS
1 YdWdtS
dSt
t
t
YSS
YSSSS
tt
tttt
jump beforejumpafter
jump beforejump beforejumpafter 1
Jump size amp jump times
Random variables
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Merton model solution Merton assumed that
The jump size Y is lognormally-distributed Can be sampled as Y=eη+γ∙g g is normal ~N(01) and ηγ are real
Jump times Poisson-distributed with mean λ Prob(n jumps)=e-
λT(λT)n n Jump times independent from jump sizes
The model has solution a weighted sum of Black-Scholes formulas
σn rn λrsquo are functions of σr and the jump-statistics given by η γ
nn
nT rTKS
n
TBS
e price Call 0
0n
-
T
TrK
S
KeT
TrK
S
SerTKSn
nnTrr
n
nnTr
nnn
22102
210
0
loglogBS 11
21 e
T
nn
222 2
21
12 12
21
T
nerrrn
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Merton model properties The model is able to produce a smile effect
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Vanna-Volga method Which model can reproduce market dynamics
Market psychology is not subject to rigorous math modelshellip
Brute force approach Capture main features by a mixture model combining jumps stochastic vols local vols etc
Buthellip Difficult to implement Hard to calibrate Computationally inefficient
Vanna-Volga is an alternative pricing ldquorecipierdquo Easy to implement No calibration needed Computationally efficient
Buthellip It is not a rigorous model Has no dynamics
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Vanna-Volga main idea The vol-sensitivities
Vega Vanna Volga
are responsible the smile impact
Practical (traderrsquos) recipie Construct portfolio of 3 vanilla-instruments which
zero out the VegaVannaVolga of exotic option at hand
Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities)
Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of
vanillas
Price
S
Price2
2
2Price
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Vanna-Volga hedging portfolio Select three liquid instruments
At-The-Money Straddle (ATM) =frac12 Call(KATM) + frac12 Put(KATM) 25Δ-Risk-Reversal (RR) = Call(Δ=frac14) - Put(Δ=-frac14) 25Δ-Butterfly (BF) = frac12 Call(Δ=frac14) + frac12 Put(Δ=-frac14) ndash ATM
KATM
KATM
K25ΔP K25ΔC
KATM
K25ΔP K25ΔC
ATM Straddle 25Δ Risk-Reversal
25Δ Butterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Vanna-Volga weights Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF
∙ BF
What are the appropriate weights wATM wRR wBF
Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes
vol-sensitivities of portfolio P = vol-sensitivities of exotic X
solve for the weights
volga
vanna
vega
volgavolgavolga
vannavannavanna
vegavegavega
volga
vanna
vega
w
w
w
BFRRATM
BFRRATM
BFRRATM
X
X
X
XAw -1
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Vanna-Volga price Vanna-Volga market price
is
XVV = XBS + wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Vanna-Volga vs market-price Can be made to fit the market price of exotics More info in
F Bossens G Rayee N Skantzos and G Delstra
Vanna-Volga methods in FX derivatives from theory to market practiseldquo
Int J Theor Appl Fin (to appear)
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Models that go the extra mile
Local Stochastic Vol model Jump-vol model Bates model
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Local stochastic vol model Model that results in both a skew (local vol) and a convexity
(stochastic vol)
For σ(Stt) = 1 the model degenerates to a purely stochastic model
For ξ=0 the model degenerates to a local-volatility model
Calibration hard
Several calibration approaches exist for example
Construct σ(Stt) that fits a vanilla market Use remaining stochastic parameters to fit eg a liquid exotic-option
market
2
1
tttt
tttttt
dWVdtVdV
dWVtSdtSdS
222LV Dupire ttt VtStS
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Jump vol model Consider two implied volatility surfaces
Bumped up from the original Bumped down from the original
These generate two local vol surfaces σ1(Stt) and σ2(Stt)
Spot dynamics
Calibrate to vanilla prices using the bumping parameter and the probability p
ptS
ptStS
dWtSSdtSdS
t
tt
ttttt
-1 prob with
prob with
2
1
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Bates model Stochastic vol model with jumps
Has exact solution for vanillas
Analysis similar to Heston based on deriving the Fourier characteristic function
More info D S Bates ldquoJumps and Stochastic Volatility
Exchange rate processes implicit in Deutsche Mark Optionsldquo Rev Fin Stud (1996) v9 pp69-107
2
1
tttt
ttttt
dWVdtd
dZdWdtSdS
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Which model is better
Good for Skew smiles
Good for simple exotics
Good for convex smiles
Allows fat-tails
Good for barrier options lt1y
Fast + accurate for simple exoticsOTKODKOhellip
Good for maturitiesgt1y
Good if product has spot amp rates as underlying
Can price most types of products (in theory)
Not good for convex smiles
Approximates numerical derivatives outside mkt quotes
Not good for Skew smiles
Often needs time-dependent params to fit term structure
Cannot be used for path-dependent optionsTARFLKBhellip
Not useful if rates are approx constant
Often unstable
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Choice of model Model should fit vanilla market (smile)
and a liquid exotic market (OT)
Model must reproduce market quotes across various tenors (term structure)
No easy answer to which model to useW Schoutens E Simons and J Tistaert A Perfect calibration Now whatldquo Wilmott Magazine March 2004
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
One-touch tables OT tables measure model success vs market price OT price asymp probability of touching barrier (discounted) Collect mkt prices for TV in the range
0-100 (away-close to barrier) Calculate model price ndash market price The better model gives model-mktasymp0
OT table
-700
-600
-500
-400
-300
-200
-100
000
100
200
300
0 02 04 06 08 1
TV price
mkt
- m
od
el
VannaVolga
LocalVol
Heston
OT tables depend on
nbr barriers
Type of underlying
Maturity
mkt conditions
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Numerical MethodsMonte Carlo Advantages
Easy to implement Easy for multi-factor
processes Easy for complex payoffs
Disadvantages Not accurate enough CPU inefficient Greeks not stableaccurate American exercise difficult Depends on quality of
random number generator
PDE Disadvantages
Hard to implement Hard for multi-factor
processes Hard for complex payoffs
Advantages Very accurate CPU efficient Greeks stableaccurate American exercise very easy Independent of random
numbers
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Monte Carlo vs PDE
Monte CarloBased on discounted average payoff over realizations of
spot
Outline of Monte Carlo simulation For each path
At each time step till maturity Draw a random number from Normal distribution N(0T) Update spot
Calculate payoff for this path Calculate average payoff across all paths
Pathsnbr
1
)(payoffPathsnbr
1
payoffE PriceOption
i
iT
Tr
TTr
Se
Se
number random
tttttt WStSSS
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Monte Carlo vs PDE
Partial Differential Equation (PDE)Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms eg with t+=t+Δt S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
PrS
P
S
PS
t
P
2
22
2
1
PrS
SPSPSP
S
SPSPS
t
tPtP
22 )()(2)(
2
1
2
)()()()(
time
Spot
today maturity
S0
K
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Issues on simulations Random numbers Barriers and hit probability Simulating american-exercise
options Likelihood ratio method
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Random numbers Simulations require at each time step a random number Statistics for example normal-Gaussian (for lognormal process)
mean=0 variance=1 This means that if we sum all random numbers we should get 0 and
stdev=1 In practise we draw uniform random numbers in [01] and convert them
to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths amp 102 steps 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should This effect is more pronounced as the number of dimensions (=number of
steps number of paths) increases
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Pseudo-random number generators RNG generate numbers in the interval [01]
With some transformations one then converts the sampling space [01] to any other that is required (eg gaussian normal space)
Random numbers are not truly random (hence ldquopseudordquo)there is a formula behind taking as input the computer clock
After a while ldquorandom numbersrdquo will repeat themselves Good random numbers have a long period before repetition
occurs ldquoMersennerdquo random numbers have a period that is a
Mersenne number ie can be written as 2n-1 for some big n (for example n=20000)
Mersenne numbers are popular due to They are quickly generated Sequences are uncorrelated Eventually (after many draws) they fill the space uniformly
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
ldquoLow-discrepancyrdquo random numbers These numbers are not random at all ldquolow discrepancyrdquo = homogenous
LDRN fill the [01] space homogenously Passing uniform numbers through the cumulative of the
probability density will produce the correct density of points
0
1
hom
og
enous
nu
mbers
form
[0
1]
Gaussian cumulative function
Non-homogenous numbers in (-infin infin)
Gaussian probability
function
Higher density of points here
ldquoPeakrdquo implies that more points should be sampled from here
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Sobolrsquo numbers Sobolrsquo numbers are low-discrepancy sequences Quality depends on nbr of dimensions = nbr Paths x nbr
Steps Uniformity is good in low dimensions Uniformity is bad in high dimensions Are convenient because hellip they are not random
Calculating the Greeks with finite difference requires the same sequence of random numbers
The calculation of the Greeks should differ only in the ldquobumpedrdquo param
S
SSSS
2
PricePrice
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Random number quality
1 2 3 4 5 6 70 0 0 0 0 0 0
05 05 05 05 05 05 05025 075 025 075 025 075 025075 025 075 025 075 025 075
0875 0875 0125 0625 0375 0375 06250375 0375 0625 0125 0875 0875 01250125 0625 0875 0875 0625 0125 03750625 0125 0375 0375 0125 0625 0875
06875 08125 08125 01875 00625 06875 0562501875 03125 03125 06875 05625 01875 0062504375 05625 00625 04375 08125 09375 0312509375 00625 05625 09375 03125 04375 0812508125 06875 04375 00625 09375 03125 0687503125 01875 09375 05625 04375 08125 0187500625 09375 06875 03125 01875 00625 0437505625 04375 01875 08125 06875 05625 09375
059375 096875 034375 090625 078125 084375 003125009375 046875 084375 040625 028125 034375 053125034375 071875 059375 065625 003125 059375 078125084375 021875 009375 015625 053125 009375 028125096875 059375 096875 078125 015625 021875 015625046875 009375 046875 028125 065625 071875 065625021875 084375 021875 053125 090625 046875 090625071875 034375 071875 003125 040625 096875 040625065625 065625 003125 034375 034375 090625 009375015625 015625 053125 084375 084375 040625 059375040625 090625 078125 009375 059375 065625 084375090625 040625 028125 059375 009375 015625 034375
Draw (n x m) table of Sobolrsquo numbers
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
2 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 10 20 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 13 40 )
0000
0100
0200
0300
0400
0500
0600
0700
0800
0900
1000
0000 0200 0400 0600 0800 1000
Pair( 20 881 )
Plot pairs of columns(12) (1020)
Non-uniform filling for large dimensions
(1340) (20881)
Nbr Steps Nbr Paths
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Barrier options
Payoff at maturity is alive if Barrier A has not been hit Barrier B has been hit
Consider a (slightly) complex barrier pattern
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Barrier options There is analytic expression for ldquosurvival probabilityrdquo
=probability of not hitting
We rewrite the pattern in terms of ldquonot-hittingrdquo events
This is equivalent to the replication formula KIAKOB = KOB ndash DKOAB
Option price = DF ∙ payoff at maturity∙ Prob(A is not hit AND B is hit)
hitnot isA ANDhit not is BProbhitnot isA Prob
hitnot isA Probhitnot isA GIVENhit not is BProb1
hitnot isA Probhitnot isA GIVENhit is BProb
hitnot is A ANDhit is BProb rule Bayes
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Barrier option replication
Prob(A is hit) = Prob(A is hit in [t1t2])∙
Prob(A is hit in [t2t3])
Prob(A is hit AND B is hit) = =Prob(A is hit in [t1t2])∙ Prob(A AND B are hit in [t2t3]) ∙ Prob(B is hit in [t3t5])
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Barrier options formula
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Barrier option formula
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
American exercise in Monte Carlo
When is it optimal to exercise the option
Naiumlve approach If at any time t Spot is out-of-the-money it is not optimal to exercise Stop Spot is in-the-money then
start new simulation from this spot if (on average) final spot finishes more in-the-money do not exercise
now if (on average) final spot finishes less in-the-money exercise now
today
K
S0
today t maturity
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Least-squares Monte Carlo Since this has to be done for every time step t
Naiumlve Monte Carlo is clearly impractical Methodology for american exercise provided by
Longstaff amp Schwartz (2001) Rev Fin Studies v14 pp113-147 Method is not exact but quite accurate (versus eg PDE) Is not hard to implement But not as CPU-efficient as standard monte carlo Central idea
Work backwards starting from maturity At each step compare immediate exercise value with expected
cashflow from continuing Exercise if immediate exercise is more valuable
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Least-squares Monte Carlo (1)
Generate spots for each path amp for each time-step Make an NpathsxNsteps table of spot paths (according to some dynamics) Make an NpathsxNsteps empty table of cashflows (CF)
Out-of-the-money
In-the-money
Npaths
Nsteps
Spot Paths
0000
0000
0000
0000
0000
0000
0000
Cashflows
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Least-squares Monte Carlo (2)
If spot at maturity is in-the-money assign for this path CF=payoff value out-of-the-money assign for this path CF=0
Out-of-the-money
In-the-money
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
CF=(Sthis path(T)-K)+
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Least-squares Monte Carlo (3)
Go one time-step backwards If spot is in-the-money option holder must decide whether to exercise now or continue
Calculate Y=discounted cashflow at next step if option is not exercised now out-of-the-money assign for this path CF=0
Spot Paths
000
000
0000
000
0000
000
000
Cashflows
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-ΔtT) ∙ CF(T)
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Least-squares Monte Carlo (4)
On the pairs Spath iYpath i pass a regression of the form
E(S) = a0+a1∙S +a2∙S2
This function is an approximation to the expected payoff from continuing to hold the option from this time point on
If E(Spath(T-Δt)) lt (Spath(T-Δt)-K)+ exercise the option at this time step Assign CF at this step = (Spath(T-Δt)-K)+ and for all larger t set CF=0
If E(Spath(T-Δt)) gt (Spath(T-Δt)-K)+ Do not exercise the option at this time step Maintain same value of cashflow at next steps
Y
S
E(S)
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Least-squares Monte Carlo (5)
Proceed similarly till the first time step and populate the matrix of cashflows
There should be one non-zero cashflow per path(the option can be exercised only once)
Callables are priced with the same idea
000000
000000
000000
000000
000000
000000
000000
exer
1
exer
paths
tCFttodayDF 1
Amerpaths
ii
N
ii S
N
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Greeks in Monte Carlo To calculate Greeks with Monte Carlo
Bump sensitivity parameter (spot vol etc) Recalculate market data with the bumped parameter (smile
curves etc) Re-run Monte Carlo Calculate Greeks as finite difference For example
This requires at least 12 Monte Carlo runs for all Greeks Not ideal for impatient traders
S
SSSS
2
PricePrice 2
PricePrice2Price
S
SSSSS
PricePriceVega
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Likelihood ratio method (1)
This method allows us to calculate all Greeks within a single Monte Carlo
Main idea
Express Greeks as payoffs Price the new ldquopayoffsrdquo with the same simulation
Note The analytics of the method simplify if spot is assumed to follow
lognormal process (as in BS) The LR greeks will not be in general the same as the finite
difference greeks This is because of the modification of the market data when using the
finite difference method
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Likelihood ratio method (2)
Consider an exotic option with a path-dependent payoff Its price will depend on all spots in the path
PDF probability density function of the spot
zi the Gaussian random number used to make the jump S i-1 Si
Probsurv the total survival probability for the spot path (given some barrier levels)
For explicit expressions for the survprob of KO or DKO see previous slides
PayoffProbPDFDF Exotic 1surv11 mmm SSSSdSdS
m
i
z
iii
m
iim
ieSt
SSS1
21
1
2211
2
1PDFPDF
ii
iii
i
it
trS
S
z
2
1 21
log
m
i
tt ii
1survsurv
1ProbProb
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
Likelihood ratio method (3)
Sensitivity with respect to a parameter α (=spot vol etc)
This is simple derivatives over analytic functions (see previous slide)
For example Delta becomes the new payoff
To be priced with the same spot path as the Payoff itself Similarly for other Greeks more lengthy expressions but doable
m
m
m
mm
mmm
S
S
S
SdS
SSdS
1surv
1surv
1
11
1surv11
Prob
Prob
1PDF
1PayoffDF
ProbPDF PayoffDF
Exotic
0
surv
surv0
1
1
10
10
Prob
Prob
1PDF
1PayoffDF
SS
S
S
tt
ttm
m
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer
References Options
ldquoOptions Futures amp other derivativesrdquo John C Hull (2008) Prentice Hall
ldquoPaul Wilmott on Quantitative Finance 3 Vol Setrdquo Paul Wilmott (2000) Wiley
Numerical methods PDE Pricing Financial Instruments The Finite
Difference Method D Tavella and C Randall (2000) Wiley
Monte Carlo ldquoMonte Carlo methods in Finance P Jaumlckel (2003) Wiley
Monte Carlo ldquoMonte Carlo methods in Financial Engineering P Glasserman (2000) Springer