Stochastic Local Volatility Models:Theory and Implementation

Artur Sepp

Bank of America Merrill Lynch

University of Leicester, UKDecember 9, 2010

1

Plan of the presentation

1) Hedging and volatility

2) Review of volatility models

3) Local volatility models with jumps and stochastic volatility

4) Calibration using Kolmogorov equations

5) PDE based methods in one dimension

6) PDE based methods in two dimensions

7) Illustrations

2

References

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

1) Sepp, A. (2007) Affine Models in Mathematical Finance: an An-alytical Approach, PhD thesis, University of Tartuhttp://math.ut.ee/~spartak/papers/seppthesis.pdf

2) 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

3

Volatility modelling

What is important for a competitive pricing and hedging model?

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 and that themodel gamma is not much different from the market implied gamma

4

Hedging and volatility I

Let us consider a vanilla option and for simplicity assume zero interestrates and dividends

We can find the option value U(σ)(t, S) by solving the celebratedBlack-Scholes-Merton (1973) partial differential equation (PDE) withlog-normal volatility parameter σ:

U(σ)t +

1

2σ2S2U

(σ)SS = 0, (1)

where t is current time and S is the spot price

To delta-hedge a short-position in this option we consider the follow-ing volatilities:σi - implied volatility for computing option value U(σi)(t, S)

σh - hedging volatility for computing option delta U(σh)S (t, S)

Note that σh might also include the vega adjustment, which is changein σi given change in the spot, so in general σi 6= σh

5

Hedging and volatility II

The delta-hedged position at time tn:

Π(tn) = S(tn)∆(σh)(tn, S(tn))− U(σi)(tn, S(tn))

One period profit-and-loss realized over time period δtn is:

δΠ(tn) =(S(tn−1) + δS(tn)

)∆(σh)(tn−1, S(tn−1))

− U(σi)(tn−1 + δtn, S(tn−1) + δS(tn))−Π(tn−1)

where δS(tn) = S(tn)− S(tn−1) and δtn = tn − tn−1

6

Hedging and volatility III Consider Taylor series of U(σi)(tn−1 +δtn, S(tn−1) + δS(tn)):

U(σi)(tn−1 + δtn, S(tn−1) + δS(tn)) ≈ U(σi)(tn−1, S(tn−1))

+ δtnΘ(σi)(tn−1, S(tn−1)) + δS(tn)∆(r)(tn−1, S(tn−1)) +1

2(δS(tn))2 Γ(r)(tn−1, S(tn−1))

where Θ(σi)(tn−1, S(tn−1)) is the option theta, and ∆(r)(tn−1, S(tn−1))

and Γ(r)(tn−1, S(tn−1)) are realized delta and gamma:

Θ(σi)(tn−1, S(tn−1)) =∂

∂tU(σi)(t, S(tn−1)) |t=tn−1

∆(r)(tn−1, S(tn−1)) =∂

∂SU(σi)(tn−1, S) |S=S(tn)

≈U(σi)(tn−1, S(tn))− U(σi)(tn−1, S(tn−1))

S(tn)− S(tn−1)

Γ(r)(tn−1, S(tn−1)) =∂

∂S∆(r)(tn−1, S) |S=S(tn)

≈∆(r)(tn−1, S(tn))−∆(r)(tn−1, S(tn−1))

S(tn)− S(tn−1)

7

Hedging and volatility IVOne period realized profit-and-loss is approximated by:

δΠ(tn) =[∆(σh)(tn−1, S(tn−1))−∆(r)(tn−1, S(tn))

]δS(tn)

− δtnΘ(σi)(tn−1, S(tn−1))−1

2(δS(tn))2 Γ(r)(tn−1, S(tn−1))

Note that option theta satisfies the BSM PDE:

Θ(σi)(t, S) = −1

2σ2i S

2Γ(σi)(t, S)

If the delta-hedging is rebalanced at discrete times tn, the realizedprofit-and-loss, P&L, Π(T ) at the option maturity is

Π(T ) =∑n

[∆(σh)(tn−1, S(tn−1))−∆(r)(tn−1, S(tn−1))

]δS(tn)

+1

2

∑n

[σ2i Γ(σi)(tn−1, S(tn−1))− V (tn)Γ(r)(tn−1, S(tn−1))

]S2(tn−1)δtn

where V (tn) is the realized variance

V (tn) =1

δtn

(S(tn)− S(tn−1)

S(tn−1)

)2

8

Hedging and volatility V. Modelling objective A:Find a pricing and hedging model that1) predicts the correct delta: ∆(σh)(tn−1, S(tn−1)) ≈∆(r)(tn−1, S(tn−1)),then the P&L will be independent from the realized price path(note that ∆(σh)(tn−1, S(tn−1)) is implied by the hedging model while

∆(r)(tn−1, S(tn−1)) represents actual change in option price observedin the market given change in the spot and associated change inimplied volatility σi)

2) predicts the correct gamma: Γ(σi)(tn−1, S(tn−1)) ≈ Γ(r)(tn−1, S(tn−1))

(note that Γ(σi)(tn−1, S(tn−1)) is also implied by the hedging model

while Γ(r)(tn−1, S(tn−1)) is the realized change in delta given changein spot and associated change in implied volatility σi)

If the model satisfies 1) and 2), realized P&L greatly simplifies to

Π(T ) =1

2

∑n

(σ2i − V (tn)

)Γ(σi)(tn−1, S(tn−1))S2(tn−1)δtn

The ”edge” comes from the spread between the implied volatilitysquared and the realized variance: σ2

i − V (tn)

9

Volatility modelling VI. Modelling objective BIf the option can be sold at implied variance σ2

i that is (much) higher

than the realized variance 1T

∫ T0 V (t′)dt′ then Π(T ) is expected to be

positive because Γ(σi)(t, S(t)) is positive for vanilla options with con-vex pay-offs (see Sepp (2011) for approximate distribution of Π(T ))

To ”materialize” the ”edge” as the P&L, it is important to have amodel that is consistent with 1) and 2)

For exotic options, considerations might be more complicated asΓ(σi)(t, S(t)) and Γ(r)(t, S(t)) might change the sign and generallyhigher order risks need to be hedged

Still, implications 1) and 2) are important for a competitive pricingand hedging model. To conclude:1) Consistency with the observed market dynamics implies stablemodel parameters and hedge ratios2) Consistency with vanilla option prices ensures that the model fitsthe price distribution implied by these prices and that the modelgamma is not much different from the market implied gamma

10

Overview I. Black-Scholes-Merton

I will start with a brief review of volatility models

Black-Scholes-Merton (1973) considered log-normal model:

dS(t) = µ(t)S(t)dt+ σS(t)dW (t), S(0) = S, (2)

where W (t) is the Brownian motion and µ(t) is the risk-neutral drift(typically, µ(t) = r(t) − d(t), where r(t) and d(t) are deterministicinstantaneous interest and dividend rates)

The constant volatility σ is independent from both the spot price andany extra factors

The model is not realistic in the presence of the skew observed in theequity markets (out-of-the money puts are relatively expensive thanout-of-the money calls)

In practice, the BSM model is as a ”static” tool: price quotation,implied volatility parametrization, benchmarking

11

Illustration for options on the S&P 500

0%

10%

20%

30%

40%

50%

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 BSM volatility, T=1month

Market volatility, T=1month

BSM volatility, T=1year

Market volatility, T=1year

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.00 0.22 0.43 0.65 0.86 1.08 1.29 1.51 1.73S', spot=S'*Forward(T)

Pro

babi

lity

dens

ity

BSM density, T=1month

Market density, T=1month

BSM density, T=1year

Market density, T=1year

Left: Market implied and BSM volatility at ATM strike for T =1monthan T =1yearRight: Corresponding market and BSM implied probability density

Market implied distribution is more skewed to the left

12

Overview II. Bachelier model, A

Bachelier (1900) proposed a normal dynamics:

dS(t) = µ(t)S(0)dt+ σnormalS(0)dW (t), S(0) = S, (3)

The implied local log-normal volatility σloc(S(t)) (approximately):

σloc(S(t)) = σnormalS(0)

S(t)

The Bachelier model produces the leverage effect: for small price thevolatility increases and vice versa

The non-zero probability of negative prices: the default event in thefinancial terms (modelled as the first time S(t) hits zero)

13

Overview II. Bachelier model, B

From the modelling point of view, the Bachelier model is more realisticthan that of Black-Scholes-Merton

However, the model is little accepted in practice: people prefer tothink in relative terms rather than in absolute

In Bachelier model, we cannot compare risks of stock A with thevolatility of 100,000% and stock B with the volatility of 10% withoutlooking at their spot prices

Now: stock A is the FTSE100 index with S(0) = 5,700 and stock Bis a penny share with S(0) = 0.1

The equivalent log-normal volatility of A is 17.54% and that of B is100.00%

14

Overview III. Jump-diffusion modelsMerton (1976) introduced log-normal dynamics with normal jumps:

dS(t) = µ(t)S(t−)dt+ σS(t−)dW (t) +((eJ − 1)dN(t)− λνdt

)S(t−)

(4)where N(t) is a Poisson process with intensity λ, J is jump amplitudewith PDF $(J) = n(η, δ), and ν is the compensator:

ν =∫ ∞−∞

eJ′$(J ′)dJ ′ − 1 = eη+1

2δ2− 1

Given jump in N(t), jump in S(t) is: ∆S(t) = S(t−)(eJ − 1) ≈ S(t−)J

Since the possibility of large jumps makes out-of-the-money puts moreexpensive, jump-diffusions introduce the implied volatility skew

However, for longer periods (more than one year) the implied distri-bution resembles to the Gaussian (by the central limit theorem)

Jump models have been extended to general Levy processes (Lewis(2001), Levendorskii-Boyarchenko (2002), Carr et al (2003), Cont-Tankov (2004), ...)

15

Overview IV. Stochastic volatility models Hull-White (1988):

dS(t) = µ(t)S(t)dt+ V (t)S(t)dW (1)(t)

dV (t) = κ(θ − V (t))dt+ εV (t)dW (2)(t)

where < dW (1)(t), dW (2)(t) >= ρdtScott (1987):

dS(t) = µ(t)S(t)dt+ eV (t)S(t)dW (1)(t)

dV (t) = κ(θ − V (t))dt+ εdW (2)(t)

Stein-Stein (1991):

dS(t) = µ(t)S(t)dt+ V (t)S(t)dW (1)(t)

dV (t) = κ(θ − V (t))dt+ εdW (2)(t)

Heston (1993):

dS(t) = µ(t)S(t)dt+√V (t)S(t)dW (1)(t)

dV (t) = κ(θ − V (t))dt+ ε√V (t)dW (2)(t)

Jumps can be included (Bates (1996), Duffie et al (2000), ...)

16

Overview V. Parametric local volatilityCox (1975) proposes the constant elasticity of the variance (CEV)model:

dS(t) = µ(t)S(S)dt+ σcev

(Sβ−1(t)

)S(t)dW (t) (5)

Rubinstein (1983) proposes the dis-placed diffusion model

dS(t) = µ(t)S(t)dt+ σdis

(β + γ

S(0)

S(t)

)S(t)dW (t) (6)

Both the CEV with β < 1 (typically, β ∈ [−5,−3] for single stocksand indexes) and dis-placed diffusions exhibit the leverage effect andthe default possibility (from my experience, the dis-placed diffusionmodel is easier to deal with)

Ingersoll (1996), Zuhlsdorff (1999) and Lipton (2000) consider thehyperbolic volatility:

dS(t) = µ(t)S(S)dt+ σhyp

(αS(t)

S(0)+ β + γ

S(0)

S(t)

)S(t)dW (t) (7)

17

Overview VI. Non-parametric local volatilityIn general, parameters of parametric local volatility models need tobe calibrated to observed market prices in least squares sense

Derman-Kani (1994), Rubinstein (1994), Dupire (1994) consider theone-dimensional diffusion model

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

with local volatility σ(loc)(t, S(t)) implied from observed market pricesof vanilla options

Andersen-Andreasen (2000) and Carr et al (2004) introduce the localvolatility with jumps

JP Morgan (1999) and Blacher (2001) consider the local volatilitymodel with stochastic variance driven by the Heston model

Lipton (2002) introduces the local stochastic volatility with jumps

Let me summarise the two main categories of the models used inpractice for exotic options

18

Overview VII. Volatility models1) Non-parametric local volatility models (Dupire (1994), Derman-Kani (1994), Rubinstein (1994))”+” are consistent with today’s market 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))”+” tend to be more in line with the market dynamics”+” equipped to model the term-structure (by mean-reversion pa-rameters) 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))aim to include ”+”’s and cross ”-”’s of the first two models

19

Overview VIII. Model specification

1) Global factors

Select factors relevant for product risk: stochastic volatility, stochas-tic interest rate, jumps, default risk, etc

(Guess) Estimate or calibrate model parameters for the dynamics ofthese factors using either historical or market data

Parameters are updated infrequently

2) Local factors

Specify local factors such as local volatility or local drift (for quantos)

Parameters of local factors are updated frequently (on the run)

20

Stochastic local volatility models

Stochastic local volatility models allow:

1) Specify the dynamics for the instantaneous volatility

2) Fit the risk-neutral distributions implied by market prices of vanillaoptions by specifying local volatility as function of the spot price

However, the model calibration requires robust numerical methods forsolving two-dimensional PDE-s

In general, a successful implementation of the model involves a decentamount of both theoretical and practical exercises

21

Calibration of parametric models I. Forward-backward equationsI apply the methods for stochastic processes and partial differentialequations dating back to Kolmogorov foundational work (1931)

Consider a 1-d problem with stochastic factor S(t) for a product thatdepends only on terminal value of S(T )PV function U(t, S;T,K), 0 ≤ t ≤ T , solves the backward equationas function of (t, S):

Ut(t, S) + LU(t, S) = −c(t, S), U(T, S) = u(S) (9)

u(S) and c(t, S) are pay-off and coupon functions, respectivelyL is the infinitesimal operator corresponding to the dynamics ofS(t) (includes volatility, drift, discounting, jumps)

Transition probability density function, G(0, S0; t, S′), aka Green’sfunction in mathematical physics, solves the forward equation asfunction of (t, S′):

Gt(t, S′)− L†G(t, S′) = 0, G(0, S′) = δ(S′ − S0) (10)

where δ() is Dirac delta function and L† is operator adjoint to L22

Calibration of parametric models II. Valuation formulaMultiply backward equation by G(0, S0; t, S) and integrate over (t, S):

−∫ T

0

∫ ∞−∞

G(0, S0; t′, S′)c(t′, S′)dS′dt′

≡∫ T

0

∫ ∞−∞

[G(0, S0; t′, S′)Ut(t

′, S′) +G(0, S0; t′, S′)LU(t′, S′)]dS′dt′

=∫ ∞−∞

[G(0, S0;T, S′)U(T, S′)−G(0, S0; 0, S′)U(0, S′)

−∫ T

0Gt(0, S0; t′, S′)U(t′, S′)dt′

]dS′+

∫ ∞−∞

∫ T0G(0, S0; t′, S′)LU(t′, S′)dt′dS′

=∫ ∞−∞

G(0, S0;T, S′)u(S′)dS′ − U(0, S0)

−∫ T

0

∫ ∞−∞

[{L†G(0, S0; t′, S′)

}U(t′, S′)−G(0, S0; t′, S′)

{LU(t′, S′)

}]dS′dt′

where, in second line, the first term is integrated by parts, and, inthe last line, terminal condition for U(T, S) and initial condition forG(0, S0; 0, S) are applied

23

Calibration of parametric models III. ConsistencyConsistency condition for adjoint operators L and L† (can be checkedby integration by parts):∫ T

0

∫ ∞−∞

[{L†G(0, S0; t′, S′)

}U(t′, S′)−G(0, S0; t′, S′)

{LU(t′, S′)

}]dS′dt′ = 0

(11)Then the PV can be computed by convolution:

U(0, S0;T,K) =∫ ∞−∞

G(0, S0;T, S′)u(S′)dS′

+∫ T

0

∫ ∞−∞

G(0, S0; t′, S′)c(t′, S′)dS′dt′(12)

This formula is known as Duhamel’s principle (19th century) inmathematical physics or Feynman-Kac formula (1949) in probability

24

Calibration of parametric models IV. Implications

Typically, for a parametric model, model parameters are assumed tobe piece-wise constant in time with jumps at times {Tn}, where {Tn}is set of maturity times of listed options

Implications for calibration by bootstrap:

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

2) Apply Duhamel’s formula (12) to compute PV-s of specified vanillainstruments

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

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

25

Calibration of parametric models V. Numerical schemesConsistency condition (11) for adjoint operators L and L† is based ontheoretical arguments:∫ T

0

∫ ∞−∞

[{L†G(0, S0; t′, S′)

}U(t′, S′)−G(0, S0; t′, S′)

{LU(t′, S′)

}]dS′dt′ = 0

It is not necessarily true for discrete numerical schemes! (see Lipton(2007) for a consistent scheme)

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

should be spacial discretisation of L†

2) Both forward and backward equations should be solved using thesame schemes

3) It is enough to develop either forward or backward scheme

4) L and L† have the same values of the operator norm, so the con-vergence of the forward scheme implies convergence of the backwardscheme and vice versa

26

Local volatility calibration I. Basic facts

Let G(t, S;T, S′) be the risk-neutral probability density at time T andstate S′ given that S(t) = S:

G(t, S;T, S′) = P[S(T ) ∈ dS′ |S(t) = S

]dS′ (13)

Assume that the discount rate is deterministic

By applying the risk-neutral valuation and Duhamel’s formula (12),un-discounted call value C(T,K) solves:

C(T,K) =∫ ∞

0(S′ −K)+G(t, S;T, S′)dS′

=∫ ∞K

(S′ −K)G(t, S;T, S′)dS′(14)

By basic calculus:

CK(T,K) = −∫ ∞K

G(t, S;T, S′)dS′

CKK(T,K) = G(t, S;T,K)(15)

27

Local volatility calibration II. Breeden-Litzenberger formula

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 (16)

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

Note that if call prices are given for all strikes and maturities,{C(market)(K,T )}, then by (15) the risk-neutral distribution satisfies(Breeden-Litzenberger (1978)):

G(market)(T,K) = C(market)KK (T,K)

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

the implied risk-neutral distribution G(t, S;T, S′) is consistent withG(market)(T,K)

28

Local volatility calibration III. Kolmogorov equation

The risk-neutral probability density G(t, S;T, S′) solves the forwardKolmogorov equation:

GT −1

2

((σ(loc,dif)(T, S′)S′)2G

)S′S′

+(µ(T )S′G

)S′

= 0

G(t, S; t, S′) = δ(S′ − S)

Multiply by (S′ −K)+ and integrate over S′

Initial condition:∫ ∞0

(S′ −K)+δ(S′ − S)dS′ = (S −K)+

Time derivative: ∫ ∞0

(S′ −K)+GTdS′ = CT (T,K)

29

Diffusion term:∫ ∞0

(S′ −K)+((σ(loc,dif)(T, S′)S′)2G

)S′S′

dS′ = σ2(loc,dif)(T,K)K2G(t, S;T,K)

= σ2(loc,dif)(T,K)K2CKK(T,K)

Drift term:∫ ∞0

(S′ −K)+(µ(T )S′G

)S′dS′ = µ(T )KCK(T,K)− µ(T )C(T,K)

As a result, we obtain the forward equation for call option prices asfunction of the ”forward” arguments T and K:

CT −1

2σ2

(loc,dif)(T,K)K2CKK + µ(T )KCK − µ(T )C = 0

C(t,K) = (S(t)−K)+

Inverting the PDE in terms of the σ2(loc,dif), we obtain Dupire equa-

tion (1994) for local volatility:

σ2(loc,dif)(T,K) =

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

2CKK(T,K)(17)

30

Jump-diffusion model I

Andersen-Andreasen (2000) consider a Merton jump-diffusion extendedto 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

(18)

where σ(loc,jd) is the local volatility corresponding to the jump-diffusion

31

Jump-diffusion model II Kolmogorov equationNow G(t, S;T, S′) solves the forward Kolmogorov equation:

GT −1

2

((σ(loc,jd)(T, S′)S′)2G

)S′S′

+(µ(T )S′G

)S′− λI(S′) = 0

G(t, S; t, S′) = δ(S′ − S)

I(S′) =∫ ∞−∞

G(S′e−J′)e−J

′$(J ′)dJ ′+ (νS′G)S′ −G

Proof: for the backward equation the jump term is given by:

I?(S′) =∫ ∞−∞

U(S′eJ′)$(J ′)dJ ′ − νS′US′ − U

Multiply by G(t, S;T, S′) and integrate over S′:∫ ∞−∞

G(t, S;T, S′)I?(S′)dS′ =∫ ∞−∞

G(t, S;T, S′)U(S′eJ′)$(J ′)dJ ′dS′

−∫ ∞−∞

νG(S′)S′US′dS′ −

∫ ∞−∞

GUdS′

=∫ ∞−∞

[∫ ∞−∞

G(t, S;T, S′e−J′)e−J

′$(J ′)dJ ′+ (νS′G)S′ −G

]U(S′)dS′

by making substitution S′eJ′ → S′ and integrating by parts

32

Jump-diffusion model IVFollowing the same steps consider:

I†(K) =∫ ∞

0(S′ −K)+I(S′)dS′

=∫ ∞

0(S′ −K)+

∫ ∞−∞

G(S′e−J′)e−J

′$(J ′)dJ ′dS′+ νKCK − (ν + 1)C

=∫ ∞−∞

C(Ke−J′)eJ

′$(J ′)dJ ′+ νKCK − (ν + 1)C

As a result, obtain the forward equation for call option prices:

CT −1

2σ2

(loc,jd)(T,K)K2CKK + µ(T )KCK − µ(T )C − λI†(K) = 0

C(t,K) = (S(t)−K)+

Inverting σ2(loc,jd) yields 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)

33

Stochastic local volatility IConsider 2-d stochastic local volatility diffusion:

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

(19)

where σ(loc,sv) is the local volatility of the diffusion with stochastic

volatility and < dW (1)(t), dW (2)(t) >= ρdt

ϑ(t, Y (t)) is the mapping function of Y (t) into the spot dynamics

34

Stochastic local volatility II. CalibrationConventional approach (for example, Ren-Madan-Qian (2007)) is touse Gyongy (1986) mimicking theorem yielding:

σ2(loc,sv)(T,K)E

[ϑ2(T, Y (T )) |S(T ) = K

]= σ2

(loc,dif)(T,K)

Gyongy theorem is purely probabilistic analysis, we follow originalLipton’s (2002) PDE-based approach applied directly in the contextof stochastic local volatility calibration

The risk-neutral probability density G(t, S, Y ;T, S′, Y ′) solves the for-ward Kolmogorov equation:

GT −1

2

((σ(loc,sv)(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,sv)(T, S′)ϑ(T, Y ′)ε(Y ′)S′G

)S′Y ′

= 0

G(t, S, Y ; t, S′, Y ′) = δ(S′ − S)δ(Y ′ − Y )

35

Stochastic local volatility III. Calibration

Consider the unconditional density:

U(t, S; t, S′) =∫ ∞−∞

G(t, S, Y ;T, S′, Y ′)dY ′

The diffusion term becomes:∫ ∞−∞

((σ(loc,sv)(T, S′)ϑ(T, Y ′)S′)2G

)S′S′

dY ′

=([∫ ∞−∞

ϑ2(T, Y ′)GdY ′]

(σ(loc,sv)(T, S′)S′)2)S′S′

=(V (T, S′)(σ(loc,sv)(T, S′)S′)2U(T, S′)

)S′S′

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

V (T, S′) =

∫∞−∞ ϑ

2(T, Y ′)G(t, S, Y ;T, S′, Y ′)dY ′

U(t, S;T, S′)

=

∫∞−∞ ϑ

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

36

Stochastic local volatility IV. Calibration

Remaining terms are trivial, yielding:

UT −1

2

(V (T, S′)(σ(loc,sv)(T, S′)S′)2U

)S′S′

+(µ(T )S′U

)S′

= 0

U(t, S; t, S′) = δ(S′ − S)

PDE for call prices:

CT −1

2(V (T,K)(σ(loc,sv)(T,K)K)2CKK + µ(T )KCK − µ(T )C = 0

The model is consistent with the Dupire local volatility if

V (T,K)σ2(loc,sv)(T,K) = σ2

(loc,dif)(T,K)

As a result, the stochastic local volatility is specified by:

σ2(loc,sv)(T,K) =

σ2(loc,dif)(T,K)

V (T,K)

37

Stochastic local volatility with jumps I

Consider a two-dimensional stochastic local volatility jump-diffusion:

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

+((eJ − 1)dN(t)− λνdt

)S(t−), S(0) = S,

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

(20)

where σ(loc, svj) is the local volatility of the diffusion with stochasticvolatility and jumps

Υ is the amplitude of the jump in Y (t) with PDF ς(Υ)

Jumps are simultaneous in S(t) and Y (t)

38

Stochastic local volatility with jumps II

The risk-neutral probability density G(t, S, Y ;T, S′, Y ′) solves the for-ward Kolmogorov equation:

GT −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′) = 0

I(S′) =∫ ∞−∞

∫ ∞−∞

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

G(t, S, Y ; t, S′, Y ′) = δ(S′ − S)δ(Y ′ − Y )

39

Stochastic local volatility with jumps III

PDE for call prices:

CT −1

2(V (T,K)(σ(loc,svj)(T,K)K)2CKK + µ(T )KCK − µ(T )C − λI†(K) = 0

where V (T,K) is the unconditional variance

Inverting σ2(loc,svj)(T,K), yields 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)

40

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

For the backward problem, L is the infinitesimal operator correspond-ing to model dynamics:

L = D+ λJ

41

1-d ProblemHere 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 we denote the discretized diffusion and integraloperators by Dn and J, respectively

The diffusion operator is space and time dependent (denoted by n)

Care must be taken for discretization of the operator for the mean-reverting process!

42

1-d Problem for diffusion problem IN - number of points in the space gridDn - discrete diffusion matrix at time tn with dimension N ×NUn - the solution vector at time tn with dimension N × 1I - the unit matrix with dimension N ×N

Time-stepping methods of the choice:

Explicit method:

Un+1 =(I + Dn+1

)Un

Implicit method: (I−Dn+1

)Un+1 = Un

θ method: (I− θDn+1

)Un+1 =

(I + θDn+1

)Un

with θ = 12 corresponding to Crank-Nicolson scheme

43

1-d Problem for diffusion problem II

The explicit method requires very fined grids and is highly unstable -it should be avoided

The Crank-Nicolson scheme is unconditionally stable and is of or-der (δS)2 and (δt)2, but might lead to negative probabilities for theforward equation

The implicit method is unconditionally stable and does not lead tonegative 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)

which is of order (δS)2 and (δt)2, and does not lead to negativeprobabilities

44

Numerics for jump-diffusions ILet me consider the forward equation with additive jumps (multiplica-tive jumps can be handled in the log-space):

JG(x) =∫ ∞−∞

G(x− j)$(J ′)dJ ′

Let me 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

45

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

46

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

47

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)

48

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)

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

49

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)

50

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)

51

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))(21)

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

52

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

53

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

54

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)

55

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

56

Illustration I. Model specificationWe specify a particular version of the LSV dynamics (20):

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(22)

where dW (1)(t)dW (2)(t) = ρdtConvenient choice of ϑ(t, Y (t)):

ϑ(t, Y (t)) = eY (t)−V[Y (t)]

where V[Y (t)] is the variance of Y (t). Then if ρ = 0:

E[ϑ2(t, Y (t)) |Y (0) = 0

]= 1

Mapping ϑ(t, Y (t)) introduces the ”volatility-of-volatility” effect with-out affecting the local volatility close to the spotSimultaneous jumps in S(t) and Y (t) are discrete with magnitudes−ν < 0 and η > 0 (note that jump variance will decrease the modelimplied correlation between spot and volatility)

57

Illustration II. Model calibration

1) Specify time and space grids

2) Compute the local volatility using Dupire equation (17) on thespecified grid (interpolating implied volatilities in strikes and maturi-ties)

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

58

Illustration III. Local volatility calibration AAt initial time t0 = 0i) initialize G0(x1(i), x2(j)) = 1x1(i)=x1(t0)1x2(j)=x2(t0)

ii) Set the conditional variance V (t1, x1(i)) = 1 so that

σ2(loc,sv)(t1, x1(i)) = σ2

(loc,dif)(t1, x1(i))

iii) Compute G1(x1(i), x2(j)) by solving the forward PDE

iv) Compute V (t1, x1(i)) by

V (t1, x1(i)) =

∑i∑j ϑ

2(t1, x2(j))G1(x1(i), x2(j))∑j G

1(x1(i), x2(j))

and set

σ2(loc,sv)(t1, x1(i)) =

σ2(loc,dif)(t1, x1(i))

V (t1, x1(i))

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

59

Illustration III. Local volatility calibration BAt time tn given 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)

60

Illustration III. Local volatility calibration C

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

61

Illustration 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

62

Illustration V. 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

63

Illustration VI. 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)

64

Illustration VII. 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!

65

Illustration VIII. Implied volatilities of options on the daily real-ized variance 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

66

Illustration IX. Options on the VIXAugment the model dynamics (22) with the third variable for therealized 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

]Valuation:i) Solve 2-d (!) backward problem for I(T, T + Tvix) as function 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

67

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)

68

Conclusions

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

I am grateful to members of the quantitative group at Bank of Amer-ica Merrill Lynch for their help and discussions during work on thisproject

Thank you for your attention!

69

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