Achieving Consistent Modeling Of VIX and EquitiesDerivatives

Artur SeppBank of America Merrill Lynch, London

artur.sepp@baml.com

Global Derivatives Trading & Risk Management 2012Barcelona

April 17-19, 2012

1

Plan of the presentation

1) Discuss model complexity and calibration

2) Emphasize intuitive and robust calibration of sophisticated volatil-ity models avoiding non-linear calibrations

3) Present local stochastic volatility models with jumps to achievejoint calibration to VIX options and (short-term) S&P500 options

4) Present two factor stochastic volatility model to fit both the short-term and long-term S&P500 option skews

2

References

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

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

2) 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

Motivation. Model complexity and calibration

The conventional approach emphasizes the importance of ”closed-form” solutions for a limited class of model chosen on the sole groundof their analytical tractability

The justification is that since the model calibration is implemented bya non-linear optimization, closed-form solution ”speed-up” the cali-bration

I prefer the opposite route:

1) Develop robust PDE methods for generic one and two factorstochastic volatility models

2) Obtain intuition about model parameters using approximation sothat non-linear fits can be avoided

4

Model specification I

Changes in the implied volatility surface can be factored as follows:

δσ(T,K) = β1 (change in short term ATM volatility)

+ β2 (change in term-structure of ATM volatility)

+ β3 (change in short term skew)

+ β4 (change in term-structure of skew)

+ ...

(1)

Factors are reliable if:1) They explain most of the variation2) Can be estimated from historical and current data in intuitive ways3) Can be explained in simple terms

5

Model specification II. Important factors for a volatility model1) Short-term ATM volatility and term-structure of ATM volatility -time-dependent level of the model ”ATM” volatility (the least of theproblem)

2) Short-term skew - jumps in the underlying and volatility factor(s)

3) Term-structure of skew and volatility of volatility - mean-reversionand volatility of volatility of volatility factor(s)

The challenge is to re-produce these factors using stochastic volatil-ity model with time-homogeneous parameters

In my presentation, I consider:

Part I - calibration of stochastic volatility with jumps to VIX skews(SV model for short-term skew, up to 6m)

Part II - calibration of two-factor stochastic volatility model (SVmodel for medium- and long-term skew, 3m-5y)

6

Part I. Joint calibration of SPX and VIX skews using jumps

I consider several volatility models to reproduce the volatility skewobserved in equity options on the S&P500 index:Local volatility model (LV)Jump-diffusion model (JD)Stochastic volatility model (SV)Local stochastic volatility model (LSV) with jumps

For each model, I analyze its implied skew for options on the VIX

I show that LV, JD and SV without jumps are not consistent with theimplied volatility skew observed in option on the VIX

I show that:Only the SV model with appropriately chosen jumps can fit the im-plied VIX skew

Importantly, that only the LSV model with jumps can fit both Equityand VIX option skews

7

Motivation IFind a dynamic model that can explain both the negative skew forequity options on the S&P500 index (SPX) and positive skew foroptions on the VIX

Implied volatility of the SP500 and VIX

10%

35%

60%

85%

110%

135%

50% 70% 90% 110% 130% 150% 170%Strike %

S&P500, 1mVIX, 1m

Implied vols of 1m options on the SPX and the VIX as functions ofstrike K% relative to 1m SPX forward and 1m VIX future, respectively

8

Motivation II. DynamicsThe VIX exhibits strong mean-reversion and jumps

Page 1

10%

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ol,

VIX

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0%

Ske

w

VIXATMSkew

Rights scale: Time series of the VIX and 1m ATM SPX impliedvolatility (ATM)Left scale: 1m 110%− 90% skew

9

The VIX I. DefinitionThe VIX is a measure of the implied volatility of SPX options withmaturity 30 days

Trading in VIX futures & options began in 2004 & 2006, respectivelyThroughout 2004-2007, the average of the VIX is 14.66%Throughout 2008-2011, the average of the VIX is 27.65%

Nowadays, VIX options are one of the most traded products on CBOEwith the average daily about 10% of all traded contracts

Formally, the VIX at time t, denoted by F (t), is the square rootof the expectation of the quadratic variance I(t, T ) of log-returnln(S(T )/S(t))

F (t) =

√1

τTE [I(t, t+ τT ) |S(t)] , τT = 30/365 (2)

For a general non-linear pay-off function u:

U(t, T ) = E [u(F (T )) |S(t)] = E[u

(√1

τTE [I(T, T + τT ) |S(T )]

)|S(t)

]10

Valuation of the VIX options I. PDE methodConsider the augmented dynamics for S(t) and I(t):

dS(t) = σ(S(t))dW (t), S(0) = S

dI(t) =

(σ(S(t))

S(t)

)2

dt, I(0) = 0(3)

1) Solve backward problem for V (t, S) = E [I(T, T + τT )] on time in-terval [T, T + τT ]:

Vt + LV = −(σ(S)

S

)2

, V (T + τT , S) = 0 (4)

where L is the backward operator:

L =1

2σ2(S)∂SS (5)

2) Solve backward problem for U(t, S) = E [u (F (T ))] on interval [0, T ]:

Ut + LU = 0 , U(T, S) = u

(√1

τTV (T, S)

)(6)

The problem is similar to valuing an option on a coupon-bearing bond

11

Valuation of the VIX options II. EnhancementSolve the forward problem for the density function of S, G(T, S′), attime T

GT − L̃G = 0 , G(0, S) = δ(S′ − S) (7)

where L̃ is operator adjoint to L:

L̃G =1

2∂SS

(σ2(S)G

)(8)

Then U(T, S) is priced by

U(0, S) =∫ ∞

0u

(√1

τTV (T, S′)

)G(T, S′)dS′ (9)

Generic implementation is done by means of finite-difference (FD)methods that allow to tackle the problem for arbitrary choice of σ(S)

When using (9) we only need to solve 2 PDE-s, backward and forward,to value VIX options with maturity T across different strikes

Stochastic volatility and jumps can be incorporated and treated byFD methods (see Sepp (2011) for a review)

12

Specific Models IDupire LV and LSV models are ”black boxes” - for any admissibleset of model parameters, they guarantee calibration to equity skew

Instead, I consider simple model specifications to analyze their cali-bration and implied VIX dynamics by fitting model parameters to:1) at-the-money(ATM) implied volatility σATM, Eq2) equity skew defined by:

SkewEq(α) ≡ σimp((1 + α)S)− σimp((1− α)S)

where σimp(K) is SPX implied vol function of strike K and α = 10%

As an output, I consider VIX skew defined by:

SkewVIX(α) ≡ σimp,VIX((1 + α)S)− σimp,VIX(S),

where σimp(K) is VIX implied vol as function of strike K

Inputs on 31 October 2011 for 1m SPX and VIX options:S(0) = 1253.3, σATM, Eq = 25%, SkewEq(10%) = −16%F (0) = 30%, σATM,VIX = 101%, SkewVIX(10%) = 12%

13

Specific Models I. CEV model A: BasicsConsider the CEV process (Cox (1975)):

dS(t) = σ

(S(t)

S(0)

)βdW (t) (10)

We can derive an approximation for implied vol at K = (1− α)S:

σimp((1− α)S) = σ

(S

S(0)

)1−β (1 +

1

2α(1− β) +O(α2)

)Thus

σATM,CEV(S) ≡ σ(

S

S(0)

)1−β, SkewEq,CEV(α) ≡ −σ

(S

S(0)

)1−βα(1− β)

Given σATM,Eq and SkewEq(α):

σ = σATM

(S

S(0)

)1−β, β = 1 +

SkewEq(α)

ασATM,Eq

14

Specific Models II. CEV model C: Dynamics of the VIXApproximate the VIX using:

F (t) ≈σ(S(t))

S(t)=σ (S(t))β

S(t)= σ (S(t))β−1

For the dynamics of F (t):

dF (t) = (β − 1)F2(t)dW (t) +1

2(β − 1)(β − 2)F3(t)dt

Observations:1), No mean-revertion

2), for β < 1, we have negative absolute correlation with the spot

15

Specific Models II. CEV model D: VIX implied volatilityWe can show that approximately:

σimp,vix((1− α)F ) = −(β − 1)F(

1−1

2α+O(α2)

)Thus:

σATM,VIX,CEV = (1− β)F , SkewVIX,CEV = (1− β)F1

2α

Using implied parameters:

σATM,VIX,CEV ≈ −SkewEq(α)

α, SkewVIX,CEV ≈ −

1

2SkewEq(α)

Both VIX ATM volatility and skew are proportional to the equity skew

Using SkewEq(10%) = −16% we get:σATM,VIX,CEV = 160% (vs 100% actual)SkewVIX,CEV = 8% (vs 12% actual)

CEV (and LV in general) tend to overestimate VIX ATM vol andunderestimate VIX skew

16

Specific Models II. CEV model E: Illustration

Implied SP500 1m Volatility

5%

15%

25%

35%

45%

55%

75% 85% 95% 105% 115% 125%

Strike %

ExactApproximateMarket

Implied VIX 1m Volatility

0%

50%

100%

150%

200%

250%

70% 90% 110% 130% 150% 170%

Strike %

ModelMarket

Left: SPXskew; Right: the VIX skewImplied model parameters: σ = 0.2474, β = −5.37

17

Specific Models III. Jump-diffusion A: BasicsMerton (1973) jump-diffusion with discrete jump in log-return ν :

dS(t) = σS(t)dW (t) + (eν − 1)S(t) [dN(t)− λdt] (11)

where N(t) is Poisson with intensity λ

We can show that to the leading order for small time-to-maturity:

σimp(K) ≈ σ −λν

σln (S/K)

The short-term skew in JD is linear in jump size and its intensity

18

Specific Models III. Jump-diffusion B: Equity implied volatilityWe have for small time-to-maturity:

σimp(S) ≈ σ, SkewEq(α) ≈ 2αλν

σJD model is overdetermined so consider expected quadratic variance:

υ = σ2 + λν2

Introduce weight wjd, 0 < wjd < 1, as proportion of variance con-

tributed by jumps (a parameter for calibration) and take υ = σ2ATM:

1 =σ2

σ2ATM

+λν2

σ2ATM

≡ (1− wjd) + wjd

Thus, obtain equation to imply λ by:

λν2

σ2ATM

= wjd ⇒ λ = wjdσ2

ATM

ν2

Accordingly:

ν =2αwjdσATM

SkewEq(α), λ =

(SkewEq)2

4α2wjd, σ ≈ σATM

19

Specific Models III. Jump-diffusion E: VIX dynamicsThe quadratic variance in the jump-diffusion model is driven by:

dI(t) = σ2dt+ ν2dN(t)

The variance swap

V (t) =1

TE[∫ T

0I(t′)dt′

]= σ2 + λν2

turns out to be a deterministic function

Thus, the model value of the VIX is constant and the implied volatilityof the VIX is zero - even though the JD model generates the equityskew!

Thus property is shared by all space-homogeneous jump-diffusionsand Levy processes!

Nevertheless, jumps are needed for calibration of VIX skew

20

Specific Models III. Jump-diffusion F: Illustration

Implied SP500 1m Volatility

10%

20%

30%

40%

50%

75% 85% 95% 105% 115% 125%

Strike %

ExactApproximateMarket

Implied VIX 1m Volatility

0%

20%

40%

60%

80%

100%

120%

140%

70% 90% 110% 130% 150% 170%

Strike %

ModelMarket

Left: SPX skew; Right: VIX skewImplied model parameters: wjd = 0.8, λ = 0.7762, ν = −0.2512,σ = 0.1769

21

Specific Models IV. CEV with jumps A: BasicsConsider process:

dS(t) = σ

(S(t)

S(0)

)βdW (t) + (eν − 1)S(t) [dN(t)− λdt] (12)

It turns out that the impact on skew from local volatility and jumpsis roughly linear and additive

Thus, specify the weight for the skew explained by jumps, wjd, andCEV local volatility wlv, wlv = 1− wjd

Using obtained results for CEV and JD models:

β = 1 +wlvSkewEq(α)

ασATM, ν =

2ασATM

SkewEq(α), λ =

wjd(SkewEq)2

4α2

σ = σATM

(S(t)

S(0)

)1−β

22

Specific Models IV. CEV with jumps B: The VIXThe variance swap can be approximated by:

V (0, T ) =1

TE

∫ T0

[dS(t′)

S(t′)

]2

dt′ ≈ σ2

(S(t)

S(0)

)2β−2

+ λν2,

The VIX is approximated using:

F (t) ≈√σ2S2β−2(t) + λν2,

Observation: in CEVJD, β becomes smaller (in absolute terms) thusthe implied vol and skew of the VIX decrease

23

Specific Models V. CEV with jumps C: Illustration

Implied SP500 1m Volatility

10%

20%

30%

40%

50%

75% 85% 95% 105% 115% 125%

Strike %

ModelMarket

Implied VIX 1m Volatility

0%

20%

40%

60%

80%

100%

120%

140%

160%

180%

70% 90% 110% 130% 150% 170%

Strike %

ModelMarket

Left: S&P500 skew; Right: the VIX skewImplied model parameters: wjd = 0.4 (fitted to match the VIX ATMvolatility), λ = 0.2484, ν = −0.3140, σ = 0.2192, β = −3.3141

24

Specific Models V. Stochastic volatility A: BasicsConsider exponential OU stochastic volatility (Scott (1987) SV model,Stein-Stein (1991) SV model is linear in Y (t)):

dS(t) = σS(t)eY (t)dW (0)(t)

dY (t) = −κY (t) + εdW (1)(t), Y (0) = 0(13)

with dW (0)(t)dW (1)(t) = ρdtκ is mean-reversion speedε is volatility-of-volatility

For equity skew, we can show approximately:

σATM ≈ σ , SkewEq(α) ≈ αρε

4σ⇒ ρ =

4σSkewEq(α)

αεFor the VIX implied volatility, we can show approximately:

σimp,vix =σε

F2(t), Skewvix(α) = −

σε

2F2(t)α

A) The ATM implied vol and the skew are proportional to σε;B) The VIX skew is negative

25

Specific Models V. Stochastic volatility D: Illustration

Implied SP500 1m Volatility

10%

20%

30%

40%

50%

75% 85% 95% 105% 115% 125%

Strike %

ModelMarket

Implied VIX 1m Volatility

0%

20%

40%

60%

80%

100%

120%

140%

160%

180%

200%

70% 90% 110% 130% 150% 170%

Strike %

ModelMarket

Left: SPX skew; Right: the VIX skewImplied model parameters: κ = 4.48, ε = 2.69, ρ = −0.85, σ =0.2194,

26

Specific Models. SummaryWe have considered several models with conclusions:

Model Equity Skew VIX Skew Mean-revertionCEV Yes Weak No

Jump-Diffusion Yes No NoCEV Jump-Diffusion Yes Yes (too steep) NoStochastic Volatility Yes No Yes

To model the VIX skew, we need three components:Local volatility (for equity and VIX skew)Stochastic volatility (mean-reverting feature of the Vix)Jumps in price and volatility (steeper equity and the VIX skew)

Two viable contenders:1) SV model with jumps (space-inhomogeneous so some analyticalsolutions possible - Sepp (2008))2) LSV model with jumps, for example, CEV or non-parametric(valuation by means of FD methods)

27

Specific Models VI. SV+Jump-Diffusion: DynamicsConsider generic LSV model with jumps:

dS(t) = σS(t)eY (t)dW (0)(t) + S(t) [(eν − 1) dN(t)− λνdt]dY (t) = −κY (t) + εdW (1)(t) + JvdN(t)

dI(t) = σ2e2Y (t)dt+ (Js)2 dN(t)

with dW (0)(t)dW (1)(t) = ρdtN(t) is Poisson process with intensity λJumps in S(t) and Y (t) are simultaneous and discrete with magni-tudes ν < 0 and η > 0

Here:σ = σ(t) - SV with jumps

σ = σ(t)(S(t)S(0)

)β- CEV+SV with jumps

σ = σ(loc,svj)(t, S(t)) - LSV with jumps (Lipton (2002))

28

Specific Models VI. SV+Jump-Diffusion: Illustration

Implied SP500 1m Volatility

10%

20%

30%

40%

50%

75% 85% 95% 105% 115% 125%

Strike %

ModelMarket

Implied VIX 1m Volatility

0%

20%

40%

60%

80%

100%

120%

140%

70% 90% 110% 130% 150% 170%

Strike %

ModelMarket

Left: SPX skew; Right: the VIX skewImplied model parameters: wsv = 0.65, wjd = 0.35 (fitted to matchthe VIX ATM volatility), λ = 0.2173, ν = −0.314, ρ = −0.85, σ =0.2227, κ = 4.48, ε = 1.7485 (= wsvε̂), η = 1.30 (fitted to match tofit the VIX skew)

29

Specific Models VI. CEV+SV+Jump-Diffusion: Illustration

Implied SP500 1m Volatility

10%

20%

30%

40%

50%

75% 85% 95% 105% 115% 125%

Strike %

ModelMarket

Implied VIX 1m Volatility

0%

20%

40%

60%

80%

100%

120%

140%

70% 90% 110% 130% 150% 170%

Strike %

ModelMarket

Left: S&P500 skew; Right: the VIX skewImplied model parameters: wjd = 0.35, wcev = 0.05, wsv = (1 −wcev)(1 − wv) = 0.6175 (fitted to match the VIX ATM volatility),β = 0.77, λ = 0.2173, ν = −0.314, ρ = −0.85, σ = 0.2227, κ = 4.48,ε = 1.6611 (= wsvε̂), η = 1.30 (fitted to match to fit the VIX skew)

30

Specific Models VI. Non-parametric LSV+Jump-Diffusion

1m

60%

70%

80%

90%

100%

110%

120%

130%

140%

150%

70% 80% 90% 100% 110% 120% 130% 140% 150% 160% 170%

K %Market, 1mModel, 1m

2m

60%

70%

80%

90%

100%

110%

120%

70% 80% 90% 100% 110% 120% 130% 140% 150% 160% 170%

K %

Market, 2mModel, 2m

3m

60%

65%

70%

75%

80%

85%

90%

95%

100%

105%

70% 80% 90% 100% 110% 120% 130% 140% 150% 160% 170%

K %

Market, 3mModel, 3m

4m

50%

55%

60%

65%

70%

75%

80%

85%

90%

70% 80% 90% 100% 110% 120% 130% 140% 150% 160% 170%

K %

Market, 4mModel, 4m

5m

40%

45%

50%

55%

60%

65%

70%

75%

80%

85%

70% 80% 90% 100% 110% 120% 130% 140% 150% 160% 170%

K %

Market, 5mModel, 5m

6m

30%

35%

40%

45%

50%

55%

60%

65%

70%

75%

70% 80% 90% 100% 110% 120% 130% 140% 150% 160% 170%

K %

Market, 6mModel, 6m

LSV model fit to VIX implied volatilities (the model is consistent withthe S&P500 implied volatilities by construction)Implied model parameters: wjd = 0.35, wlv = 0.05, wsv = 0.6175,λ = 0.2173, ν = −0.314, ρ = −0.80, κ = 4.48, ε = 1.6611, η = 1.30

31

ConclusionsFor calibration of LSV model to SPX and VIX skews, a careful choicebetween the stochastic volatility, local volatility and jumps is neces-sary:

1) Local volatility: weight wlv - calibration parameter (typicallysmall)

2) Price jumps: weight wjd - calibration parameter (fitted to VIXATM volatility), λ and ν determined from given equity skew

3) Volatility jumps: η - calibration parameter (fitted to VIX skew)

4) SV parameters: SV weight wsv = 1 − wlv − wjd - calibrationvariables, κ and ε from historical (implied) data

VIX options give extra information for calibration of LSV models

32

Part II. Joint calibration of medium- and long-term SPX skewsusing two-factor stochastic volatility model

Observation: One-factor stochastic volatility model with time-independentparameters can only calibrate toA) either medium-term skews (up to 1 year) using large mean-reversionand vol-of-volB) or long-term skews (from 1 year) using low mean-reversion andvol-of-vol

Idea: introduce a two-factor stochastic volatility model with one fac-tor for short-term skew and second factor for long-term skew

Challenge: find a proper weight (time-dependent) for the two factors

Bergomi (2005) assumes constant weights - I found it difficult tocalibrate his model with constant weights

33

Skew decay IConsider 110%−90% market skew, Skew(Tn), for maturities {1m,2m, ...,60m}

Compute skew decay rate: Rmarket(Tn) = Skew(Tn)Skew(Tn−1), n = 2, ...,60

In addition, consider the following test functions:

Fα(Tn) = (Tn)−α

with decay rate Rα(Tn) = Fα(Tn)Fα(Tn−1) =

(TnTn−1

)−αWe observe:1) for small T , the skew decay α ≈ 0.25

2) for large T , the skew decay α ≈ 0.50

Experiment - compute weighted average:

Fα1,α2(Tn) = ω(Tn)T−α1n + (1− ω(Tn))T−α2

n , ω(t) = e−κt , κ = 2

with skew decay rate: Rα1,α2(Tn) =Fα1,α2(Tn)Fα1,α2(Tn−1)

34

Skew decay II

0.85

0.87

0.89

0.91

0.93

0.95

0.97

0.99

2m 6m 10m 14m 18m 22m 26m 30m 34m

T

Ske

w d

ecay

rat

e

Market rate skew

rate 1/T^0.5

rate 1/T^0.25

weighted sum

For short maturities, the skew is proportional to T−0.25

For longer maturities, the skew is proportional to T−0.50

The decay of the market skew is reproduced using ”weighted” skew:

F0.25,0.50(Tn) = ω(Tn)T−0.25n + (1− ω(Tn))T−0.5

n , ω(t) = e−κt , κ = 2

35

Two-factor SV model IStart with one-factor SV dynamics for S(t) and SV factor Y (t):

dS(t)/S(t) = µ(t)dt+ σ exp {Y (t)} dW (0)

dY (t) = −κY (t)dt+ εdW (1), dW (0)dW (1) = ρ

Then extend to two-factor SV dynamics :

dS(t)/S(t) = µ(t)dt+ σ exp {ω(t)Y1(t) + (1− ω(t))Y2(t)} dW (0)

dY1(t) = −κ1Y1(t)dt+ ε1dW(1) , dY2(t) = −κ2Y2(t)dt+ ε2dW

(2)

dW (0)dW (1) = ρ01, dW (0)dW (2) = ρ02, dW (1)dW (2) = ρ12 ≡ ρ01ρ02Here:Y1(t) - SV factor for short-term volatility;Y2(t) - SV factor for long-term volatility;ω(t), 0 < ω(t) < 1, - weight between short-term and long-term factor:

ω(t) = exp{−

1

2(κ1 + κ2) t

}We expect: κ1 >> κ2, ε1 >> ε2, |ρ01| > |ρ02|

36

Two-factor SV model II. CalibrationThe model is implemented using a 3-d PDE solver with a predictor-correctorTypically, we use N1 = 100 per year in time dimension, N2 = 200 inspot dimension, N3 = N4 = 25 in volatility dimensions

For calibration, we solve the forward PDE to compute the density ofthe underlying price and value European calls and puts at once (takesabout 1-minute to calibrate to ATM volatility up to 5 years)

To simplify the calibration we consider two quantities at given matu-rity objects: the ATM volatility and 90%− 110% skew

1) calibrate two sets of parameters {κ, ε, ρ} for 1-factor SV model:the first one - to short term skew (up to 1 year)the second one - to long-term skew (from 1y to 5y)2) Use these parameters for the 2-factor SV model with weight ω(t)

For all three models, piece-wise constant volatilities {σ(Tn)} are cal-ibrated so that the model matches given market ATM volatility atmaturity times {Tn}

37

Two-factor SV model III. Illustration of calibrationCalibrated parameters for SPX (top) and STOXX 50 (bottom)

Short-term Long-term 2-f modelκ 3.50 0.65ε 1.95 0.95ρ -0.85 -0.80

ρ12 0.6812 (κ1 + κ2) 2.08

Short-term Long-term 2-f modelκ 2.40 0.55ε 1.40 0.70ρ -0.85 -0.80

ρ12 0.6812 (κ1 + κ2) 1.48

38

Two-factor SV model IV. Calibration to SPXTerm structure of skews for 1-f SV model calibrated up to 1y skews,1-f SV model calibrated from 1y to 5y skews, 2-factor SV model

-12%

-10%

-8%

-6%

-4%

-2%

0%

3m 9m 15m 21m 27m 33m 39m 45m 51m 57m

Market SkewModel Skew

-12%

-10%

-8%

-6%

-4%

-2%

0%

3m 9m 15m 21m 27m 33m 39m 45m 51m 57m

Market Skew

Model Skew

-12%

-10%

-8%

-6%

-4%

-2%

0%

3m 9m 15m 21m 27m 33m 39m 45m 51m 57m

Market Skew

Model Skew

39

Two-factor SV model V. Calibration to STOXX 50Term structure of skews for 1-f SV model calibrated up to 1y skews,1-f SV model calibrated from 1y to 5y skews, 2-factor SV model

-9%

-8%

-7%

-6%

-5%

-4%

-3%

-2%

-1%

0%

3m 9m 15m 21m 27m 33m 39m 45m 51m 57m

Market SkewModel Skew

-9%

-8%

-7%

-6%

-5%

-4%

-3%

-2%

-1%

0%

3m 9m 15m 21m 27m 33m 39m 45m 51m 57m

Market Skew

Model Skew

-9%

-8%

-7%

-6%

-5%

-4%

-3%

-2%

-1%

0%

3m 9m 15m 21m 27m 33m 39m 45m 51m 57m

Market Skew

Model Skew

40

Illustration VI. Model implied vol across strikes for SPXA) 2-factor SV model with piecewise-constant volatility fits the term-structure of market skews, unlike 1-factor SV modelB) Small discrepancies in model and market implied volatilities acrossstrikes are eliminated by introducing a parametric local volatility withextra parameter to match the skew across strikes

3m

10%

15%

20%

25%

30%

35%

40%

45%

70% 75% 80% 85% 90% 95% 100% 105% 110%

K %

Imp

lied

Vo

l

Market 3mModel 3m

6m

10%

15%

20%

25%

30%

35%

40%

70% 75% 80% 85% 90% 95% 100% 105% 110% 115%

K %

Imp

lied

Vo

l

Market 6mModel 6m

9m

10%

15%

20%

25%

30%

35%

40%

70% 75% 80% 85% 90% 95% 100% 105% 110% 115%

K %

Imp

lied

Vo

l

Market 9mModel 9m

1y

10%

15%

20%

25%

30%

35%

70% 75% 80% 85% 90% 95% 100% 105% 110% 115%

K %

Imp

lied

Vo

l

Market 12mModel 12m

3y

10%

15%

20%

25%

30%

35%

70% 75% 80% 85% 90% 95% 100% 105% 110% 115%

K %

Imp

lied

Vo

l

Market 36mModel 36m

5y

10%

15%

20%

25%

30%

35%

70% 75% 80% 85% 90% 95% 100% 105% 110% 115%

K %

Imp

lied

Vo

l

Market 60mModel 60m

41

Illustration VII. Model implied density for SPX

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0.00 0.18 0.37 0.55 0.73 0.91 1.10 1.28 1.46 1.65 1.83 2.01 2.20

Normalized Spot

Den

sity

3m 6m 9m 12m 15m

18m 21m 24m 27m 30m

33m 36m 39m 42m 45m

48m 51m 54m 57m 60m

Observe heavy tails for longer-maturities with concentration aroundS = 0 (zero is assumed to be an absorbing barrier for my PDE solver)

42

Illustration VIII. The term structure of implied volatility-of-volatility

40%

60%

80%

100%

120%

140%

160%

180%

3m 6m 1y 2y 3y 4y 5y

T

Imp

lied

Vo

lVo

l 2-Factor SV1-Factor SV

The term structure of implied volatility-of-volatility (implied log-normalvol for ATM option on the realized variance) in 1- and 2-factor SVmodel

43

Conclusions

I have presented:

1) Local stochastic volatility model with jumps for modeling ofshort-term skews and illustrated its calibration to VIX options skews

I have shown that jumps are needed to fit the model to both SPXand VIX skews

2) 2-factor stochastic volatility model for modeling of medium-and long-dated skews

I have shown that only two-factor stochastic volatility model can fitboth medium- and long-dated skews

For both models, I have tried to emphasize an intuitive approach tomodel calibration and avoid using ”blind” non-linear calibrations

44

Acknowledgment

During my work I benefited from interactions with:

Alex Lipton on his universal local stochastic volatility with jumps(Lipton (2002))

Hassan El Hady, Charlie Wang and other members of BAML GlobalQuantitative Analytics

The opinions and views expressed in this presentation are those ofthe author alone and do not necessarily reflect the views and policiesof Bank of America Merrill Lynch

Thank you for your attention!

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ReferencesBergomi, L. (2005), “Smile Dynamics 2”, Risk, October, 67-73Cox, J. C. (1975), Notes on Option Pricing I: Costant Elasticity ofVariance Diffusions Unpublished Note, Stanford UniversityLipton, A. (2002), “The vol smile problem”, Risk, February, 81-85Merton, R. (1973), “Theory of Rational Option Pricing”, The BellJournal of Economics and Management Science 4(1), 141-183Scott L. (1987) “ Option pricing when the variance changes ran-domly: Theory, estimation and an application,” Journal of Financialand Quantitative Analysis, 22, 419-438Sepp, A. (2008) “Vix option pricing in a jump-diffusion model,” Risk,April, 84-89Sepp, A. (2011) “Efficient Numerical PDE Methods to Solve Cali-bration and Pricing Problems in Local Stochastic Volatility Models”,Presentation at ICBI Global Derivatives in ParisStein E., Stein J. (1991) “Stock Price Distributions with StochasticVolatility: An Analytic Approach”, Review of Financial Studies, 4,727-752

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