An Idiot’s Guide to Option Pricing Bruno Dupire Bloomberg LP [email protected] CRFMS, UCSB...
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Transcript of An Idiot’s Guide to Option Pricing Bruno Dupire Bloomberg LP [email protected] CRFMS, UCSB...
An Idiot’s Guide to Option Pricing
Bruno DupireBloomberg LP
CRFMS, UCSBApril 26, 2007
Bruno Dupire 2
Warm-up
%30][
%70][
Black
Red
P
P
Black if$0
Red if100$
Roulette:
A lottery ticket gives:
You can buy it or sell it for $60Is it cheap or expensive?
Bruno Dupire 3
Naïve expectation
Buy6070
Bruno Dupire 4
Replication argument
Sell6050
“as if” priced with other probabilities
instead of
OUTLINE
1. Risk neutral pricing
2. Stochastic calculus
3. Pricing methods
4. Hedging
5. Volatility
6. Volatility modeling
Bruno Dupire 6
Addressing Financial Risks
•volume•underlyings•products•models•users•regions
Over the past 20 years, intense development of Derivatives
in terms of:
Bruno Dupire 7
$
TSK
To buy or not to buy?
• Call Option: Right to buy stock at T for K
$
TSK
$
TSK
TO BUY NOT TO BUY
CALL
Bruno Dupire 8
Vanilla Options
European Call:Gives the right to buy the underlying at a fixed price (the strike) at some future time (the maturity)
TSKPayoffPut
European Put:Gives the right to sell the underlying at a fixed strike at some maturity
0,max)(Payoff Call KSKS TT
Bruno Dupire 9
Option prices for one maturity
Bruno Dupire 10
Risk Management
Client has risk exposure
Buys a product from a bank to limit its risk
Risk
Not Enough Too Costly Perfect Hedge
Vanilla Hedges Exotic Hedge
Client transfers risk to the bank which has the technology to handle it
Product fits the risk
Risk Neutral Pricing
Bruno Dupire 12
Price as discounted expectation
!?
Option gives uncertain payoff in the futurePremium: known price today
Resolve the uncertainty by computing expectation:
Transfer future into present by discounting
?!
Bruno Dupire 13
Application to option pricing
Risk Neutral ProbabilityPhysical Probability
o TTT
rT dSKSSe ))((Price
Bruno Dupire 14
Basic Properties
0)(price0 AA
)(price)(price)(price BABA
Price as a function of payoff is:
- Positive:
- Linear:
Price = discounted expectation of payoff
Bruno Dupire 15
gives 1 in state
Option A gives
Toy Model
nss ,...,1
ix is
1 period, n possible states
in state
iA is
iii
iii
iii
xq
AxAAxA
)(price)(price
If , 0 in all other states,
where price(1)price)(price ii AA is a discount factor
jj
ii A
Aq
)(price
)(price1and0
iii qq is a probability:
Bruno Dupire 16
FTAP
Fundamental Theorem of Asset Pricing
1) NA There exists an equivalent martingale measure
2) NA + complete There exists a unique EMM
Cone of >0 claimsClaims attainable from 0
Separating hyperplanes
Bruno Dupire 17
Risk Neutrality Paradox
• Risk neutrality: carelessness about uncertainty?
• 1 A gives either 2 B or .5 B1.25 B• 1 B gives either .5 A or 2 A1.25 A• Cannot be RN wrt 2 numeraires with the same probability
Sun: 1 Apple = 2 Bananas
Rain: 1 Banana = 2 Apples
50%
50%
Stochastic Calculus
Bruno Dupire 19
Modeling Uncertainty
Main ingredients for spot modeling• Many small shocks: Brownian Motion
(continuous prices)
• A few big shocks: Poisson process (jumps)
t
S
t
S
Bruno Dupire 20
Brownian Motion
10
100
1000
• From discrete to continuous
Bruno Dupire 21
Stochastic Differential Equations
tW
),0(~ stNWW st
dWdtdx ba
At the limit:
continuous with independent Gaussian increments
SDE:
drift noise
a
Bruno Dupire 22
Ito’s Dilemma
)(xf
dWdtdx b a
Classical calculus:
expand to the first order
Stochastic calculus:
should we expand further?
dxxfdf )('
Bruno Dupire 23
Ito’s Lemma
dtdW 2)(
dWbdtadx
dtbxfdxxf
dxxfdxxf
xfdxxfdf
2
2
)(''2
1)('
)()(''2
1)('
)()(
At the limit
for f(x),
If
Bruno Dupire 24
Black-Scholes PDE
• Black-Scholes assumption
• Apply Ito’s formula to Call price C(S,t)
• Hedged position is riskless, earns interest rate r
• Black-Scholes PDE
• No drift!
dWdtS
dS
dtCS
CdSCdC SStS )2
(22
dtSCCrdSCdCdtCS
C SSSSt )()2
(22
)(2
22
SCCrCS
C SSSt
SCC S
Bruno Dupire 25
P&L
St t
St
Break-even points
t
t
Option Value
St
CtCt t
S
Delta hedge
P&L of a delta hedged option
Bruno Dupire 26
Black-Scholes Model
If instantaneous volatility is constant :
dWdtS
dS
Then call prices are given by :
)2
1))/(ln(
1(
)2
1))/(ln(
1(
0
00
TrTKST
NKe
TrTKST
NSC
rT
BS
No drift in the formula, only the interest rate r due to the hedging argument.
drift:
noise, SD:
tSt
tSt
Pricing methods
Bruno Dupire 28
Pricing methods
• Analytical formulas
• Trees/PDE finite difference
• Monte Carlo simulations
Bruno Dupire 29
Formula via PDE
• The Black-Scholes PDE is
• Reduces to the Heat Equation
• With Fourier methods, Black-Scholes equation:
)(2
22
SCCrCS
C SSSt
TddT
TrKSd
dNeKdNSC rTBS
12
20
1
210
,)2/()/ln(
)()(
xxUU2
1
Bruno Dupire 30
Formula via discounted expectation
• Risk neutral dynamics
• Ito to ln S:
• Integrating:
• Same formula
dWdtrS
dS
dWdtrSd )
2(ln
2
])[(])[()
2(
0
2
KeSEeKSEepremiumTWTrrT
TrT
TT WTrSS )
2(lnln
2
0
Bruno Dupire 31
Finite difference discretization of PDE
• Black-Scholes PDE
• Partial derivatives discretized as
)(),(
)(2
22
KSTSC
SCCrCS
C
T
SSSt
2)(
),1(),(2),1(),(
2
),1(),1(),(
)1,(),(),(
S
niCniCniCinC
S
niCniCniC
t
niCniCniC
SS
S
t
Bruno Dupire
Option pricing with Monte Carlo methods
• An option price is the discounted expectation of its payoff:
• Sometimes the expectation cannot be computed analytically:– complex product– complex dynamics
• Then the integral has to be computed numerically
P EP f x x dxT0
the option price is its discounted payoffintegrated against the risk neutral density of the spot underlying
Bruno Dupire
Computing expectationsbasic example
•You play with a biased die
•You want to compute the likelihood of getting
•Throw the die 10.000 times
•Estimate p( ) by the number of over 10.000 runs
Bruno Dupire
Option pricing = superdie
Each side of the superdie represents a possible state of the financial market
• N final values
in a multi-underlying model
• One path
in a path dependent model
• Why generating whole paths?
- when the payoff is path dependent
- when the dynamics are complexrunning a Monte Carlo path simulation
Bruno Dupire
Expectation = Integral
Unit hypercube Gaussian coordinates trajectory
Gaussian transform techniques discretisation schemes
A point in the hypercube maps to a spot trajectorytherefore
EP f x S S dx g y dy
Ng x
T t tdd d
ixi
d
.Pr ,...,,1
,1
10
0
1
Bruno Dupire 36
Generating Scenarios
Bruno Dupire 37
Low Discrepancy Sequences
dimensions1 & 2
Halton Faure Sobol
dimensions20 & 25
dimensions51 & 52
Hedging
Bruno Dupire 39
To Hedge or Not To Hedge
Daily Position
Daily P&L
Full P&L
Big directional risk HedgeDelta Small daily amplitude risk
S
P&L
Unhedged
0Hedged
Bruno Dupire 40
The Geometry of Hedging
• Risk measured as • Target X, hedge H
• Risk is an L2 norm, with general properties of orthogonal projections
• Optimal Hedge:
TPLSD
ttt HXPL
HXHXRisk TTvar
H
HXHXH
infˆ
Bruno Dupire 41
The Geometry of Hedging
Bruno Dupire 42
Super-replication
•Property:
Let us call:
Which implies:
E XY E X E Y 2 2
yx
yx
xy
PP
YPXPXY
YPXPYX
2
:Portfolio by the dominated is XY so
,0 , and allFor
22
2
P X
P Yx
y
:
:
price today of
price today of
2
2
price XYP P P P
P PP P
y x x y
x yx y
2
Bruno Dupire 43
A sight of Cauchy-Schwarz
Volatility
Bruno Dupire 45
Volatility : some definitions
Historical volatility :
annualized standard deviation of the logreturns; measure of uncertainty/activity
Implied volatility :
measure of the option price given by the market
Bruno Dupire 46
Historical Volatility
• Measure of realized moves• annualized SD of logreturns
t
tt S
Sx 1ln
2
1
2
1
252ii t
n
it xx
n
Bruno Dupire 47
Historical volatility
Bruno Dupire 48
Implied volatility
Input of the Black-Scholes formula which makes it fit the market price :
Bruno Dupire 49
Market Skews
Dominating fact since 1987 crash: strong negative skew on Equity Markets
Not a general phenomenon
Gold: FX:
We focus on Equity Markets
K
impl
K
impl
K
impl
A Brief History of Volatility
Bruno Dupire 51
Evolution theory of modeling
constant deterministic stochastic nD
Bruno Dupire 52
A Brief History of Volatility
– : Bachelier 1900
– : Black-Scholes 1973
– : Merton 1973
– : Merton 1976
Qtt
t
t dWtdtrS
dS )(
Qtt dWdS
Qt
t
t dWdtrS
dS
dqdWdtkrS
dS Qt
t
t )(
Bruno Dupire 53
Local Volatility Model
Dupire 1993, minimal model to fit current volatility surface
2
,2
2
2 2,
),(
K
CK
KC
rKTC
TK
dWtSdtrS
dS
TK
Qt
t
t
Bruno Dupire 54
sought diffusion(obtained by integrating twice
Fokker-Planck equation)
1D Diffusion
s
Risk Neutral
Processes
Compatible with Smile
The Risk-Neutral Solution
But if drift imposed (by risk-neutrality), uniqueness of the solution
Bruno Dupire 55
European prices
Localvolatilities
Localvolatilities
Exotic prices
From simple to complex
Bruno Dupire 56
Stochastic Volatility Models
Heston 1993, semi-analytical formulae.
tttt
ttt
t
dZdtbd
dWdtrS
dS
)(
222
The End