a consistent pricing model for index options and volatility derivatives

8/9/2019 A Consistent Pricing Model for Index Options and Volatility Derivatives

1/36Electronic copy available at: http://ssrn.com/abstract=1474691

A Consistent Pricing Model for

Index Options and Volatility Derivatives

Rama Cont

Center for Financial Engineering

Columbia University, New York

[email protected]

Thomas Kokholm

Aarhus School of Business

Aarhus University

[email protected]

September 17, 2009

Abstract

We propose and study a flexible modeling framework for the jointdynamics of an index and a set of forward variance swap rates writ-ten on this index, allowing options on forward variance swaps andoptions on the underlying index to be priced consistently. Our modelreproduces various empirically observed properties of variance swapdynamics and allows for jumps in volatility and returns.

An affine specification using Levy processes as building blocks leadsto analytically tractable pricing formulas for options on variance swapsas well as efficient numerical methods for pricing of European op-tions on the underlying asset. The model has the convenient fea-ture of decoupling the vanilla skews from spot/volatility correlationsand allowing for different conditional correlations in large and smallspot/volatility moves.

We show that our model can simultaneously fit prices of Europeanoptions on S&P 500 across strikes and maturities as well as optionson the VIX volatility index. The calibration of the model is done intwo steps, first by matching VIX option prices and then by matchingprices of options on the underlying.

Presented at the 19th Annual Derivatives Securities and Risk Management Conference

2009 and the Nordic Finance Network Workshop 2009.A first draft of this paper was completed while Thomas Kokholm held a visiting

scholar position at the Graduate School of Business (GSB), Columbia University, and hewishes to thank Bjrn Jrgensen and the accounting division at GSB for placing theirfacilities at his disposal. Moreover, thanks to Bjarne Astrup, Peter Lchte Jrgensen andElisa Nicolato for comments.


2/36Electronic copy available at: http://ssrn.com/abstract=1474691

Contents

1 Introduction 1

1.1 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Variance Swaps and Forward Variances 5

2.1 Variance Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Forward Variances . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Options on Forward Variance Swaps . . . . . . . . . . . . . . 7

3 A Model for the Joint Dynamics of Variance Swaps and the

Underlying Index 7

3.1 Variance Swap Dynamics . . . . . . . . . . . . . . . . . . . . 83.2 Dynamics of the Underlying Asset . . . . . . . . . . . . . . . 93.3 Pricing of Vanilla Options . . . . . . . . . . . . . . . . . . . . 113.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4.1 Normally Distributed Jumps . . . . . . . . . . . . . . 133.4.2 Exponentially Distributed Jumps . . . . . . . . . . . . 14

3.5 Impact of Jumps on the Valuation of Variance Swaps . . . . . 15

4 The VIX Index 16

4.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 VIX Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Implementation 19

5.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Conclusion 21

A Characteristic Functions 25

A.1 Normally Distributed Jumps . . . . . . . . . . . . . . . . . . 25A.2 Exponentially Distributed Jumps . . . . . . . . . . . . . . . . 26

B Proof of proposition 2 27

C Tables 28


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1 Introduction

Volatility indices such as the VIX index and derivatives written on suchindices have gained popularity in markets as tools for hedging volatility riskand as market-based indicators of volatility. Variance swap contracts [19] are

increasingly used by market operators to take a pure exposure to volatilityor hedge the volatility exposure of options portfolios.The existence of a liquid market for volatility derivatives such as VIX

options, VIX futures and a well developed over-the-counter market for op-tions on variance swaps, and the use of variance swaps and volatility indexfutures as hedging instruments for other derivatives have led to the need fora pricing framework in which volatility derivatives and derivatives on theunderlying asset can be priced in a consistent manner. In order to yieldderivative prices in line with their hedging costs, such models should bebased on a realistic representation of the joint dynamics of the underlyingasset and variance swaps written on this, while also be able to match theobserved prices of the liquid derivatives futures, calls, puts and varianceswaps used as hedging instruments.

In principle, any continuous-time model with stochastic volatility and/or jumps implies some joint dynamics for variance swaps and the underlyingasset price. Broadie and Jain [8] studied the valuation of volatility deriva-tives in the Heston model; Carr et al. [11] study the pricing of volatilityderivatives in models based on Levy processes. However, in many com-monly used models the dynamics implied for variance swaps is unrealistic[10, 5]: for example, exponential Levy models imply a constant varianceswap term structure while one-factor stochastic volatility models predictperfect correlation among movements in variance swaps at all maturities.

Also, as pointed out by Bergomi [5, 7], the joint dynamics of forward

volatilities and the underlying asset is neither explicit nor tractable in mostcommonly used models, which makes parameter selection/calibration diffi-cult and does not enable the user to choose a parameter to take a view onforward volatility. As a result, classical models such as the Heston modelor (time changed) exponential-Levy models are unable to match empiricalproperties of variance swaps and VIX options [5, 7, 6, 10, 21]. Some of theseissues can be tackled using multi-factor stochastic volatility models [10, 21]but these models remain incapable of reproducing finer features of the datasuch as the magnitude of the VIX option skew or the different conditionalcorrelations in large and small spot/volatility moves (see Table 1).

A new modeling approach, recently proposed by Bergomi [5, 7, 6] (see

also related work of Buhler [10] and Gatheral [21]) is one in which, insteadof modeling instantaneous volatility, one models directly the (forward)variance swaps for a discrete tenor of maturities. This approach, which canbe seen as the analog of the LIBOR market model for volatility modeling,turns out to be quite flexible and allows to deal with the issues raised above

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while retaining some tractability.These models are based on diffusion dynamics where market variables

are driven by a multidimensional Brownian motion. Recent price historyacross most asset classes has pointed to the importance of discontinuities inthe evolution of prices; this anecdotal evidence is supplemented by an in-

creasing body of statistical evidence for jumps in price dynamics [1, 2, 3, 17].Volatility indices such as the VIX have exhibited, especially during the re-cent crisis, large fluctuations which strongly point to the existence of jumps,or spikes, in volatility. Figure 1, which depicts the daily closing levels ofS&P 500 and the VIX volatility index from September 22nd, 2003 to Febru-ary 27th, 2009, reveals to the simultaneity of large drops in the S&P 500with spikes in volatility, which corresponds to the well-known leverage ef-fect. Comparing the relative changes in the two series (Figure 2) revealsthat, while there is already a negative correlation in small changes in theseries, large changes jumps exhibit an even stronger negative correlation,close to -1 (see Table 1). These observations are confirmed by a recent

study of Tauchen and Todorov [27], who find significant statistical evidenceof simultaneous jumps of opposite sign in the VIX and the underlying in-dex. These empirical facts need to be accounted for in a realistic modelfor variance swap dynamics. They are also important from a pricing per-spective: Broadie and Jain [9] find that, for a wide range of models andparameter specifications, the effect of discrete sampling on the valuationof volatility derivatives is typically small while the effect of jumps can besignificant. Jumps in volatility are also important in order to produce thepositive skew of implied volatilities of VIX options [6, 21].

1.1 Contribution

Following the approach proposed by Bergomi [5, 7, 6], the present workproposes an arbitrage-free modeling framework for the joint dynamics offorward variance swap rates along with the underlying index, which

1. captures the information in index option prices by matching the indeximplied volatility smiles.

2. is capable of reproducing any observed term structure of variance swaprates.

3. captures the information in options on VIX futures by matching theirprices/ implied volatility smiles.

4. implies a realistic joint dynamics of spot and forward implied volatili-ties, allowing in particular for jumps in volatility and returns.

5. allows for the spot/volatility correlation and the implied volatilityskews (of vanilla options) to be parametrized independently.

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6. is able to handle the term structure of vanilla skew separately fromthe term structure of volatility of volatility.

7. is tractable and enables efficient pricing of vanilla options, which is akey point for calibration and implementation of the model.

We address these different issues, while retaining tractability, by introducinga common jump factor which affects variance swaps and the underlying indexwith opposite signs. Describing these jumps in terms of a Poisson randommeasure leads to an analytically tractable framework, where VIX optionsand calls/puts on the underlying index can be simultaneously priced usingFourier-based methods. Our framework is in fact a class of models andallows for various specifications of the jump size distribution; we give twoworked-out examples and detail their implementation.

The difference between our modeling framework and the Bergomi model[7] is mainly the ability to meet points 5), 6) and 7) above. Also, thanks to asemi-analytic representation of call option prices, our model also satisfies the

tractability property 1), allowing efficient calibration to the whole impliedvolatility surface. This is an advantage over the Bergomi model where onlythe short-term implied volatility smile can be matched [5, 7].

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

22-Sep-03 25-May-04 27-Jan-05 29-Sep-05 05-Jun-06 07-Feb-07 10-Oct-07 13-Jun-08 17-Feb-09

Date

S&P

500

5

15

25

35

45

55

65

75

85

22-Sep-03 25-May-04 27-Jan-05 29-Sep-05 05-Jun-06 07-Feb-07 10-Oct-07 13-Jun-08 17-Feb-09

Date

VIX

Figure 1: Time series of the VIX index (bottom) depicted together with theS&P 500 (top) covering the period from September 22nd, 2003 to February27th, 2009.

1.2 Outline

Section 2 describes variance swap contracts, forward variance swap ratesand options on variance swaps. The model is described in Section 3 and two

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0.06 0.05 0.04 0.03 0.02 0.01 0 0.01 0.02 0.03

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 2: Daily relative changes in the VIX (vertical axis) vs daily relativechanges in the S&P 500 index (horizontal axis), September 22nd, 2003 toFebruary 27th, 2009.

Table 1: Conditional correlations between the daily returns on S&P 500 andthe VIX from September 22nd, 2003 to February 27th, 2009.

Abs. Return Unconditional < 0.5% > 0.5% < 1% > 1% < 5% > 5%

Correlation -0.74 -0.45 -0.78 -0.66 -0.79 -0.76 -0.93Observations 1368 677 691 1031 337 1348 20

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model specifications are presented. Section 4 discusses the VIX index andthe connection between forward variance swap rates and VIX index futures.Section 5 implements the two specifications of the model and examines theirperformance in jointly matching VIX options and options on the S&P 500.Section 6 concludes.

2 Variance Swaps and Forward Variances

Consider an underlying asset whose price Sis modeled as a stochastic process

(St)t0 on a filtered probability space

,F, {Ft}t0 ,P

, where {Ft}t0represents the history of the market. We assume the market is arbitrage-freeand prices of traded instruments are represented as conditional expectationswith respect to an equivalent pricing measure Q. We shall neglect in thesequel corrections due to stochastic interest rates.

2.1 Variance Swaps

The annualized realized variance of a (price) process S over a time gridt = t0 < ... < tk = T is given by

RVt,T =M

k

ki=1

log

StiSti1

2, (1)

where M is the number of trading days per year.A variance swap (VS) with maturity T initiated at t < T pays the

difference between the annualized realized variance of the log-returns RVt,Tless a strike called the variance swap rate VTt , determined such that thecontract has zero value at the time of initiation t.

For any semimartingale S, as supi=1,...,k |ti ti1| 0 the realized vari-ance converges to the quadratic variation of the log price:

ki=1

log

StiSti1

2Q [log S]T [log S]t .

Approximating the realized variance by the quadratic variation of the logreturns is justified when the sampling frequency is daily, as is the case inmost variance swap contracts, to warrant replacing the realized variance byits continuous counterpart [9]. The fixed leg paid in a variance swap is thenapproximated by

VTt = 1T tE ([log S]T [log S]t | Ft) , (2)

and we refer to this quantity as the (spot) variance swap (VS) rate prevailingat date t for the maturity T. An example of a variance swap term structureis given in Figure 3.

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0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

a ug -0 8 se p- 08 ok t- 08 n ov -0 8 de c- 08 j an -0 9 f eb -0 9 ma r-0 9 a pr -0 9 maj -0 9 j un -09

Figure 3: The term structure of 1 month forward variance swaps for theS&P 500 on August 20th, 2008.

2.2 Forward Variances

The forward variance, quoted at date t for the period [T1, T2] is the strikethat sets the value of a forward variance swap running from T1 to T2 to zeroat time t; it is given by

VT1,T2t =1

T2 T1E

[log S]T2 [log S]T1 | Ft

(3)

=(T2 t) VT2t (T1 t) VT1t

T2 T1 , (4)

where t < T1 < T2. The last equality follows easily by substitution of (2). Inparticular, as noted in [5, 6] forward variances have the martingale propertyunder the pricing measure.

Choosing t < s < T1 < T2 it follows by substitution of (3) and the useof the law of iterated expectations that

E

VT1,T2s | Ft

= VT1,T2t (5)

which shows that forward variances are martingales under the pricing mea-sure Q.

Assume that a set of settlement dates is given

T0 < T1 < ... < Tn

known as the tenor structure and that the interval between two tenor datesis fixed, i = Ti+1 Ti = (equal to 30 days if we use the tenor structureof the VIX futures). We define the forward variance over the time interval[Ti, Ti+1] as

Vit = VTi,Ti+1t . (6)

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2.3 Options on Forward Variance Swaps

A call option with strike K and maturity T1 on a forward variance swapfor the period [T1, T2] gives the holder the option to enter at date T1 intoa variance swap running from T1 to T2 with some predetermined strike K.Hence, the value at time T1 is

(T1, T1, T2, K) =

eT2T1

rsdsE (RVT1,T2 K | FT1)+

=eT2T1

rsds

VT1,T2T1 K+

, (7)

which gives us the time t value

(t, T1, T2, K) = eT2t

rsdsE

VT1,T2T1 K

+| Ft

. (8)

From this expression it is clear that having tractable dynamics for the for-

ward variance swap rates enable the pricing of options on variance swaps.

3 A Model for the Joint Dynamics of Variance

Swaps and the Underlying Index

Our goal is to construct a model which

allows for an arbitrary initial variance swap term structure. allows to specify directly the dynamics of the observable forward vari-

ance swaps Vit on a discrete tenor of maturities (Ti, i = 1..n).

allows for flexible modeling of the variance swap curve: for example,it should be able to accommodate the fact that variance swaps at thelong end of the maturity curve have a lower variability than those atthe short end.

allows for jumps in volatility [27] and in the underlying asset [1, 2, 3,17].

allows for a flexible specification of the spot price dynamics. is analytically tractable i.e. leads to efficient numerical methods for

pricing/calibration of calls/puts both on the underlying and volatility

derivatives such as options on forward variance swaps and options onforward volatility.

In order to achieve these goals, we first specify the dynamics of (a discretetenor of) forward variance swaps (Section 3.1) using an affine specification

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which allows Fourier-based pricing of European-type volatility derivatives.Once the dynamics of forward variance swaps has been fixed, we specify ajump-diffusion dynamics for the underlying asset which is compatible withthe variance swap dynamics (Section 3.2). Presence of a jump componentas well as a diffusion component in the underlying asset allows us to satisfy

this compatibility condition while simultaneously matching values/impliedvolatilities of options on the underlying asset (Section 3.3).

3.1 Variance Swap Dynamics

Given that the forward variance swap rate is a (positive) martingale underthe pricing measure, we model it as

Vit = Vi0 e

Xit

= Vi0 exp

t0

isds +

t0

ek1(Tis)dZs +

t0

R

ek2(Tis)xJ(dx ds)

,

(9)

where J(dxdt) is a Poisson random measure with compensator (dx) dt andZ a Wiener process independent of J. The martingale condition imposes

it = 1

22e2k1(Tit)

R

exp

ek2(Tit)x

1

(dx) . (10)

For t > Ti we let Vit = V

iTi

. This specification allows for jumps and theexponential functions inside the integrals allow to control the term structureof volatility of volatility. For example, ifk1 is large and k2 small, the diffusionZresults mostly in fluctuations at the short end of the curve, while the jumpsimpact the entire variance swap curve. Correlated Brownian factors can beadded if more flexibility in the variance swap curve dynamics is desired,e.g. correlation between movements in the short and the long end of thecurve. Likewise, functional forms other than exponentials can be used in(9), although the exponential specification is quite flexible as we will observein the examples.

Expressions such as (8) can be evaluated using Fourier-based methods[13, 24, 18] given the characteristic function ofXiTi , which in this case has asimple form:1

E

eiuXi

Ti

= exp

1

2u2

Ti

02e2k1(Tis)ds + iu

Ti

0isds

+Ti0

R

exp

iuek2(Tis)x

1 (dx) dt . (11)1We hope that the reader will not confuse the complex number i with the index in the

forward variance swap rate Vit .

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3.2 Dynamics of the Underlying Asset

Once the dynamics of forward variance swaps Vit for a discrete set of ma-turities Ti, i = 1..n has been specified, this imposes some constraints on the(risk neutral) dynamics of the underlying asset (St)t0. S should be suchthat

1. the discounted spot price St = exp(t0 (rs qs) ds)St is a (positive)

martingale, where qt is the dividend yield.

2. the dynamics of the spot price is compatible with the specificationof Vit , which puts the following constraint on the quadratic variationprocess [ln S] of the log-price:

E[ [ln S]Ti+1 [ln S]Ti |Ft] = Vit (12)To these constraints we add a third requirement, namely that:

3. the model values of calls/puts on S match the observed prices acrossstrikes and maturities.

Typically we need at least two distinct parameters/degrees of freedom in thedynamics of the underlying asset in order to accommodate points 2) and 3)above.

The Bergomi models [5] and [7] propose to achieve this by introducinga random local volatility function which is reset at each tenor date andchosen such at time Ti to match the observed value of V

iTi

. This procedureguarantees coherence between the variance swaps and the underlying assetdynamics but leads to a loss of tractability: even vanilla call options needto be priced by Monte Carlo simulation for maturities T > T1.

We adopt here a different approach which allows a greater tractabilitywhile simultaneously allowing for jumps in the volatility and the price. Infact, as we will see, introducing jumps is the key to tractability.

The underlying asset is driven by

a Brownian motion W, correlated with the diffusion component Zdriving the variance swaps: < W, Z >t= t.

a jump component, which is driven by the same Poisson randommeasure J which drives jumps in the variance. However, we allowdifferent jump amplitudes in the underlying and the forward variance.

Presence of a jump component as well as a diffusion component in the un-

derlying asset allows us to satisfy this compatibility condition while simul-taneously matching values/implied volatilities of options on the underlyingasset (Section 3.3).

For each interval [Ti, Ti+1) we introduce a stochastic diffusion coefficienti and a function ui(x, V

i) which expresses the size of the jump in the

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underlying asset in terms of the jump size x in forward variance level Vi andVi itself. We will give in section 3.4 some simple and flexible specificationsfor this function ui but most developments below hold for arbitrary choiceof ui.

We will present here the detailed computations in the (typically useful)

case where T = Tm, for m = 1,...,n, but the analysis can be easily general-ized to any maturity. The dynamics of the underlying asset is then specifiedas

STm = S0 exp

Tm0

(rs qs) ds +m1i=0

i (Ti+1 Ti) + i

WTi+1 WTi

+m1i=0

Ti+1Ti

R

ui

x, ViTi

J(dx ds)

,

(13)

where i =

12

2i

Reui

x,Vi

Ti 1 (dx) and the is are stochasticand revealed at time Ti by relation (16) below to match the known variance

swap value ViTi at this time point. The drift term i is also stochastic andFTi-measurable. The random jump measure J in the index dynamics is thesame as that in the VS dynamics, so the index and the variance swaps jumpsimultaneously. ui is a deterministic function of x and V

iTi

chosen to matchthe observed volatility smiles implied by prices of options on the spot. W iscorrelated with Z with a correlation .

As far as pricing of vanilla instruments is concerned, the model can beviewed simply as a flexible (and analytically tractable) parameterization ofthe joint distribution of forward variance swap rates and the underlying asseton a set of tenor dates. At this level the only assumption we are making isthat this joint distribution is infinitely divisible [26] and our model is just aflexible parameterization of the Levy triplet of the distribution. This makesit particularly easy to price any payoff which involves these variables onlyat tenor dates.

The assumption of a common jump factor which affects the varianceswaps and the underlying in opposite directions is not only analytically con-venient but quite realistic from an empirical perspective. As revealed inTable 1, while the unconditional correlation of daily returns of the VIX in-dex and the S&P 500 from September 22nd, 2003 to February 27th, 2009is 74%, the conditional correlation between the two sub-series for dailymoves in the S&P 500 less than 0.5% drops to

0.45 while the conditional

correlation of moves greater than 5%, which can be interpreted as jumps, is93%, close to -100%. This observation is also in agreement with the find-ings reported by Tauchen and Todorov [27] using non-parametric methods.This feature, which has no equivalent in diffusion-based stochastic volatilitymodels such as [7, 21], is a generic property of our framework.

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Given the dynamics (13) of the underlying asset, the quadratic variationover the time interval [T0, Tm] is given by

[log S]Tm [log S]T0 =m1

i=02i (Ti+1 Ti) +

Ti+1

Ti Rui

x, ViTi

2

J(ds dx) .

(14)By taking expectation on (14) the variance swap rate in (2) equals

VTmT0 =m1i=0

Ti+1 TiTm T0

E

2i | FT0

+ E

R

ui

x, ViTi2

(dx) | FT0

,

and the forward variance at time t for the interval [Ti, Ti+1] equals

Vit = E

2i | Ft

+ E

R

ui

x, ViTi2

(dx) | Ft

. (15)

Equation (15) has to hold at all times, but since V

i

t is a martingale we justneed to ensure that at time Ti

ViTi = 2i +

R

ui

x, ViTi2

(dx) . (16)

Having fixed (dx) it is seen that any observed forward variance structurecan be matched by an appropriate choice of the is which leaves the param-eters in ui free to calibrate to option prices.

3.3 Pricing of Vanilla Options

Let us now explain the procedure for pricing European calls and puts on S

in this framework. The aim is to compute in an efficient manner the valueof call options of various strikes and maturities at, some initial date t = 0:

C(0, S0, K , T m) = eTm

0 rsdsE[(STm K)+|F0]. (17)

Denote by F(Z,J)t the filtration generated by the Wiener process Z and thePoisson random measure J. By first conditioning on the factors driving thevariance swap curve and using the iterated expectation property

C(0, S0, K , T m) = eTm

0 rsdsE[E[(STm K)+|F(Z,J)Tm ] |F0] (18)

we obtain a mixing formula for valuing call options:Proposition 1 The value C(0, S0, K , T m) of a European call option withmaturity Tm and strike K is given by

C(0, S0, K , T m) = E[CBS (UTm, K , T m; )], (19)

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where CBS(S,K,T; ) denotes the Black-Scholes formula for a call optionwith strike K and maturity T:

2 =1

Tm

m1

i=02i

1 2

(Ti+1 Ti) , (20)

and UTm is aF(Z,J)Tm -measurable random variable given by

UTm = S0 exp

m1i=0

1

22i

2 +

R

eui

x,Vi

Ti

1

(dx)

(Ti+1 Ti)

ZTi+1 ZTi

i +

Ti+1Ti

R

ui(x, ViTi

)J(dx ds)

(21)

and the expectation in (19) is taken with respect to the law of (Z, J).

Proof. Conditional on F(Z,J)Tm , the increments of Z and the paths of thevariance swap rates, and in particular ViTi , thus the corresponding i areknown for i = 0, 1,...,m from equation (16). Moreover

WTi+1 WTi d=

ZTi+1 ZTi

+

(1 2)

WTi+1 WTi

,

where the Wiener processes Z and W are independent. The process St cantherefore be decomposed as

STm = UTm expTm

0

(rs qs) ds +m1

i=0 1

22i 1

2

(Ti+1 Ti)+i

1 2

WTi+1 WTi

,

where 2i = ViTi

Rui

x, ViTi2

(dx) and

UTm = S0 exp

m1i=0

1

22i

2 +

R

eui

x,Vi

Ti

1

(dx)

(Ti+1 Ti)

ZTi+1 ZTi

i +Ti+1

Ti Rui(x, V

iTi

)J(dx ds)

is F(Z,J)Tm -measurable. In distribution we have

STmd= UTme

Tm

0(rsqs)ds+Tm+WTm ,

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where is given in (20) and

=1

Tm

m1i=0

12

2i

1 2 (Ti+1 Ti) , (22)where we notice that = 12 ()2. Hence, given F(Z,J)Tm , the inner con-ditional expectation in (18) reduces to the evaluation of the Black-Scholesformula

E[(STm K)+|F(Z,J)Tm ] = CBS (UTm, K , T m; ) ,where depends on all the uis through (16) and (20).

This result has interesting consequences for pricing and calibration ofvanilla contracts. Note that the outer expectation can be computed byMonte Carlo simulation of the Z and J: with N simulated sample paths forZ and J we obtain the following approximation

CN =1

N

Nk=1

CBSU(k)Tm , K , T m; (k)N C(0, S0, K , T m) . (23)Since the averaging is done over the variance swap factors Z and J, this isa deterministic function of the parameters in the uis. This will prove usefulwhen calibrating the model using option data, since we do not have to runthe N Monte Carlo simulations for each calibration trial.

Equation (23) thus allows to calibrate the model to the entire impliedvolatility surface in an efficient manner, in contrast to the Bergomi modelwhere it is only possible to calibrate to at-the-money slopes of the impliedvolatilities (ATM skews).

3.4 Examples

Different classes of models in the above framework can be obtained by vari-ous parameterizations of the Levy measure (dx) describing the jumps andfor the functions ui appearing in (13). In this paper we implement twomodel parameterizations; normal jumps and double exponential jumps inthe variance swaps and the index.

3.4.1 Normally Distributed Jumps

In the first example, we specify the Levy measure as

(dx) = 1

2e

(xm)2

22 dx, (24)

where m and 2 is the mean and variance in a normal distribution and the intensity of the jumps. With (dx) specified the characteristic function

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in (11) can be calculated (see (41) in Appendix A) and prices of options onthe variance swaps can be computed by Fourier transform methods.

We now consider the following specification for ui, which relates thejumps in the variance swap to the jumps in the underlying asset

uix, ViTi = ViTiVi0

1

2

bix , (25)

but any functional form can be used as long as (16) leads to positive valuesfor i.

This gives us the is at time Ti

2i = ViTi V

iTi

Vi0

R

(bix)2 (dx)

= ViTi ViTiVi0

b2i m2 + b2i

2

.

In order to achieve non-negative values for 2i we require

b2i m2 + b2i

2 Vi0 . (26)

3.4.2 Exponentially Distributed Jumps

To introduce an asymmetry in the tail of the jump size distribution we canuse a double exponential distribution [23]:

(dx) =

p+e+x1x0 + (1 p) e|x|1x


17/36

jumps in the VIX futures. Together with negative correlation between Zand W, this feature enables the model to generate positive skews for theimplied volatility of VIX options and negative skews for the implied volatil-ity of calls and puts on the underlying index, as observed in empirical data.This property is achieved without the need to add extra volatility factors, as

proposed recently in e.g. [10, 20, 21] where multi-factor mean-reverting dy-namics on volatility and volatility of volatility are imposed to accommodatethis behavior, resulting in a loss of tractability.

3.5 Impact of Jumps on the Valuation of Variance Swaps

As noted above, recent price history across most asset classes has pointedto the importance of discontinuities in the evolution of prices; this is sup-plemented by an increasing body of statistical evidence for jumps in pricedynamics [1, 2, 3, 17]. Before moving on we would like to digress on the effectof jumps in the underlying index dynamics when pricing variance swaps.

As noted by Neuberger [25], if the underlying asset follows continuousdynamics, its quadratic variation and hence also the payoff from a varianceswap can be replicated by continuous rebalancing of a position in the un-derlying asset and a static position in a log contract, which in turn can bereplicated by a static portfolio of calls and puts [12, 19]:

1

T t ([log S]T [log S]t) =2

T tT

t

dStSt

log STSt

and then under discrete sampling

M

k

k

i=1 logSti

Sti12

= 2M

k

k

i=1 Si

Si1 log

Si

Si1 .The left hand side is the realized variance paid and the right hand side isthe payoff from the hedging strategy. Figure 4 depicts the day to day profit-loss from holding a long position in the replicating portfolio and a shortposition in the three month variance swap. Variance Point (VP) is definedas realized variance multiplied by 10,000. As can be seen, the replicatingportfolio performs relatively well most of the days, but days in which theindex moved considerably show an error of up to -1.5 VP. Table 2 showsthe error from holding the portfolio relative to the realized variance over thethree month p eriod. The maximum error from following this strategy arisesin the June 2008 - September 2008 variance swap and results in an error of0.243%. These four real examples indicate that the error from assumingcontinuous dynamics have been historically small.

Sometimes this is taken as an argument that jumps are not significantwhen variance swaps are priced. This can very well be a false conclusion,

15


18/36

which can be realized from the following. When assuming continuous dy-namics, the error that arises when valuing variance swap rates is given [14]by

2T

tE

Tt

R

ey 1 y y

2

2

(dy)ds | Ft

,

where y is the jump size in the index. That this expectation is small un-der the physical measure does not entail that the mean of the error underthe pricing measure should be zero. This common fallacy is akin to sayingthat since crashes occur infrequently deep out-of-the-money puts should bepriced at zero: recent experience shows that such mistaken assertions canbe costly. In fact adding jumps with negative mean value to the log-pricedynamics should increase the value of the variance swap rate, since beinglong a variance swap can be seen as an insurance against downward move-ments in the underlying asset and therefore have a risk premium attachedto it [15].

4 The VIX Index

As mentioned, the market for options on the VIX index is well developed,and unless data on options on variance swaps is available, information stem-ming from this market should be used when the dynamics of the varianceswaps are specified. For completeness we describe the VIX index in thefirst subsection, but the reader is referred to [16] and [14] for a detailed dis-cussion. In the second subsection we derive our choice for the VIX futuresdynamics given the dynamics of the forward variance swaps .

4.1 Description

The Chicago Board Options Exchange (CBOE) introduced the volatilityindex, VIX, in 1993, and it has since become the industry benchmark formarket volatility. The VIX index provides investors with a quote on theexpected market volatility over the next 30 calendar days. In September2003 the VIX was revised such that the index is now calculated on thebasis of put and call options on S&P 500 instead of the S&P 100 index.Furthermore, the index was changed to calculate the expected volatility onthe basis of options in a wide range of strike prices, where the original VIXindex was based on at-the-money strikes only.

The general formula used in the VIX calculation at time t is

V STt =2

T ti

KiK2i

erTt(Tt)Q (Ki, T; t) 1

T t Ft

K0 12 , (29)

where T is time of expiration, Ki the strike price of the ith out-of-the money

16


19/36

option, Ki the resolution of the strike grid:

Ki =Ki+1 Ki1

2.

For the lowest strike, Ki is the difference between the lowest and second

lowest strikes. This also applies to the highest strike. rTt is the risk-free

interest rate to expiration, Q (Ki, T; t) the midpoint between the bid andask prices on the option with strike Ki and maturity T, either a call ifKi > Ft or a put if Ki < Ft, where Ft is the forward S&P 500 index price.K0 is the first strike below the forward index level Ft. Normally, no optionswill expire in exactly 30 days. Therefore, CBOE interpolates b etween V ST1tand V ST2t

V IXt = 100

365

30

(T1 t) V ST1t

NT2 30NT2 NT1

+ (T2 t) V ST2t30 NT1

NT2 NT1

,

(30)

where NT1 and NT2 denote the actual number of days to expiration for thetwo options maturities, NT1 < 30 < NT2 .The VIX is often presented as an indicator of the expected volatility over

the next 30 days. It is not immediately clear from either (29) or (30) thatthis is in fact the case. For notational simplicity assume that the jumpsexhibit finite variation. Then it can be shown that the realized variance ofreturns can be decomposed into three components

[log S]T [log S]t =2

T tFt

0

1

K2(K ST)+ dK

+

Ft

1

K2(ST K)+ dK+

T

t 1

Fs 1

Ft dFsTt

R

ey 1 y y

2

2

J(dy ds)

, (31)

where J is a Poisson random measure with Levy measure (dx), drivingthe jumps in the underlying asset. Notice how the realized variance canbe replicated by trading in options and futures up to a discontinuous jumpcomponent.

Taking expectations with respect to the pricing measure in (31) we obtain

VTt =2

T t eT

trsds

0

Q (K, T; t)

K2dK

2T tE

Tt

R

ey 1 y y22 (dy) ds | Ft . (32)

Equation (29) is a discretization of the first term in (32). The extra term

1Tt

FtK0

12

in (29) is a contribution due to the discretization around Ft.

17


20/36

The square of the VIX index is thus a model free estimate of the expectedvolatility over the next 30 days given continuous dynamics of the underlyingbut under general price processes the error term is given by the last term in(32):

V IX2t = Vt+30dayst +

2

T tE Tt Rey 1 y y2

2 (dy) ds | Ft .(33)

4.2 VIX Futures

Let V IXit denote the VIX futures price for the interval [Ti, Ti+1] seen attime t. For t = Ti we have

V IXiTi2

= ViTi + 2

R

(eui

x,Vi

Ti

1 ui

x, ViTi

ui x, ViTi22

)(dx) .

We also have the martingale property for t < Ti

V IXit = E V IXiTi |FtBy Jensens inequality for convex functions

V IXit2 E V IXiTi2 |Ft

= Vit + 2E

R

eui

x,Vi

Ti

1 ui

x, ViTi

ui x, ViTi22

(dx) | Ft

,

(34)

since Vit is a martingale. Equation (34) shows that there is a convexitycorrection connecting VIX futures to forward variance swap rates.

To obtain a tractable framework, we first characterize the dynamics ofVit then propose an approximation which has a closed-form characteristic

function.

Proposition 2 The process

Vit has the multiplicative decompositionVit =

Vi0M

itA

it

where Ait is a finite variation process and Mit is an exponential martingale

given by

Mit = exp

t

0isds +

1

2 t

0ek1(Tis)dZs +

t

0 R1

2ek2(Tis)xJ(dx ds)

,

(35)

where

it = 1

82e2k1(Tit)

R

exp

1

2ek2(Tit)x

1

(dx) . (36)

18


21/36

The proof of this result is given in Appendix B.Using this result we approximate V I Xit with

V IXit V IXi0Mit = V I Xi0 exp(Yit ) . (37)

where Y

i

t = ln M

i

t . This approximation leaves the initial level of the V IX

i

and the volatility of V IXi untouched and thus is relevant for pricing VIXderivatives. As in the case of the forward variance swaps, once the Levymeasure (dx) is specified, the characteristic function for the exponent in(37) can be easily computed, so options on the VIX index can be priced byFourier transform methods.2

5 Implementation

In this section the model specifications described in Section 3.4 are imple-mented on data on the VIX and the S&P 500 implied volatility smiles.

5.1 Data

We assess the performance of the model using prices from August 20th, 2008on a range of VIX put and call options for five maturities, VIX futures forthe same maturities, the dividend yield on S&P 500, call and put options onS&P 500 for six maturities and for the various maturities we also have thecorresponding prices of the futures on S&P 500, from which we also derivea discount curve. We also observe 3 month forward variance swap rates forvarious maturities. The variance swap rates have been converted to forward1 month variance swap rates by simple linear extrapolation and these aredepicted in Figure 3. Options for which the bid price is zero were removed.

5.2 Calibration

The calibration of the model consists of three steps:

1. Determine the parameters controlling the variance swap dynamics bycalibration to VIX options using the characteristic function in (42) or(44) and Fourier transform methods.

2. Use the parameters from the first step to simulate N paths of thevariance swaps and store the increments of Z, the jump times andjump sizes along with the ViTis.

3. Calibrate to options on the index recursively by use of (23).

2The characteristic functions for the specifications in Section 3.4 are given in AppendixA.

19


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In the first step we compute model prices by the fast Fourier transformtechnique described in [13] using the revised method in [18]. The calibrationis performed by minimizing the sum of squared errors weighted by the inversebid-ask spread across all maturities and strikes on out-of-the-money optionson the VIX futures

SE =

options

1

QAsk QBid (QMarket,Mid QModel)2 , (38)

using a gradient-based minimization algorithm.The resulting parameters for the two specifications are shown in Table 3

and it also includes the resulting average relative percentage bid-ask errorfrom the calibration,

Error =1

# {options}

options

max

(QModel QAsk)+ , (QBid QModel)+

QMarket,Mid.

(39)We see how both model specifications are able to achieve very low calibrationerror. Note the high value of k1 compared to k2, which implies (in therisk-neutral dynamics) that the diffusion component Z is mainly causingfluctuations at the short end of the variance swap curve, while the jumpsimpact the entire curve.

Figures 5 and 6 depict the (Black) implied volatility for the bid, ask, midand model prices of VIX options as a function of moneyness. We observethat model prices fit very well within the bid-ask spread for almost all theobservations in both specifications. This performance should be comparedwith flat model implied volatilities in the Bergomi model [5] and downwardsloping volatilities in the Heston model [22]. The Bergomi model [7] is able

to generate positive skews on volatility of volatility via the use of a Markovfunctional mapping for the forward variances, but it should be noted thatthe model can only price vanilla options on the underlying index by MonteCarlo simulation.

Step 2 is done by discretization of (9). In this example we have used2 105 simulated sample paths. Simulation of the jumps is straightforward,since they arrive at constant intensity and the jump sizes are computed bydraws from a normal/exponential distribution times the scaling ek2(Ti),where is the jump time. For details on how to simulate the increments inthe variance swaps due to the diffusion the reader is referred to [5].

In step 3 we have fixed the correlation parameter between the continuous

components to = 0.45.3

Then, the last step is implemented step-wise bymatching the model prices (23) to market prices on the shortest maturity

3Instead one could also specify a correlation i on each interval and include that pa-rameter in the calibration for each interval. We tried this, but the performances of themodels did not improve with this added flexibility.

20


23/36

out-of-the-money options. Again, this is achieved by minimizing the SE(38). This yields b0. Then b1 are found in the same way by calibrating tothe next maturing options by the use ofb0 for the first interval in (21). Theremaining bis are estimated in a similar manner.

The estimated parameters for the two different specifications of the jump

size distributions are shown in Table 4 along with the pricing error in (39)for each maturity. The mean and standard deviations of the jumps before

scaling with

ViTi/Vi0

12 are also included. The values of bi for i = 0,..., 3

relate to the monthly distributions of the index up to the maturity of theoptions expiring on December 19th, 2008. For the last two option maturi-ties the interval between the expirations are three months and hence bi fori = 4, 5 relate to the three month distributions of the index. Double expo-nential jumps perform slightly better on the first interval, but there is nota strong difference between the two specifications, showing that the modelperformance is not very sensitive to the choice of the jump size distribution.

Figures 7 and 8 show the result of the calibration to the S&P 500 options.

The fit of the two specifications are practically indistinguishable and overallthe calibrations perform well for both with somewhat less success on theshortest maturity.

We conclude this section with an examination of the size of the jump er-ror term when valuing the forward variance swap rates Vi under the assump-tion of continuous dynamics given the models implemented in this paper aretrue. Recall that the error is given by

i = 2E

R

eui

x,Vi

Ti

1 ui

x, ViTi

ui x, ViTi22

(dx) | F0

.

(40)

In Table 5 the absolute errors implied by the calibration of the two modelspecifications are computed for each interval. Moreover, the errors relativeto the initial forward variance swap value are reported. For the first intervalit is 2.2%/2.3% and then it roughly increases as i increases. The reportederrors for the last two intervals are taken as the average of the three monthlyerrors relative to the average of the three monthly forward variances coveringthe same intervals.

6 Conclusion

We have presented a model for the joint dynamics of a set of forward vari-

ance swap rates along with the underlying index. Using Levy processesas building blocks, this model leads to a tractable pricing framework forvariance swaps, VIX futures and vanilla call/put options, which makes cal-ibration of the model to such instruments feasible. This tractability featuredistinguishes our model from previous attempts [5, 4, 21] which only allow

21


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for full Monte Carlo pricing of vanilla options.Our model reproduces salient empirical features of variance swap dynam-

ics, in particular the strong negative correlation of large index moves withlarge moves in the VIX and the positive skew observed in implied volatilitiesof VIX options, by introducing a common jump component in the variance

swaps and the underlying asset. Using two different specifications for thejump size distribution (Levy measure) we have illustrated the feasibility ofthe numerical implementation, as well as the capacity of the model to matcha complete set of market prices of vanilla options and options on the VIX.Our model can be used to price and hedge various payoffs sensitive to for-ward volatility, such as cliquet or forward start options, as well as volatilityderivatives, in a manner consistent with market prices of simpler instrumentssuch as calls, puts or variance swaps which are typically used for hedgingthem.

References

[1] Y. Ait-Sahalia and J. Jacod, Estimating the Degree of Activity ofJumps in High Frequency Financial Data, Annals of Statistics, forth-coming, (2008).

[2] T. Andersen, L. Benzoni, and J. Lund, An Empirical Investiga-tion of Continuous-Time Equity Return Models, Journal of Finance, 57(2002), pp. 12391284.

[3] O. Barndorff-Nielsen and N. Shephard, Variation, Jumps, Mar-ket Frictions and High-Frequency Data in Financial Econometrics, inAdvances in Economics and Econometrics: Theory and Applications,

Ninth World Congress, R. Blundell, W. K. Newey, and T. Persson,eds., Econometric Society Monographs, Cambridge Univ Press, 2007,pp. 328372.

[4] L. Bergomi, Smile Dynamics, Risk, 17 (2004), pp. 117123.

[5] , Smile Dynamics II, Risk, 18 (2005), pp. 6773.

[6] , Dynamic properties of smile models, in Frontiers in QuantitativeFinance: Volatility and Credit Risk Modeling, R. Cont, ed., Wiley,2008, ch. 3.

[7] , Smile Dynamics III, Risk, 21 (2008), pp. 9096.

[8] M. Broadie and A. Jain, Pricing and Hedging Volatility Derivatives,The Journal of Derivatives, 15 (2008), pp. 724.

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[9] , The Effect of Jumps and Discrete Sampling on Volatility andVariance Swaps, International Journal of Theoretical and Applied Fi-nance, 11 (2008), pp. 761797.

[10] H. Buehler, Consistent Variance Curve Models, Finance and Stochas-

tics, 10 (2006), pp. 178203.[11] P. Carr, H. Geman, D. Madan, and M. Yor, Pricing Options on

Realized Variance, Finance and Stochastics, 9 (2005), pp. 453475.

[12] P. Carr and D. Madan, Towards a Theory of Volatility Trading, inVolatility: New Estimation Techniques for Pricing Derivatives, R. Jar-row, ed., Risk Publications, 1998.

[13] P. Carr and D. Madan, Option Valuation using the Fast FourierTransform, Journal of Computational Finance, 2 (1999), pp. 6173.

[14] P. Carr and L. Wu, A Tale of Two Indices, Journal of Derivatives,

13 (2006), pp. 1329.

[15] , Variance Risk Premiums, Review of Financial Studies, 22 (2009),pp. 13111341.

[16] CBOE, VIX: CBOE Volatility Index ,http://www.cboe.com/micro/vix/vixwhite.pdf, (2003).

[17] R. Cont and C. Mancini, Nonparametric Tests for Analyzing theFine Structure of Price Fluctuations, Financial Engineering Report2007-13, Columbia University, 2007.

[18] R. Cont and P. Tankov, Financial Modelling with Jump Processes,Chapman & Hall/CRC, 2004.

[19] K. Demeterfi, E. Derman, M. Kamal, and J. Zou, A Guide toVolatility and Variance Swaps, Journal of Derivatives, 6 (1999), pp. 932.

[20] D. Duffie, J. Pan, and K. Singleton, Transform Analysis andAsset Pricing for Affine Jump-diffusions, Econometrica, 68 (2000),pp. 13431376.

[21] J. Gatheral, Consistent Modeling of SPX and VIX Options, in Bache-lier Congress, July 2008.

[22] S. Heston, A Closed-Form Solution for Options with StochasticVolatility with Applications to Bond and Currency Options, Review ofFinancial Studies, 6 (1993), pp. 327343.

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[23] S. Kou, A Jump-Diffusion Model for Option Pricing, ManagementScience, 48 (2002), pp. 10861101.

[24] A. Lewis, A Simple Option Formula For General Jump-DiffusionAnd Other Exponential Levy Processes, tech. report, OptionCity.net,

http://optioncity.net/pubs/ExpLevy.pdf, 2001.[25] A. Neuberger, The Log Contract: A New Instrument to Hedge

Volatility, Journal of Portfolio Management, 20 (1994), pp. 7480.

[26] K. Sato, Levy Processes and Infinitely Divisible Distributions, Cam-bridge University Press, 1999.

[27] V. Todorov and G. Tauchen, Volatility Jumps, working paper,Duke University, 2008.

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28/36

A.2 Exponentially Distributed Jumps

Variance Swaps

The characteristic function in (11) takes the form

E eiuXiTi = exp12

2iu 1 e2k1Ti

2k1 1

22u2 1 e

2k1Ti

2k1

iuTi0

p+

+ ek2(Tis)+

(1 p) + ek2(Tis)

1

ds

+

Ti0

p+

+ iuek2(Tis)+

(1 p) + iuek2(Tis)

1

ds

.

(43)

VIX Futures

Recall from (37) that V IXit V IXi0Mit where Mit = exp(Yit ) is a positiveexponential martingale. In this specification the characteristic function ofYiTi is given by

E

eiuYi

Ti

= exp

1

82iu

1 e2k1Ti2k1

18

2u21 e2k1Ti

2k1

iuTi0

p+

+ 12ek2(Tis)+

(1 p) +

12ek2(Tis)

1

ds

+

Ti0

p+

+ 12 iuek2(Tis)+

(1 p) +

12iue

k2(Tis) 1

ds

.

(44)

26


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B Proof of proposition 2

We can express Vit in (9) as

Vit = Vi0 +

t

0

Visek1(Tis)dZs

+t0

R

Vis

exp

ek2(Tis)x

1

J(ds dx)

t0

R

Vis

exp

ek2(Tis)x

1

(dx) ds.

Applying the Ito formula to

Vit we obtainVit =

Vi0 +

1

2

t0

Vis

22e2k1(Tis)

1

4

Vis

32

ds

t

0 R Vis expek2(Tis)x 11

2 Vis 1

2 (dx) ds

+

t0

Visek1(Tis)

1

2

Vis

12 dZs

+

t0

R

Vis

12

exp

1

2ek2(Tis)x

1

J(ds dx)

=

Vi0 +1

2

t0

Vise

k1(Tis)dZs

t0

R

Vis

1

2

exp

ek2(Tis)x

1

1

82e2k1(Tis)

(dx) ds

+ t

0 RVisexp1

2

ek2(Tis)x 1J(ds dx) .This can be written as

Vit =

Vi0 exp

t0

isds + 12t0

ek1(Tis)dZs (45)

+

t0

R

1

2ek2(Tis)xJ(dx ds)

,

where

it =

1

42e2k1(Tit) 1

2 R exp

ek2(Tit)x

1

(dx) .

The stochastic integrals in (45) being processes with independent incre-ments, a straightforward application of the exponential formula for Poissonrandom measures [18, Prop. 3.6.] yields the multiplicative decompositionin Proposition 2 where M is given by (35)-(36).

27


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31/36

Table 4: Model parameters calibrated from the S&P 500 volatility smiles on

August 20th, 2008 with the correlation between the two Brownian compo-nents set to -0.45.

i 0 1 2 3 4 5

Gaussian jumps

bi -0.151 -0.159 -0.152 -0.173 -0.187 -0.193bim -0.081 -0.086 -0.082 -0.093 -0.101 -0.104|bi| 0.037 0.039 0.038 0.043 0.046 0.048Error (%) 8.8 0.6 1.1 1.8 1.9 2.7

Double exponential jumps

bi -0.153 -0.159 -0.149 -0.169 -0.184 -0.192 bip+ bi(1p) -0.030 -0.031 -0.029 -0.033 -0.036 -0.038b2ip

2++

b2i(1p)

2

12

0.034 0.035 0.033 0.037 0.040 0.042

Error (%) 6.9 0.8 1.4 2.0 2.1 2.7

Table 5: The errors arising from approximating the index dynamics witha continuous process given the model specifications described in this paperare true.

i 0 1 2 3 4 5

Gaussian jumps

i 102 0.100 0.119 0.133 0.157 0.225 0.252iVi0

(%) 2.2 2.4 2.5 2.8 3.8 4.1

Double exponential jumps

i

102 0.105 0.120 0.131 0.157 0.218 0.252

iVi0

(%) 2.3 2.4 2.4 2.8 3.6 4.1

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-2

-1.5

-1

-0.5

0

0.5

1

1.5

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64VP

Sep 2007 - Dec 2007Dec 2007 - Mar 2008

Mar 2008 - Jun 2008

Jun 2008 - Sep 2008

Figure 4: The profit-loss from day to day in variance points (VP) fromholding a long position of the hedge portfolio and a short variance swap forfour different three month variance swaps. The x-axis is trading days duringthe three month period.

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0.8 1 1.2 1.4 1.6 1.80.4

0.6

0.8

1

Expiry: 170908

1 1.5 2 2.5

0.2

0.4

0.6

0.8

1Expiry: 221008

1 1.5 2 2.50.2

0.4

0.6

0.8

1Expiry: 191108

1 1.5 2 2.50.2

0.4

0.6

0.8

1Expiry: 171208

1 1.5 2 2.50.2

0.4

0.6

0.8

1Expiry: 210109

Mid

Model

Bid

Ask

Figure 5: VIX implied volatility smiles on August 20th 2008 for themodel with normally distributed jump sizes plotted against moneynessm = K/V IXt on the x axis.

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0.8 1 1.2 1.4 1.6 1.80.4

0.6

0.8

1

Expiry: 170908

1 1.5 2 2.5

0.2

0.4

0.6

0.8

1Expiry: 221008

1 1.5 2 2.50.2

0.4

0.6

0.8

1Expiry: 191108

1 1.5 2 2.50.2

0.4

0.6

0.8

1Expiry: 171208

1 1.5 2 2.50.2

0.4

0.6

0.8

1Expiry: 210109

Mid

Model

Bid

Ask

Figure 6: VIX implied volatility smiles on August 20th 2008 for the modelwith double exponentially distributed jump sizes plotted against moneynessm = K/V IXt on the x axis.

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0.7 0.8 0.9 1 1.10.1

0.2

0.3

0.4

0.5Expiry: 190908

0.7 0.8 0.9 1 1.1 1.2

0.1

0.2

0.3

0.4

0.5Expiry: 171008

0.8 0.9 1 1.1 1.20.1

0.2

0.3

0.4

0.5Expiry: 211108

0.4 0.6 0.8 1 1.2 1.40.1

0.2

0.3

0.4

0.5Expiry: 191208

0.8 1 1.2 1.40.1

0.2

0.3

0.4

0.5Expiry: 200309

0.5 1 1.50.1

0.2

0.3

0.4

0.5Expiry: 190609

Mid

Model

Bid

Ask

Figure 7: S&P 500 implied volatility smiles on August 20th 2008 for themodel with normally distributed jump sizes plotted against moneyness m =K/St on the x axis.

33


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0.7 0.8 0.9 1 1.10.1

0.2

0.3

0.4

0.5Expiry: 190908

0.7 0.8 0.9 1 1.1 1.2

0.1

0.2

0.3

0.4

0.5Expiry: 171008

0.8 0.9 1 1.1 1.20.1

0.2

0.3

0.4

0.5Expiry: 211108

0.4 0.6 0.8 1 1.2 1.40.1

0.2

0.3

0.4

0.5Expiry: 191208

0.8 1 1.2 1.40.1

0.2

0.3

0.4

0.5Expiry: 200309

0.5 1 1.50.1

0.2

0.3

0.4

0.5Expiry: 190609

Mid

Model

Bid

Ask

Figure 8: S&P 500 implied volatility smiles on August 20th 2008 for themodel with double exponentially distributed jump sizes plotted against mon-eyness m = K/St on the x axis.

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a consistent pricing model for index options and volatility derivatives

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