Efficient Numerical PDE Methods to Solve Calibration andPricing Problems in Local Stochastic Volatility Models

Artur SeppBank of America Merrill Lynch, London

artur.sepp@baml.com

Global Derivatives Trading & Risk Management 2011Paris

April 12-15, 2011

1

Plan of the presentation

1) Volatility modelling

2) Local stochastic volatility models: stochastic volatility, jumps,local volatility

3) Calibration of parametric local volatility models using partial dif-ferential equation (PDE) methods

4) Calibration of non-parametric local volatility volatility models withjumps and stochastic volatility using PDE methods

5) Numerical methods for PDEs

6) Illustrations using SPX and VIX data

2

References

Some theoretical and practical details for my presentation can befound in:

1) Sepp, A. (2012) An Approximate Distribution of Delta-HedgingErrors in a Jump-Diffusion Model with Discrete Trading and Trans-action Costs, Quantitative Finance, 12(7), 1119-1141http://ssrn.com/abstract=1360472

2) Sepp, A. (2008) VIX Option Pricing in a Jump-Diffusion Model,Risk Magazine April, 84-89http://ssrn.com/abstract=1412339

3) Sepp, A. (2008) Pricing Options on Realized Variance in the He-ston Model with Jumps in Returns and Volatility, Journal of Compu-tational Finance, Vol. 11, No. 4, pp. 33-70http://ssrn.com/abstract=1408005

3

Literature

Local volatility: Dupire (1994)

Local volatility with jumps: Andersen-Andreasen (2000)

Local stochastic volatility with jumps: Lipton (2002)

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Volatility modelling I

What is important for a competitive pricing and hedging volatilitymodel?

1) Consistency with the observed market dynamics implies stablemodel parameters and hedges

2) Consistency with vanilla option prices ensures that the modelfits the risk-neutral distribution implied by these prices

3) Consistency with market prices of liquid exotic options

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Volatility modelling II. Volatility models1) Non-parametric local volatility models (Dupire (1994), Derman-Kani (1994), Rubinstein (1994))”+” LV models are consistent with vanilla prices by construction”-” but they tend to be poor in replicating the market dynamics ofspot and volatility (implied volatility tends to move too much givena change in the spot, no mean-reversion effect)”-” especially, it is impossible to tune-up the volatility of the impliedvolatility as there is simply no parameter for that!

2) Parametric stochastic volatility models (Heston (1993))”+” SV models tend to be more aligned with the market dynamics”+” introduce the term-structure (by mean-reversion parameters)and the volatility of the variance (by vol-of-vol parameters)”-” need a least-squares calibration to today’s option prices”-” unfortunately, any change in any of the parameters of the mean-reversion or the vol-of-vol requires re-calibration of other parameters

3) Stochastic local volatility models (JP Morgan (1999), Blacher(2001), Lipton (2002), Ren-Madan-Qian (2007))LSV models aim to include ”+”’s and cross ”-”’s of first two models

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Volatility modelling III. LSV model specification

1) Global factors - specify the dynamics for the instantaneous volatil-ity, include jumps or default risk if this is relevant for product risk

(Guess) Estimate or calibrate model parameters for the dynamics ofthe instantaneous volatility and jumps using either historical or marketdata

Parameters are updated infrequently

2) Local factors - specify local factors for either parametric or non-parametric local volatility

Parameters of local factors are updated frequently (on the run) tofit the risk-neutral distributions implied by market prices of vanillaoptions

3) Mixing weight - specify mixing weight between stochastic andlocal volatilities, adjust vol-of-vol and correlation accordingly

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LSV model dynamics. We consider a specific version of the LSVdynamics with jumps under pricing measure:

dS(t) = µ(t)S(t−)dt+ σ(loc,svj)(t−, S(t−))ϑ(t−, Y (t−))S(t−)dW (1)(t)

+((e−ν − 1)dN(t) + λνdt

)S(t−), S(0) = S

dY (t) = −κY (t)dt+ εdW (2)(t) + ηdN(t), Y (0) = 0(1)

S(t) is the underlying and Y (t) is the factor for instantaneous stochas-tic volatility with dW (1)(t)dW (2)(t) = ρdt

σ(loc,svj)(t, S(t)) is the local volatility (to be considered later)

ϑ(t, Y (t)) is volatility mapping with V[Y (t)] being variance of Y (t):

ϑ(t, Y (t)) = eY (t)−V[Y (t)]

For ρ = 0, E[ϑ2(t, Y (t)) |Y (0) = 0

]= 1, so that ϑ(t, Y (t)) introduces

”volatility-of-volatility” effect without affecting the local volatilityclose to the spot

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Stochastic volatility factor IQ: Can we observe the dynamics of stochastic volatility factor Y (t)?A: Consider the model implied ATM volatility squared:

V (t) = σ20e

2Y (t)−2V[Y (t)] ≈ σ20(1 + 2Y (t))

Thus, given daily time series of implied ATM volatility σATM(tn):

Y (tn) =1

2

σ2ATM(tn)

σ2ATM

− 1

where σ2

ATM is the average ATM volatility during the period

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Left: One-month ATM implied volatility for SPX; Right: Y(t)9

Stochastic volatility factor II. Estimation

To obtain historical estimates for the mean-reversion and vol-of-vol,apply regression model:

Y (tn) = βY (tn−1) + θςn

where ςn is iid normals

Thus:κ = −252 ln(β)/2

ε =√

252θ(2κ)/(1− e−2κ/252)

For SPX (using data for last 4 years):R2 = 74%κ = 4.49ε = 2.30

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Stochastic volatility factor III. Fit to implied SPX volatilities (23-March-1011) using flat local volatility (log-normal SV model) andhistorical parameters with ρ = −0.8Conclusion: pure stochastic volatility model is adequate to describelonger terms skew but need local factors to ”correct” mis-pricings

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Jumps I

We assume that jumps in S(t) and Y (t) are simultaneous and dis-crete with magnitudes −ν < 0 and η > 0

Q: Why do we need jumps in addition to local stochastic volatility?A: Market prices of exotics (eg cliques) imply very steep forward skewsso the value of vol-of-vol parameter must be very large (I will showan example in a bit)

However, local stochastic volatility with a large vol-of-vol parametercannot be calibrated consistently to given local volatility

Calibration to both exotics and vanilla can be achieved by addinginfrequent but large negative jumps

Q: Why simultaneous jumps?A: Realistic, do not decrease the implied spot-volatility correlation

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Jumps IIQ: Why discrete jumps?A: Check instantaneous correlation between X(t) = lnS(t) and Y (t):

ς ≡< dX(t), dY (t) >√

< dX(t), dX(t) >√< dY (t), dY (t) >

=ρεσ(loc,svj)(t, S(t))ϑ(t, Y (t)) + νηλ√

(σ(loc,svj)(t, S(t))ϑ(t, Y (t)))2 + ν2λ√ε2 + η2λ

In equity markets, ς∗ ∈ [−0.5,−0.9]If Y (t)→∞ or λ→ 0 then ς → ρIf Y (t)→ 0 or λ→∞ then ς → sign(ν) = −1 if ν < 0

For discrete jumps, the correlation is the smallest possible: ς ≈ −1For exponential jumps: ς → 0.5sign(ν) = sign(ν) = −0.5 if ν < 0

Jump variance leads to de-correlation (not good for stability of ρ)

Experience: even for 1-d jump-diffusion Merton model, adding jumpvariance to model calibration leads to less stable calibration

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Jumps III. Q: How to estimate jump parameters?Consider a jump-diffusion model under the historic measure:

dS(t)/S(t) = µdt+ σdW (t) +(eJ − 1

)dN(t), S(t0) = S,

N(t) is Poisson process with intensity λ

Jumps PDF is normal mixture: w(J) =∑Ll=1 pln(J; νl, υl),

∑Ll=1 pl =

1L, L = 1,2, .., is the number of mixturespl, 0 ≤ pl ≤ 1, is the probability of the l-th mixtureνl and υl are the mean and variance of the l-th mixture, l = 1, .., L

Cumulative probability function can be represented as weighted sumover normal probabilities (see Sepp (2011))

Can be applied for estimation of jumps

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Jumps IV. Empirical estimation for the S&P500 using last 10 yearsFit empirical quantiles of sampled daily log-returns to model quantilesby minimizing the sum of absolute differences

The best fit, keeping L as small as possible, is obtained with L = 4Model estimates: σ = 0.1348, λ = 46.4444, υl is uniforml pl νl

√υl plλ expected frequency

1 0.0208 -0.0733 0.0127 0.9641 every year2 0.5800 -0.0122 0.0127 26.9399 every two weeks3 0.3954 0.0203 0.0127 18.3661 every six weeks4 0.0038 0.1001 0.0127 0.1743 every five years

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Jumps V. Implied Calibration of Jumps

Remove intermediate jumps (l = 2,3) and large positive jumps (l = 4)- the stochastic local volatility takes care of ”moderate” skew

Jump magnitude from historical estimation ν = −0.0733 is too smallto fit one-month SPX skew

An estimate implied from one-month SPX options is (approximately)ν = −0.25

Assume only large negative jumps with λ = 0.2

The magnitude of jump in volatility η is calibrated to match VIXskews

Typically η = 1.0 implying that ATM volatility will double following ajump

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Local VolatilityUsing local volatility we fit vanilla options while having flexibility tospecify the dynamics for stochastic volatility and jumps

Local volatility parametrization:

1) Parametric local volatility

Specify a functional form for local volatility (CEV, shifted-lognormal,quadratic, etc)

Calibrate by boot-strap and least-squares

Good for single stocks where implication of Dupire non-parametriclocal volatility is problematic and unstable due to lack of quotes

2) Non-parametric local volatility

Fit the model local volatility to Dupire local volatility

Good for indices with liquid option market

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Parametric local volatility IConsider the following parametric form:

σloc(t, S) = σatm(t)σskew(t, S)σsmile(t, S) (2)

where σatm(t) is at-the-money forward volatility

Skew function is specified as ratio of two CEVs:

σskew(t, S) =(S/S0)β(t)−1 + a

(S/S0)β(t)−1 + b

=1

2

((1 + q) + (1− q) tanh

(1 + q

1− q(β(t)− 1) ln(S/S0)

))where β(t) is the skew parameter, q is weight parameter, 0 < q < 1

Smile function is quadratic:

σsmile(t, S) =√

1 + (α(t) ln(S/S0))2

where α(t) is smile parameter

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Parametric local volatility IIQ: What is intuition about the model parameters and initial guessesfor model calibration?A: From the given market implied volatilities, compute ATM volatility,skew and smile as follows:ϑatm(T ) = σ(T,K = S(T ))ϑskew(T ) ∝ σ(T,K = S(T )+)− σ(T,K = S(T )−)ϑsmile(T ) ∝ σ(T,K = S(T )+)− 2σ(T,K = S(T )) + σ(T,K = S(T )−)

Consider local volatility at S = S0 to obtain:

σatm(t) =2ϑatm(t)

1 + q

β(t) =2ϑskew(t)

ϑatm(t)+ 1

α2(t) =2ϑsmile(t)

ϑatm(t)

(3)

σatm(t) is proportional to the ATM volatilityβ(t) is the skew relative to ATM volatilityα(t) is the smile relative to ATM volatility

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Calibration of parametric models I. Backward equationConsider general pricing equation for options on S(t) under stochasticlocal volatility modelPV function U(t, S, Y ;T,K), 0 ≤ t ≤ T , solves the backward equationas function of (t, S, Y ):

Ut + LU(t, S, Y ) = 0

U(T, S, Y ) = u(S)(4)

where u(S) is pay-off functionL is infinitesimal backward operator:

LU(t, S, Y ) =1

2(σ(loc,svj)(t, S)ϑ(t, Y )S)2USS + µ(t)SUS

+1

2ε2(Y )UY Y + θ(Y )UY

+ ρσ(loc,svj)(t, S)ϑ(t, Y )ε(Y )SUSY + λI(S)

I(S) =∫ ∞−∞

∫ ∞−∞

U(SeJ , Y + Υ)$(J)ς(Υ)dJdΥ− νSUS − U

For generality, jump sizes in S and Y have PDFs $(J) and ς(Υ),respectively; ν is jump compensator

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Calibration of parametric models II. Forward equationTransition probability density function, G(0, S0, Y0; t, S′, Y ′) solves theforward equation as function of (t, S′, Y ′):

Gt(t, S′, Y ′)− L†G(t, S′, Y ′) = 0,

G(0, S′, Y ′) = δ(S′ − S0)δ(Y ′ − Y0)(5)

where δ() is Dirac delta function

L† is forward operator adjoint to L:

L†G(t, S′, Y ′) = −1

2

((σ(loc,svj)(T, S′)ϑ(T, Y ′)S′)2G

)S′S′

+(µ(T )S′G

)S′

−1

2

(ε2(Y ′)G

)Y ′Y ′

+(θ(Y ′)G

)Y ′

−(ρσ(loc,svj)(T, S′)ϑ(T, Y ′)ε(Y ′)S′G

)S′Y ′− λI(S′)

I(S′) =∫ ∞−∞

∫ ∞−∞

G(S′e−J , Y ′ −Υ)e−J$(J)ς(Υ)dJdΥ + (νS′G)S′ −G

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Calibration of parametric models III Option PV can be computedby convolution (Duhamel’s or Feynman-Kac formula):

U(0, S0, Y0;T,K) =∫ ∞−∞

∫ ∞−∞

G(0, S0, Y0;T, S′, Y ′)u(S′)dS′dY ′ (6)

Calibration by bootstrap: For parametric local volatility, model pa-rameters are assumed to be piece-wise constant in time with jumpsat times {Tm}, {Tm} is set of maturity times of listed options

1) Given calibrated set of piece-wise constant model parameters attime Tm−1 and known values of G(Tm−1, S, Y ), make a guess for pa-rameters at time Tm and compute G(Tm, S, Y )

2) Compute PV-s of vanilla options using discrete version of (6)

3) By changing parameters for time Tm, minimize the sum of squareddifferences between model and market prices

4)After convergence, store G(Tm, S, Y ) and go to the next time slice

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Calibration of parametric models IV. Numerical schemesConsistency condition for adjoint operators L and L† (can be checkedby integration by parts) is based on theoretical arguments:∫ T

0

∫ ∞−∞

∫ ∞−∞

[{L†G(0, S0, Y0; t′, S′, Y ′)

}U(t′, S′, Y ′)

−G(0, S0, Y0; t′, S′, Y ′){LU(t′, S′, Y ′)

}]dS′dY ′dt′ = 0

(7)

It is not necessarily true for discrete numerical schemes (Lipton (2007))

Implications for numerical schemes:1) For adjoint operators, if M is spacial discretisation of L then MT

should be spacial discretisation of L†2) If MT is diagonally dominant with positive diagonal and non-positive off-diagonal elements (for 1-d problems with zero correla-tion), then (MT )−1 > 0 and computed probabilities are non-negativeand sum up to one (Andreasen (2010), Nabben (1999))3) Both forward and backward equations should be solved using thesame schemes4) L and L† have the same values of operator norm, so the conver-gence of the forward scheme implies convergence of backward scheme

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Calibration of parametric models V. Non-linear least square fitModel parameters: Θ = (θ1, ..., θq), Model prices: U(Θ) = (u1, ..., un)

Jacobian: Jn×q = (Jn′q′): Jn′q′ =∂un′∂θq′

(computed numerically)

Market prices: M = (m1, ...,mn), Weight (diagonal) matrix: Wn×n

Gauss-Newton normalized equations:

(JTWJ)(∆Θ) = JTW(∆U), ∆U = M−U(Θ(k)), Θ(k+1) = Θ(k)+∆Θ

Use SVD for stability, dimensionality is small so calibration is fast

Levenberg-Marquardt method:

(JTWJ + λI)(∆Θ) = JTW(∆U)

λ is Marquardt parameter to make the matrix diagonally-dominantLarge value of λ - the steepest descent (tendency to be slow)How to set λ? (NR in C (Press (1988)) use deterministic value)One possibility: λ = 1/trace(JTWJ)

Experience: for carefully chosen functional form of the local volatilitywith robust initial estimates, Gauss-Newton works fast

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Calibration of parametric models VI. IllustrationImplied volatilities for parametric volatility with vol-of-vol reduced by50%: the fit is acceptable

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Local volatility calibration I. Diffusion model

Consider a one-dimensional diffusion:

dS(t) = µ(t)S(t)dt+ σ(loc,dif)(t, S(t))S(t)dW (t), S(0) = S (8)

where σ(loc,dif) is the local volatility of the diffusion

Calibration objective: how we should specify σ(loc,dif)(T, S′)?

In terms of call option prices, we have Dupire equation (1994):

σ2(loc,dif)(T,K) =

CT (T,K) + µ(T )KCK(T,K)− µ(T )C(T,K)12K

2CKK(T,K)(9)

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Local volatility calibration II. Jump-diffusion model. Andersen-Andreasen (2000) consider Merton jump-diffusion with local volatility:

dS(t) =µ(t)S(t−)dt+ σ(loc,jd)(t, S(t−))S(t−)dW (t)

+((eJ − 1)dN(t)− λνdt

)S(t−), S(0) = S

(10)

Here σ2(loc,jd) is specified by Andersen-Andreasen equation (2000):

σ2(loc,jd)(T,K) =

CT (T,K) + µ(T )KCK(T,K)− µ(T )C(T,K)− λI†(K)12K

2CKK(T,K)

= σ2(loc)(T,K)−

λI†(K)12K

2CKK(T,K)(11)

I†(K) =∫ ∞−∞

C(Ke−J′)eJ

′$(J ′)dJ ′+ νKCK − (ν + 1)C (12)

Calibration: i) Given Dupire local volatility σ(loc,dif)(T,K) solve for-ward equation for call prices C(T,K) using FD for 1-d problemii) Compute I†(K) and imply σ(loc,jd)(T,K)

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Local volatility calibration III. Stochastic local volatility:

dS(t) = µ(t)S(t)dt+ σ(loc,sv)(t, S(t))ϑ(t, Y (t))S(t)dW (1)(t), S(0) = S,

dY (t) = θ(Y (t))dt+ ε(Y (t))dW (2)(t), Y (0) = Y

The model is consistent with the Dupire local volatility if

σ2(loc,sv)(T,K) =

σ2(loc,dif)(T,K)

V (T,K)(13)

where V (T, S′) is the conditional variance:

V (T, S′) =

∫∞−∞ ϑ

2(T, Y ′)G(t, S, Y ;T, S′, Y ′)dY ′∫∞−∞G(t, S, Y ;T, S′, Y ′)dY ′

(14)

Calibration:(More details to follow)i) Stepping forward in time, solve for density G(t, S, Y ;T, S′, Y ′) usingFD for 2-d problemii) Compute V (T,K) and imply σ2

(loc,sv)(T,K) using (13)

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Local volatility calibration IV. Stochastic local volatility withjumps specified by dynamics (1)Local stochastic volatility for jump-diffusion, σ2

(loc,svj)(T,K), is spec-

ified by Lipton equation (2002):

σ2(loc,svj)(T,K) =

CT (T,K) + µ(T )KCK(T,K)− µ(T )C(T,K)− λI†(K)12V (T,K)K2CKK(T,K)

=1

V (T,K)

σ2(loc,dif)(T,K)−

λI†(K)12K

2CKK(T,K)

=σ2

(loc,jd)(T,K)

V (T,K)(15)

Calibration: i) Given Dupire local volatility σ(loc,dif)(T,K) solve for-ward equation for call prices C(T,K) using FD for 1-d problem andimply σ(loc,jd)(T,K) using (11)

ii) Stepping forward in time, solve for density G(t, S, Y ;T, S′, Y ′) usingFD for 2-d problem with jumpsiii) compute V (T,K) using (14) and imply σ2

(loc,sv)(T,K) using (15)

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PDE based methods

Non-parametric local stochastic volatility can only be implementedusing numerical methods for partial integro-differential equations

These methods should be flexible to handle jumps in one and twodimensions and the correlation term for two dimensions

I will review some of the available methods

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1-d Problem. For the backward problem:

L = D+ λJHere D is the diffusion-convection operator:

DU(t, S) ≡1

2σ2(t, S)S2USS + µ(t)SUS + λU

J is the integral operator:

JU(t, S) ≡∫ ∞−∞

U(t, SeJ′)$(J ′)dJ ′

For the forward problem, we consider the operator L† adjoint to L:

L† = D†+ λJ †

For both problems, denote the discretized diffusion and integral op-erators by Dn and J, respectively

i) The diffusion operator is space and time dependentii) Care must be taken for discretization of the operator for the mean-reverting process

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1-d Problem for diffusion problem. N - number of spacial stepsDn - discrete diffusion matrix at time tn with dimension N ×NUn - the solution vector at time tn with dimension N × 1Time-stepping with θ-method:(

I− θDn+1)Un+1 =

(I + θDn+1

)Un

i) θ = 0 - explicit method requires very fined grids and is highlyunstable - it should be avoidedii) θ = 1/2 - Crank-Nicolson scheme is unconditionally stable and isof order (δS)2 and (δt)2, but might lead to negative densitiesiii) θ = 1 - implicit method is unconditionally stable and does not leadto negative densities, but it is of order (δt) and becomes less accuratefor longer maturities

My favourite is the predictor-corrector scheme:(I−Dn+1

)U = Un(

I−Dn+1)Un+1 = Un +

1

2Dn+1(Un − U)

with order of (δS)2 and (δt)2; does not lead to negative densities

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Numerics for jump-diffusions IConsider the forward equation with additive jumps (multiplicativejumps can be handled in the log-space):

JG(x) =∫ ∞−∞

G(x− j)$(J ′)dJ ′

Consider discrete negative jumps with size −η, η > 0:

JG(x) = G(x+ η)

Discretization is a simple linear interpolation:

JGi = wGk + (1− w)Gk+1

where k = min{k : xk+1 ≥ xi + ν} and w =xk+1−(xi+ν)xk+1−xk

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Numerics for jump-diffusions IIIn the matrix form, for uniform spacial grid:

J =

0 0 ... w 1− w 0 ... 0 00 0 ... 0 w 1− w ... 0 0...0 0 ... 0 0 0 ... w 1− w0 0 ... 0 0 0 ... 0 1...0 0 ... 0 0 0 ... 0 0

J(p,N) = 1, where p = min{p : xp + ν ≥ xN}

In general, for two-sided jumps, J is a full matrix:The implicit method for the integral term leads to O(N3) complexityand is not practicalThe explicit method leads to O(N2) complexity but needs extracare for stability

For exponential jumps, Lipton (2003) develops recursive scheme withO(N) complexity

34

Numerics for jump-diffusions III

In case of discrete jumps, we have two-banded matrix with p rowsthat have non-zero elements

Consider solving a linear system:

(I− βJ)X = R

where β = λδt, 0 < β < 1

We can solve this system by back-substitution with cost of O(N)operations

35

Numerics for jump-diffusions IVConsider θ and θJ schemes for the diffusion and the jump parts:(

I− θDn+1 − θJλn+1J)Un+1 =

(I + θDn+1 + θJλn+1J

)Un

where λn+1 = (tn+1 − tn)λ(tn+1)

The accuracy with θ = θJ = 12 is supposed to be O(dx2) +O(dt2)

However, the matrix in the lhs will be full: the cost to invert is O(N3)

Implicit-explicit method:(I−Dn+1

)Un+1 =

(I + λn+1J

)Un

with accuracy of O(dx2) +O(dt)

θ-explicit method:(I− θDn+1

)Un+1 =

(I + θDn+1 + λn+1J

)Un

with accuracy of O(dx2) +O(dt)

36

Numerics for jump-diffusions V. Andersen-Andreasen (2000)scheme

Make the first half step with θ = 1 and θJ = 0 and the second halfstep with θ = 1 and θJ = 1:(

I−1

2Dn+1

)U =

(I +

1

2λn+1J

)Un(

I−1

2λn+1J

)Un+1 =

(I +

1

2Dn+1

)U

with accuracy of O(dx2) +O(dt2)

When J is full, the second equation can be solved by the applicationof the discrete Fourier transform (DFT) with the cost of O(N logN)

Beware of deficiencies of the DFT

For discrete jumps, the second equation solved with cost of O(N)operations

37

Numerics for jump-diffusions VI. d’Halluin et al (2005) scheme

Consider fixed point iterations:

Set V 1 = Un

Iterate for p = 1, .., p (p = 2 is good enough):(I− θDn+1

)V p+1 =

(I + θDn+1

)Un + λn+1JV

p,

Set Un+1 = V p

The accuracy is O(dx2) +O(dt)

38

Numerics for jump-diffusions VII

My favourite is the implicit scheme with predictor-corrector appliedtwice:

U(0) =(I + Dn+1 + λn+1J

)Un(

I−Dn+1)U = U(0)(

I−Dn+1)Un+1 = U(0) +

1

2(Dn+1 + λn+1J)(U − Un)

The accuracy is O(dx2) +O(dt2)

39

Numerical Methods for Two Dimensional Problem I

We consider the forward problem for U(t, x1, x2):

UT −MU = 0

U(0, x1, x2) = δ(x1 − x1(0))δ(x2 − x2(0))(16)

where

M = D1 +D2 + C+ λJD1 and D2 are 1-d diffusion-convection operators in x1 and x2 direc-tions, respectivelyC is the correlation operatorJ is the integral operator for joint jumps in x1 and x2

Let D1 and D2 denote the discretized 1-d diffusion-convection oper-ators in x1 and x2 directions, respectively

C and J are the discretized correlation and integral operator, respec-tively

40

Douglas-Rachford (1956) scheme

Make a predictor and apply two orthogonal corrector steps:

U(0) = (I + C + λJ + D1 + D2)Un

(I− θD1)U = U(0) − θD1Un

(I− θD2)Un+1 = U − θD2Un

In the second line, for each fixed index j we apply the diffusion oper-ator in x1 direction; and solve the tridiagonal system of equations toget the auxiliary solution U(·, x2(j))

In the third line, keeping i fixed, we apply the diffusion step in x2direction and solve the system of tridiagonal equations to get thesolution Un+1(x1(i), ·) at time tn+1

Complexity is O(N1N2) per time step

41

Craig-Sneyd (1988) scheme

Start as with Douglas-Rachford scheme make a second predictor andagain apply two orthogonal corrector steps:

U(0) = (I + C + λJ + D1 + D2)Un

(I− θD1)U(1) = U(0) − θD1Un

(I− θD2)U(2) = U(1) − θD2Un

U(3) = U(0) +1

2(C + λJ)(U(2) − Un)

(I− θD1)U(4) = U(3) − θD1Un

(I− θD2)Un+1 = U(4) − θD2Un

42

Hundsdorfer-Verwer (2003) scheme (in’t Hout-Foulon (2008))

Similar to Craig-Sneyd scheme with predictor including D1 and D2and the second corrector applied on U(2):

U(0) = (I + C + λJ + D1 + D2)Un

(I− θD1)U(1) = U(0) − θD1Un

(I− θD2)U(2) = U(1) − θD2Un

U(3) = U(0) +1

2(C + λJ + D1 + D2)

(U(2) − Un

)(I− θD1)U(4) = U(3) − θD1U

(2)

(I− θD2)Un+1 = U(4) − θD2U(2)

43

Discretisation of integral term

Direct methods are infeasible because of O(N21N

22 ) complexity

DFT method (Clift-Forsyth (2008)) has O(N1N2 logN1N2) complex-ity but suffers from problems associated with the DFT

Explicit methods with O(N1N2) complexity are available for discreteand exponential jumps (Lipton-Sepp (2011))

The simplest case is if jumps are discrete with sizes η1 and η2:

JG = G(x1 − η1, x2 − η2)

This term is approximated by bi-linear interpolation with the secondorder accuracy leading to the O(N1N2) complexity

44

Summary I. LSV model calibration

1) Specify time and space grids

2) Compute the local volatility using Dupire equation (9) on the spec-ified grid (interpolating implied volatilities in strikes and maturities)

3) Calibrate local stochastic volatility on the specified grid

4) Use backward PDE methods or Monte-Carlo simulations of thelocal stochastic volatility model to compute values of exotic options(see Andreasen-Huge (2010) for consistent schemes)

45

Summary II. Local volatility calibration at time tngiven Gn−1(x1(i), x2(j)) and V (tn−1, x1(i))i) Compute the local variance by:

σ2(loc,sv)(tn, x1(i)) =

σ2(loc,dif)(tn, x1(i))

V (tn−1, x1(i))

ii) Compute Gn(x1(i), x2(j)) by solving the forward PDE

iii) Compute V (tn, x1(i)) by

V (tn, x1(i)) =

∑i∑j ϑ

2(tn, x2(j))Gn(x1(i), x2(j))∑j G

n(x1(i), x2(j))

and set

σ2(loc,sv)(tn, x1(i)) =

σ2(loc,dif)(tn, x1(i))

V (tn, x1(i))

D) Repeat B) and C) and go to the next time step

For stochastic volatility with jumps we use σ2(loc,jd)

46

Summary III. Predictor-corrector schemes

When we use predictor-corrector schemes:

i) Compute the predictor step using local stochastic volatility fromprevious time step

ii) Update local stochastic volatility and compute the corrector stepwith new volatility

iii) After the corrector step, compute new local stochastic volatilityand go to the next step

47

Summary IV. Discrete dividends

At ex-dividend time tn:

S(tn+) = S(tn−)−Dnwhere Dn is the cash dividend at time tn

Note that this corresponds to the negative jump in S(t) with a con-stant magnitude Dn at deterministic time tn

Therefore the developed method for discrete negative jumps are read-ily applied for discrete dividends

It is important that σ2(loc,dif) is consistent with the discrete dividends

48

Illustration I. Local volatilities for options on the S&P 500

0%

10%

20%

30%

40%

50%

60%

70%

0.2 0.2 0.3 0.3 0.4 0.5 0.6 0.8 1.0 1.2S', spot = S'*forward(T)

Loca

l Vol

atili

ty

Dupire Local Volatility, T=1month

SV Local Volatility, T=1month

0%

10%

20%

30%

40%

50%

60%

70%

80%

0.2 0.2 0.3 0.3 0.4 0.5 0.6 0.8 1.0 1.2S', spot = S'*forward(T)

Loca

l Vol

atili

ty

Dupire Local Volatility, T=1year

SV Local Volatility, T=1year

Left: Dupire local volatility and LSV local volatilities at T=1 monthRight: ... at T=1 year

LSV local volatility is an adjustment to the skew implied by thestochastic volatility part and is mostly flat

49

Illustration II. Forward volatilities for options on the S&P 500

10%

15%

20%

25%

30%

35%

40%

45%

50%

55%

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5K, Strike=K*Forward(T)

Imp

lied

BS

M v

ola

tilit

y

6m Spot skew, Local vol

6m Spot skew, LSV

6m-6m Skew if S(6m)>1.1S(0), Local vol

6m-6m Skew if S(6m)>1.1S(0), LSV

10%

15%

20%

25%

30%

35%

40%

45%

50%

55%

60%

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5K, Strike=K*Forward(T)

Imp

lied

BS

M v

ola

tilit

y

6m Spot skew, Local vol

6m Spot skew, LSV

6m-6m Skew if S(6m)<0.9S(0), Local vol

6m-6m Skew if S(6m)<0.9S(0), LSV

6m-6m skew: the implied volatility for a vanilla call option startingin 6 month and maturing in one year from now with strike fixed in 6months: K = S(T = 6m)Left: 6m-6m skew if S(T = 6m) > 1.1S(0): the implied volatility forthe forward-start option conditional that S(T = 6m) > 1.1S(0)Right: 6m-6m skew if S(T = 6m) < 0.9S(0): the implied volatilityfor the forward-start option conditional that S(T = 6m) < 0.9S(0)

50

Illustration III. Implications

Conditional that spot moves up, the LSV model preserves the shapeof the volatility skew while the pure local volatility does not

Conditional that spot moves down, the pure local volatility impliesthat the ATM volatility goes too high and the skew flattens unlikethe LSV model

The LSV is more closer to the actual dynamics!

51

Illustration IV. Quarterly 95−105% spread clique on SPX, T = 2y:

u =∑n

max

{min

{S(tn)

S(tn−1)− 1, c

}, −f

}c = f = 5%

Marking model1 5.42% considered as ”market price”1-d LocalVol 0.90% tiny, no fit to steep forward skew!

1-d LocalVol+jump3 2.40% still smallHeston 4.33% under-price

Heston+jump 2 6.08% over-priceLSV, 70% 3 3.78% too small

LSV, 100% 3 4.48% under-price, in line with HestonLSV, 130% 3 4.77% still under-price !

LSV+jump2 70% 3 5.37% closest to the marketLSV+jump2 100% 3 6.37% over-price

1internal ”local beta” model for cliques developed by Alex Langnau21 jump per 5 years: λ = 0.2, ν = −25%3vol-of-vol is set by x% ∗ ε, ε = 2.3

52

Illustration V. Implied volatilities of options on the daily realizedvariance of the S&P 500

0%

40%

80%

120%

160%

200%

23% 58% 92% 127% 162% 196% 231%

K, Strike=K*ExpectedVariance(T)

Imp

lied

log

-no

rmal

vo

ltili

ty

LSV, T=3m

Local Vol, T=3m

0%

40%

80%

120%

15% 37% 59% 81% 103% 125% 147%

K, Strike=K*ExpectedVariance(T)

Imp

lied

log

-no

rmal

vo

ltili

ty

LSV, T=1year

Local Vol, T=1year

Left: Implied log-normal volatilities on options on the daily realizedvariance of the S&P 500 with maturity T=3 monthRight: ... at T=1 year

LSV local volatility implies higher volatility of the realized variance adthe vol-of-vol parameter can be ”tuned-up” to match the market

53

Options on the VIX IAugment the model dynamics (1) with the third variable for the re-alized quadratic variance of S(t):

dI(t) =(σ(loc,svj)(t−, S(t−))ϑ(t−, Y (t−))

)2dt+ ν2dN(t), I(0) = 0

Consider the expected variance realized over period [T, T + Tvix]:

I(T, T + Tvix) =1

TvixE[∫ T+Tvix

TdI(t′)

]where Tvix = 30/365.25 is the annualized tenor of the VIX

A call option on the VIX maturing at T is computed by:

C(T,K) = E[(I(T, T + Tvix)−K

)+| I(T, T ) = 0

]

54

Options on the VIX II. ValuationConsider

I(T, T+Tvix) =1

TvixE[∫ T+Tvix

TdI(t′)

]≈

1

Tvix

∑tn∈[T,T+Tvix]

(E [V (tn)] + ν2λ

)dtn

where V (t) is instantaneous variance:

V (t) =(σ(loc,svj)(t, S(t))ϑ(t, Y (t))

)2

U(t, S, Y ) ≡ I(T, T+Tvix) solves backward problem (4), t ∈ [T, T+Tvix]:

Ut + LU(t, S, Y ) = −1

Tvix

(V (t) + ν2λ

)U(T + Tvix, S, Y ) = 0

Valuation: i) Solve 2-d backward problem for I(T, T + Tvix) as func-tion of (S, Y )ii) Compute G(t, S, Y ;T, S′, Y ′) and evaluate C(T,K) by convolutioniii) Can price call with different strikes at a time

55

Options on the VIX III. VIX Implied volatilities

20%

21%

22%

23%

24%

1m 2m 3m 4m

VIX

Fu

ture

s

LSVLSV-JumpMarket

1M

80%

100%

120%

140%

160%

90% 100% 110% 120% 130% 140% 150%

LSVLSV-JumpMarket

2M

80%

100%

120%

140%

160%

90% 100% 110% 120% 130% 140% 150%

LSVLSV-JumpMarket

3M

60%

80%

100%

120%

90% 100% 110% 120% 130% 140% 150%

LSVLSV-JumpMarket

4M

60%

80%

100%

120%

90% 100% 110% 120% 130% 140% 150%

LSVLSV-JumpMarket

Conclusion: consistency with options on the SPX does not lead toconsistency with options on the VIX; Need extra flexibility for jumpsin volatility (time and/or space dependency)

56

Conclusions

I have presented theoretical and practical grounds for stochastic localvolatility models and highlighted details of model implementation

The presentation is based on my experience in implementation of theLSV model with jumps for BAML equity derivatives library

During this project I benefited from interactions with Alex Lipton,Hassan El Hady and other members of BAML quantitative analytics

Thank you for your attention!

57

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