stochastic volatility of volatility and variance risk premia

Stochastic dynamics of variance risk premia through stochasticvolatility of volatility

Ole E. Barndorff-NielsenThe T.N. Thiele Centre for Mathematics in Natural Science,

Department of Mathematical Sciences,& CREATES, School of Economics and Management

Aarhus University& Technische Universitat Munchen: Institute for Advanced Studies

Almut E. D. VeraartCREATES, School of Economics and Management,

Aarhus University

January 20, 2011

Abstract

This paper introduces a new class of stochastic volatility models which allows for stochasticvolatility of volatility. Such models are given by volatility modulated non–Gaussian OrnsteinUhlenbeck processes. We study the probabilistic properties of such models both under the physicaland under the risk neutral probability measure, where we focus in particular on the role of thevolatility of volatility component in the integrated variance. Further, we show that the volatility ofvolatility component can be used to account for the leverage effect in a novel way and we discusshow long memory can be introduced into our new model class.

The main contribution of the paper is that we derive an explicit formula for the variance riskpremium in our new stochastic volatility model. That formula reveals that any stochastic dynamicsof the variance risk premium have to come from a stochastic volatility of volatility component —provided the physical and the risk neutral probability measures are related through a structurepreserving change of measure. This result indicates that (stochastic) volatility of volatility is akey component in a stochastic volatility model since it essentially drives the stochastic dynamicsof the variance risk premium. These results are in accordance with recent empirical findings onthe dynamics of variance risk premia.

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2 MODELLING FRAMEWORK

1 Introduction

This paper introduces a new class of stochastic volatility models which allows for stochastic volatil-ity of volatility. In particular, we propose to model the squared stochastic volatility process by ageneralised version of a non–Gaussian Ornstein Uhlenbeck (OU) process, extending the model byBarndorff-Nielsen and Shephard (2001, 2002), where the generalisation is given by the fact that thebackground driving Levy process of the OU process is modulated by an additional stochastic volatilitycomponent, which we refer to as stochastic volatility of volatility.

We show that our new class of models is highly analytically tractable, in particular, momentsand autocorrelation functions can be easily computed and we can even derive an explicit formulafor the integrated squared volatility process which is one of the key objects of interest in financialeconometrics since it can be regarded as the accumulated stochastic variance over a certain period.

However, solely the fact that our new model has advantageous analytical properties is clearly notenough to justify that it is a good choice for a stochastic volatility model. Hence we will motivateour modelling choice further by showing that our new model accounts for the leverage effect and forstochastic volatility and simultaneous long memory in a novel way. The key finding, however, is thenthat the additional stochastic volatility of volatility component is directly linked to the variance riskpremium (VRP), meaning that in the absence of the stochastic volatility of volatility component theVRP would be purely deterministic (at least under the conditions used in this paper) which wouldcontradict recent empirical studies which reveal stochastic dynamics of the VRP, see for instanceTodorov (2010), Carr and Wu (2009), Drechsler and Yaron (2008), Bollerslev et al. (2009).

The focus in this paper on the new class of models reflects our viewpoint that a concrete fullspecification of what is meant by volatility of volatility has to be relative to a given or chosen basicvolatility model so that volatility of volatility means variation beyond what is contained in the basemodel. (Thus, for instance, the quadratic variation of the volatility in the base model is not volatilityof volatility in the sense of that tenet.)

The outline of the remaining part of the paper is given as follows. First of all, we introduce the gen-eral modelling framework in Section 2 and relate it to the recent literature on (multi–factor) stochasticvolatility models. Section 3 is then devoted to the main probabilistic properties of the volatility mod-ulated OU processes. In particular, we will show how the leverage effect and additional long memorycan be introduced into our new modelling framework. The influence of the additional volatility ofvolatility on deriving option prices is then studied in Section 4. Next, Section 5 contains the key resultwhich is given by the explicit link between the variance risk premium and the stochastic volatility ofvolatility component. Furthermore, we derive explicit formulae for the VRP for particularly relevantchoices of the volatility of volatility component. Finally, Section 6 concludes and gives an outlook onfuture research. The proofs of our main results are relegated to the Appendix.

2 Modelling framework

Throughout the paper, we assume that the logarithmic asset price Y = (Yt)t≥0 is given by an Itosemimartingale

dYt = atdt+ σt−dWt + dJt, (1)

which is defined on a probability space (Ω,F , (Ft)t≥0,P), where a = (at)t≥0 is a predictable driftprocess, σ = (σt)t≥0 is a cadlag stochastic volatility process and J = (Jt)t≥0 is the jump componentof the Ito semimartingale. Note that an Ito semimartingale is defined as a semimartingale whose

2


characteristics are absolutely continuous with respect to the Lebesgue measure (see e.g. Jacod (2008)).So, for the jump component, we assume that

Jt =∫ t

0

∫Rκ(δ(s, x))(µ(ds, dx)− ν(ds, dx)) +

∫ t

0

∫R

(δ(s, x)− κ(δ(s, x)))µ(ds, dx),

where ν(ds, dx) = dsFs(dx) and δ is a predictable map from Ω × R+ × R on R such that thepredictable random measure Fs(dx) is the restriction to R\0 of the image of the Lebesgue measureon R by the map x 7→ δ(ω, t, x), and µ is a Poisson random measure with predictable compensator ν.Also, κ is a continuous truncation function, which is bounded with compact support and κ(x) = x ona neighbourhood of x.

The variation of financial markets, which is often referred to as squared volatility, is usually mea-sured by means of the quadratic variation of the logarithmic price process. In our modelling frame-work, the quadratic variation (QV) (denoted by [·]) is given by

[Y ]t = σ2+t +

∑0≤s≤t

(∆Js)2 , (2)

where σ2+t =

∫ t0 σ

2sds is the integrated squared stochastic volatility process and where ∆Js = Js −

Js− denotes the jump of J at time s. Taking the square root of the quadratic variation√

[Y ]t leads toa measure of the volatility of the asset price. In the following, we will work with a specific new modelfor the squared volatility process which is given by a volatility modulated non–Gaussian Ornstein–Uhlenbeck process.

2.1 Stochastic volatility by volatility modulated non–Gaussian Ornstein–Uhlenbeckprocesses

Barndorff-Nielsen and Shephard (2001, 2002) proposed to model the squared volatility σ2 by a non–Gaussian Ornstein–Uhlenbeck (OU) process. In the following, we will refer to such a model as BNSmodel. The BNS model is highly analytical tractable and fits high frequency financial data well. Here,we extend that line of thinking by generalising such OU processes to allow for an additional stochasticvolatility of volatility component. In the following, we will call this new class of processes volatilitymodulated non–Gaussian Ornstein–Uhlenbeck (VMOU) processes. They are defined as follows.

Let σ2t := V

(i)t , for i ∈ 1, 2, for two stochastic processes V (i) = (V (i)

t )t≥0, where

dV(i)t = −λV (i)

t dt+ dL(i)t , (3)

where λ > 0 is a constant, the memory parameter, and L(i) is the background driving volatilitymodulated Levy process. The additional volatility can be introduced either by stochastic proportionalor by stochastic temporal scaling, which results in the following two cases.

Model 1:

L(1)t =

∫ t

0ωλs−dLλs; (4)

(Lt)t≥0 is a Levy subordinator and (ωt)t≥0 denotes a stationary non–negative, cadlag stochasticvolatility process, which is assumed to be independent of L.

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Model 2:

L(2)t = dLτλt ; (5)

(Lt)t≥0 is a Levy subordinator, τ = (τt)t≥0 is a cadlag time change process with stationaryincrements, which is independent of L. Further, we assume that τ0 = 0.

The processes ω2 and τ can be interpreted as the stochastic variability of variance, in Model 1 andModel 2, respectively. Clearly, when ωt ≡ 1 or τt = t, we obtain the well–known BNS model.Both ω and τ can be driven by a Brownian motion or (and) a jump process. E.g. we can think of ωbeing a square root diffusion, see Cox et al. (1985), or an IG–OU process, see Barndorff-Nielsen andShephard (2001).

For the time change case, we sometimes work under the following assumption.

(T) In Model 2, the time change is absolutely continuous, i.e. τt =∫ t

0 ξsds for a positive, stationarystochastic process ξ.

It is important to note that while stochastic proportional and temporal scaling are, under suitableregularity assumptions, equivalent in a Brownian framework and for stable Levy processes, see Veraartand Winkel (2010), this is however not the case for general Levy processes and, in particular, not fora general Levy subordinator – the case we study here. While the size of the jumps is scaled by astochastic factor in the former case, the speed at which the jumps occur is determined by the timechange component in the latter case.

Remark It is important to note that there are at least four different ways to include an additionalstochastic component in a non–Gaussian OU process, two of which have been presented above. Al-ternatively, one could make the memory parameter λ stochastic and could study models of the formdVt = −λt−Vt−dt + dLt, where λt is a positive stationary process. Closely related to the latter isthe concept of supOU processes and the need to model (quasi) long range dependence, see Barndorff-Nielsen and Shephard (2002), and these two ideas could be combined in a single construction.

Still another possibility would be to let dVt = Vt−dUt + dLt, where (U,L) is a bivariate Levyprocess, see e.g. Behme et al. (2011). However, this appears less appealing in the present modellingcontext. This is due to the fact that when we work with one of the generalisations described in (4) and(5), we get explicit formulae for the integrated process V +

t =∫ t

0 Vsds, which we describe in moredetail below. This is not the case if we include the additional stochastic factor as a random memoryprocess λt. Since the integrated process is a key quantity in financial econometrics since it describesthe accumulated variance over a certain time period, we decide in favour of a model, where we canget an explicit formula for V +.

2.2 Relation to affine models

Before we turn to study general model properties both under the physical and under the risk neutralprobability measure, we focus on an important example which falls into the second model category.

The class of affine model has recently attracted a lot of research attention since such models arehighly analytically tractable and hence very useful in practical applications. Hence a natural questionarising when defining new stochastic volatility models is whether they fall into the class of affinemodels.

Duffie et al. (2003) gave a rigorous definition of regular affine processes, which form a subclassof time–homogeneous Markov processes. Intuitively speaking such processes are characterised by the

4


fact that the “logarithm of the characteristic function of the transition distribution [. . . ] is affine withrespect to the initial state [. . . ]. The coefficients defining this affine relationship are the solutions ofa family of ordinary differential equations (ODEs) that are the essence of the tractability of regularaffine processes”, see Duffie et al. (2003, p. 985), and also Kallsen (2006) for a survey. It turns outthat SV models of the first type are generally not affine (for a stochastic ω), whereas SV modelssatisfying (5) are affine as soon as the time change is affine. A natural choice would be to work withan absolutely continuous time change, where the corresponding density process is affine. E.g. thedensity process could be given by a square root diffusion or a non–Gaussian OU process. Let us studya concrete example in the following.

Example A stochastic volatility model which accounts for stochastic variance of variance and theleverage effect, i.e. the (possibly negative) correlation between the asset price and the volatility, couldbe defined by

dYt = atdt+√V

(2)t− dWt + ρdLλτt ,

dV(2)t = −λV (2)

t dt+ dLλτt ,

dτt = ξtdt,

dξt = α(β − ξt)dt+ γ√ξtdBt,

where ρ ≤ 0,B denotes a Brownian motion (independent ofW ), α, β, γ > 0 denote positive constantsand the other quantities are defined as before. Note that this model belongs to the class of affinemodels since the the BNS model itself is affine and the new (additional) time change is given by thetime integral of an affine process, see Keller-Ressel (2008).

2.3 Multifactor stochastic volatility models

Before we study our new stochastic volatility model in more detail, we would like to briefly set it intoperspective to other stochastic volatility models in the recent literature.

The first generation of SV models was established with the focus on accounting for the well–know stylised facts of asset returns such as time varying volatility, volatility clusters, the existence ofa leverage effect, see Nelson (1991), i.e. the (typically negative) correlation between asset returns andtheir volatility, see e.g. Barndorff-Nielsen and Shephard (2007, Forthcoming); Ghysels et al. (1996);Shephard and Andersen (Forthcoming); Shephard (2005) for a review. Also, the existence of im-plied volatility smiles and skews derived from option prices clearly indicated that SV is an essentialcomponent in an asset pricing model.

In a next step, the classical stochastic volatility models were extended to allow for jumps, longmemory, long run components and non–linear mean–reversion etc., see e.g. Comte and Renault(1998).

In particular, the class of multifactor SV models is important to mention in this context. Theyform a very natural generalisation of the classical one factor SV models and are very successful, inparticular, in the context of option pricing, see e.g. Christoffersen et al. (2009, 2008).

Clearly, various additional random factors can be introduced in a stochastic volatility model indifferent ways. The classical multifactor SV models usually work with a linear combination of SVmodels. However, an extra source of randomness could also be added to reflect a stochastic leveragecomponent, see Veraart and Veraart (Forthcoming).

Alternatively, we can work with a richer random structure in the volatility itself, the approach wepursue in this paper. This route has also been taken earlier by Meddahi and Renault (2004). They

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3 MODEL PROPERTIES UNDER THE PHYSICAL PROBABILITY MEASURE

introduced a semiparametric class of volatility models which is characterized by an autoregressivedynamic of the stochastic variance, which is called square–root stochastic autoregressive volatility(SR-SARV) and which is shown to be closed under temporal aggregation.

It is clear that there are various ways to construct multifactor SV models (both in discrete andin continuous time) and the question of which approach is preferable will crucially depend on thespecific application.

This paper is devoted to studying the uncertainty in the volatility itself. This is an important aspectin finance, since an investor faces at least two sources of uncertainty, when investing in a security: theuncertainty about the return (which is described by the return variance) and the uncertainty about thereturn variance itself, see Carr and Wu (2009).

In order to quantify the uncertainty in the volatility, we introduce an additional source of random-ness representing stochastic volatility of volatility in our new VMOU model, which can be regarded asa two factor stochastic volatility model. Here we view stochastic volatility of volatility as expressingthe possibility or fact that there is greater variability i.e. more volatility in the data structure understudy than might initially be surmised. In modelling terms this means that we consider the initialthinking as embodied in a (classical) SV model and want to describe the extra variability by a furthersource of randomness.

This raises the question whether there is actually a need for such a stochastic volatility of volatilitycomponent. One way to study this question would be to construct an appropriate statistical test, e.g. byextending the work by Dette et al. (2006), see also Section 3.4 for more details. However, here wewill proceed differently. It turns out that the risk associated with the uncertainty in the varianceis measured by the variance risk premium (VRP) and we will establish the link between the VRPand our new volatility of volatility component in the following. This will show why the additionalvolatility of volatility component is a crucial ingredient in our SV model.

3 Model properties under the physical probability measure

In order to fully understand the implications of such a stochastic volatility model, we study variousproperties of the volatility and the integrated volatility process under the physical probability measureP first, before we turn our attention to an analysis under the risk neutral probability measure.

3.1 Marginal invariance under time–wise rescaling

The model specifications of the VMOU processes, as defined in Section 2.1, have the property thatthe marginal distribution of V (i) is independent of the parameter λ; thus λ can properly be interpretedas the memory parameter. Otherwise put, indicating the dependence on λ by writing V (i) (λ) we havethat the law of V (i)

t (λ) (which by the stationarity does not depend on t) equals the law of V (i)λt (1)

This follows directly by substitution in the defining integrals.

3.2 Properties of the squared volatility process V (i)

From standard arguments, we deduce the following representation:

V(i)t = V

(i)0 e−λt +

∫ t

0e−λ(t−s)dL(i)

s .

6


Then, the stationary version of V (i) can be written as

V(i)t =

∫ t

−∞e−λ(t−s)dL(i)

s ,

where L is suitably extended to the negative half line (see Barndorff-Nielsen and Shephard (2001)).Note that a VMOU process of the first type is a special case of a Levy semistationary (LSS)

process, which has recently been introduced by Barndorff-Nielsen et al. (2010) in the context ofmodelling energy spot prices.

Clearly, the Levy subordinator L is a Markov process. However, the Markov property is notpreserved under stochastic integration. In particular, V is no longer a Markov process. However, thebivariate process (V (1), ω) satisfies the Markov property if ω is itself a Markov process.

Regarding the time–change model, also in that case, V (2) is generally not a Markov process any-more. However, if the Levy subordinator L is time–changed by an independent Levy subordinator τ ,then the Markov property is preserved.

Note that we can easily derive an expression for the increment process of V (i). In particular, wehave for any h ≥ 0 that

V(i)t+h − V

(i)t =

(e−λh − 1

)V

(i)t +

∫ t+h

te−λ(t+h−s)dL(i)

s . (6)

Due to the high analytical tractability of our new model, the moments and autocorrelation structurecan be easily derived, see Section A.1 in the Appendix.

We see that (volatility modulated) non–Gaussian OU processes (as well as the often used CIRprocess (Cox et al. (1985), Heston (1993)) have an exponentially declining autocorrelation functionand, hence, do not allow for longer memory in the volatility process. Since many empirical studiesreveal that medium or long memory is an important property of volatility, Barndorff-Nielsen andShephard (2002) introduced the concept of a superposition of short memory processes (as mentionedbefore), which results in a process which can have a more slowly decaying autocorrelation functionand, hence, reflects the empirical findings more appropriately. Clearly, this concept can also be appliedin the context of VMOU processes. A detailed treatment is given in Section 3.6.

3.3 Properties of the integrated squared volatility process V (i)+

In the finance literature, integrated squared volatility is regarded as the main object of interest sinceit essentially measures the accumulated (continuous) variance over a certain period of time (usually aday). So, this section analyses main properties of this key quantity in our new modelling framework.In the following, we will use the notation V (i)+ = (V (i)+

t )t≥0 for the integrated process

V(i)+t =

∫ t

0V (i)s ds.

Also, we define

ελ(t) :=1λ

(1− e−λt

).

First of all, we derive a representation result for the integrated process.

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Proposition 1 The integrated process can be written as

V(i)+t = ελ(t)V (i)

0 +∫ t

0ελ(t− s)dL(i)

s =1λ

(L

(i)t + V

(i)0 − V (i)

t

).

The proof of the above Proposition is straightforward and, therefore, not given here.These different representations of V (i)+ are interesting, since they shed some light on the joint

behaviour of V (i) and V (i)+. In particular, we can deduce some results on co–jumps and cointegration(as introduced by Granger (1981)). Clearly, V (i)

t and L(i)t have identical jumps (breaks), they co–

break, i.e. ∆V (i)t = ∆L(i)

t , but V (i) and L(i) are not cointegrated . However, V (i)+ and L(i) are infact cointegrated since

λV(i)+t − L(i)

t = V(i)

0 − V (i)t .

I.e. we have found a linear combination of the non–stationary processes V (i)+ and L(i) which isstationary. So, roughly, for large t, λV (i)+

t will have the same distribution as L(i)t , where the error in

this approximation is a stationary process, which is given by V (i)0 − V (i)

t . Now we can clearly seewhat kind of influence the stochastic variance of variance has in the new modelling set up: While thelong–run behaviour of integrated volatility in the classical BNS model is described by the backgrounddriving Levy process, our new model allows for a greater flexibility in the sense that it can allow forprocesses which have stationary, but not necessarily independent increments in the long run behaviourof the integrated variance. In particular, the long–run behaviour of integrated volatility can exhibitvolatility clusters itself due to the new component of volatility of volatility. This reveals an importantaspect of the additional volatility of volatility component.

Also, since L(i) is a nonnegative process, the integrated process V (i)+ is bounded below by thequantity ελ(t)V (i)

0 .Note that we can use formula (6) to derive a representation result for the increments of the inte-

grated process: For h ≥ 0 we get

V(i)+t+h − V

(i)+t = ελ(h)V (i)

t +∫ t+h

tελ(t+ h− s)dL(i)

s . (7)

The various cumulants of the integrated process can now be easily derived using the representationresult from Proposition 1.

3.4 Some comments on model identification

When introducing an additional volatility component in an OU model, we have to ask ourselveswhether such models are in fact identified and whether we can test from real data that an additionalvolatility component is present in the data.

The first question can be addressed via the characteristic functional of the VMOU processes.In Section 4 we will answer the second question by linking the volatility of volatility component

to the variance risk premium. However, without using any risk neutral information available to usthrough option prices, how can we distinguish between a non–Gaussian OU process and a volatilitymodulated non–Gaussian OU process statistically?

One way to answer this question is to focus on the quadratic variation of the VMOU process,which is given, respectively, by[

V (1)]t

=∫ t

0ω2λs−d[L]λs, and

[V (2)

]t

= [L]τλt .

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We know that for a Levy subordinator L, [L] is again a Levy subordinator.In the context of Model 1, we see that [V (1)] has independent increments if ω is deterministic.

As soon as ω is a stochastic process, the independent increment property generally does not hold anylonger. Also, if ω was a deterministic function which is not just a constant, then [V (1)] does not havestationary increments any longer.

A practical implication of these results is that, in principal, we can estimate the quadratic variationof the spot volatility process V (1) (based on a spot volatility estimator, see e.g. Aıt-Sahalia and Jacod(2009); Bandi and Reno (2008); Kristensen (2010); Lee and Mykland (2008); Veraart (2010)) and teststatistically whether the estimated [V (1)] has independent and stationary increments.

In the framework of Model 2, suppose as above, that τ is an absolutely continuous time changesatisfying condition (T). In that case, [L]τ is generally not a Levy subordinator anymore since theindependent increment property is violated. As before, this property may be tested statistically basedon an empirical estimate of [V (2)], which is constructed based on a spot volatility estimator.

3.5 Leverage through stochastic volatility of volatility

Next, we focus on the fact that the additional stochastic volatility of volatility component can beused for introducing the leverage effect into stochastic volatility models in a novel way. The (usuallynegative) correlation between asset returns and volatility has been found in many empirical studies,see e.g. Black (1976), Christie (1982) and Nelson (1991) among others and, more recently, by Harveyand Shephard (1996), Bouchaud et al. (2001), Tauchen (2004, 2005), Yu (2005) and Bollerslev et al.(2006).

So far, leverage type effects have usually been introduced by directly correlating the driving pro-cess of the volatility with the driving process of the asset prices (as e.g. in the Heston (1993) model).Introducing leverage in the BNS model is slightly more complicated since the volatility is driven bya subordinator and the price is driven by a Brownian motion which are inherently independent fromeach other (by the Levy – Khinchine formula). Hence Barndorff-Nielsen and Shephard (2001) sug-gested to add a jump component to the asset price, which is given by the subordinator which drivesthe volatility multiplied by a (negative) constant. Hence, such a structure assumes linear dependencebetween asset price and volatility. However, having an additional random factor in the stochasticvolatility model, i.e. the stochastic variance of variance, makes it possible to introduce leverage typeeffects indirectly and independently of the fact whether we want to have a jump component in themodel for the logarithmic asset price. In order to illustrate this, let us look at a small example.

For simplicity, we discard jumps in the price process in the following and show that a leverageeffect can be solely introduced by a diffusion component.

Proposition 2 We consider the following model

dPt =√V

(1)t− dWt,

dV(1)t = −λV (1)

t dt+ ωλtdLλt,

dωt = α(β − ωt)dt+ γ√ωtBt,

for parameters λ, α, β, γ > 0 and a Brownian motion B = (Bt)t≥0 with d[B,W ]t = ρdt, forρ ∈ [−1, 1] \ 0 and all the other quantities defined as above. For this model, we get

Cov(Pt, V(1)t ) = E(PtV

(1)t ) 6≡ 0 and Cov(Pt, P 2

t ) = E(P 3t

)6≡ 0.

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The proof is given in the Appendix.So, we see that we can have a non–zero correlation between the asset price and the squared volatil-

ity, even if the volatility is jump driven and there are no jumps in the logarithmic asset price.Similarly, when we study the time change case, see e.g. the example in Section 2.2, we can

introduce a leverage effect in the diffusion part by correlating the Brownian motion driving the assetprice with the one driving the time change.

3.6 Introducing long memory

In this section, we show how long(er) memory can be incorporated into the class of VMOU processes.This is done by extensions of the idea of supOU processes, as introduced by Barndorff-Nielsen (2001)and further discussed in Barndorff-Nielsen and Shephard (2003), Barndorff-Nielsen and Leonenko(2005), Fasen and Kluppelberg (2007), Barndorff-Nielsen and Stelzer (2011), Barndorff-Nielsen andStelzer (Forthcoming).

The long(er) memory can be introduced by randomising the memory parameter λ using the con-cept of Levy bases. A Levy basis is an independently scattered random measure whose values areinfinitely divisible. The foundation of the theory of such measures were laid by Rajput and Rosinski(1989). For a recent account of the definition and basic properties of Levy bases see Section 1.3 ofBarndorff-Nielsen et al. (Forthcoming).

3.6.1 Model 1

Let us start with Model 1, where the additional volatility component enters as a multiplicative factor,i.e.

V(1)t =

∫ t

−∞e−λ(t−s)ωλs−dLλs.

Let L be the Levy basis on R× R+ which has characteristic quadruplet

(0, 0, ν(dx), dtλπ(dλ)). (8)

Here ν is the Levy measure of L, π denotes a probability measure on (0,∞) and dtλπ(dλ) (with(t, λ) ∈ R× R+) is the control measure of L, see Barndorff-Nielsen (2010) for more details.

Then we define

V(1)t =

∫ ∞0

∫ t

−∞e−λ(t−s)ωλs−L(ds, dλ),

which can be regarded as an extension of V (1)t , where we associate a distribution with the memory

parameter λ through the Levy basis L. The verification of the existence of the double integral canbe carried out along the lines in Barndorff-Nielsen and Stelzer (Forthcoming). The process V (1) isstrictly stationary, as follows for instance from calculating the cumulant functional of V (1).

For a square integrable V (1) the conditional autocovariance function given ω may be rewritten as

E(

(V (1)0 − E(V (1)

0 |ω))(V (1)t − E(V (1)

t |ω))|ω)

=∫ ∞

0

∫ 0

−∞e−λt+2λsω2

λsdsλπ(dλ)

=∫ ∞

0e−λtπ(dλ)

∫ 0

−∞e2λsω2

λdsλds =∫ ∞

0e−λtπ(dλ)

∫ 0

−∞e2uω2

udu︸︷︷︸:=X(1)

= X(1)

∫ ∞0

e−λtπ(dλ),

10


where the random variable X(1) and its law do not depend on λ.Long memory of V (1) can be obtained by choosing π to be the probability measure of the Gamma

law, see the example below.

3.6.2 Model 2

Next, we study how we can introduce long memory in VMOU processes of type 2. As in the case ofModel 1, we will randomise the memory parameter λ via a Levy basis. However, since the additionalstochastic volatility enters in form of a time change rather than as a stochastic proportional, the ap-proach is somewhat different and leads to integration with respect to a random measure more generalthan a Levy basis.

Recall that

V(2)t =

∫ t

−∞e−λ(t−s)dLτλs .

Let T the random measure associated with the stochastic process τ , so that for intervals (a, b] wehave T ((a, b]) = τ (b) − τ (a). (If τ was a Levy subordinator, then T would be the correspondingLevy basis.) We introduce a random measure M on R× R+ characterised by requiring that Mconditionally on T is the Levy basis on R× R+ having characteristic quadruplet

(0, 0, ν(dx), T (λdt)π(dλ)). (9)

Here ν and π are as above and the control measure is T (λdt)π(dλ). (This construction is analo-gous that of extended subordination by meta–time changes of Levy bases as introduced in Barndorff-Nielsen (2010) and Barndorff-Nielsen and Pedersen (2010).)

Then, define the process V (2) by

V(2)t =

∫ ∞0

∫ t

−∞e−λ(t−s)M(ds, dλ).

That this determines a well–defined strictly stationary process can be verified by calculating the char-acteristic functional of V (2).

Under square integrability, the conditional autocovariance function given τ takes the form

E(

(V (2)0 − E(V (2)

0 |τ))(V (2)t − E(V (2)

t |τ))|τ)

=∫ ∞

0

∫ 0

−∞e−λt+2λsT (λds)π(dλ)

=∫ ∞

0e−λtπ(dλ)

∫ 0

−∞e2λsT (λds) =

∫ ∞0

e−λtπ(dλ)∫ 0

−∞e2uT (du)︸︷︷︸

:=X(2)

= X(2)

∫ ∞0

e−λtπ(dλ),

where the random variable X(2) and its law do not depend on λ.

3.6.3 An important example

We have seen that the conditional autocovariance function of V (i) is given by

X(i)

∫ ∞0

e−λtπ(dλ) = X(i)π(λ),

for i = 1, 2 and where π(λ) denotes the Laplace transform of π.In the following example, we show how we can obtain long memory in such a modelling frame-

work.

11

4 MODEL PROPERTIES UNDER THE RISK NEUTRAL MEASURE

Example Let us assume that π is the Gamma law Γ(2H, 1) for H > 0. The corresponding densityfunction is given by

fΓ(2H,1)(x) =1

Γ(2H)x2H−1e−x, for x > 0.

Then ∫ ∞0

e−λtπ(dλ) =1

Γ(2H)

∫ ∞0

e−xte−xx2H−1dx = (t+ 1)−2H .

Note in particular that V (i) exhibits second order long range dependence ifH ∈ (12 , 1) forH := 1−H .

(For the original supOU processes this same choice of π was discussed in Barndorff-Nielsen (2001),Barndorff-Nielsen and Shephard (2001). A concrete application is given in Barndorff-Nielsen andShephard (Forthcoming).)

Focusing on Model 2, we have seen that additional volatility and long memory can be introducedin a non–Gaussian OU process by an extended subordination approach, where the additional volatilityenters through a time change τ , with associated random measure T, and the long memory can beobtained by a suitable choice of the probability measure π. These two measures have to be combined,as described in the characteristic quadruplet defined in (9), in order to obtain both long memory andadditional stochastic volatility.

4 Model properties under the risk neutral measure

After we have studied the new stochastic volatility model under the physical probability measure, wewill focus on important aspects of the new model under the risk neutral probability measure, whichwe denote by Q. Note that the formulae derived for the moments of the new model stay the sameunder a structure preserving measure (using the risk neutral parameters).

In particular, we will study how option prices can be derived based on our new model.

4.1 Option pricing

A popular method for computing option prices is based on the Laplace or Fourier transform of theasset price. Such methods have been introduced by Heston (1993) and have subsequently been studiedby Carr and Madan (1999); Lee (2004); Lewis (2001); Nicolato and Venardos (2003); Raible (2000)amongst others.

The main idea here is that if c denotes the Laplace transform of the payoff function c of an option,i.e.

c(z) =∫ ∞−∞

e−zxc(x)dx,

see Raible (2000) for explicit forms, and if the Laplace transform of the logarithmic asset price, whichwe denote by φ, is known, then the option price Ct (with time of maturity given by t + h for h ≥ 0)can be computed using Fourier inversion, specifically

Ct =e−rh

2πi

∫ θ+i∞

θ−i∞φ(z)c(z)dz,

where r ≥ 0 denotes the risk–free interest rate and θ is a constant belonging to the set where both cand φ are defined (provided such a constant exists).

12

4 MODEL PROPERTIES UNDER THE RISK NEUTRAL MEASURE

Here, we will show that a transform–based method can be used for computing option prices whenthe logarithmic asset price is given by the generalised BNS model where the squared volatility pro-cess is a volatility modulated non–Gaussian OU process. In order to do that, we derive the Laplacetransform of the integrated volatility process and of the log–price process, which are both obtained insemi–analytic form.

Throughout this section, we will work with the following more specific model for the asset price(for i = 1, 2) which is given by S(i)

t = S(i)0 exp

(Y

(i)t

)for

dY(i)t = (µ+ βV

(i)t )dt+

√V

(i)t− dWt, (10)

where µ, β ∈ R and V (i) is defined as before.Here we assume that the model is formulated under the risk neutral probability measure Q.Before we derive the Laplace transformation of the price process, we formulate a condition which

ensures that the discounted asset price e−rtS(i)t is a (local) martingale, where we follow closely Nico-

lato and Venardos (2003). Applying Ito’s formula, we obtain the following dynamics for the assetprice S(i):

dS(i)t = S

(i)t−

(b(i)t dt+

√V

(i)t− dWt

), where b

(i)t = µ+

(β +

12

)V

(i)t .

Hence, the discounted asset price is a (local) martingale if and only if

b(i)t − r = 0, for i = 1, 2. (11)

Assuming the martingale condition holds, the Laplace transformation of the price process can bedetermined in the form given in the following Proposition, the proof of which is given in the Appendix.

Proposition 3 Let φ(i)(θ) = EQt (exp(θY (i)

t+h)), for h ≥ 0 and for i = 1, 2. Then

φ(1)(θ) = exp(θYt + θµh+

(βθ +

θ2

2

)ελ(h)V (1)

t

)EQt

[exp

(λ

∫ t+h

tψL(f (1)(s, θ))ds

)],

where f (1)(s, θ) := (βθ+θ2/2)ελ(t+h−s)ωλs and where ψL denotes the log–transformed Laplacetransform of L. For Model 2, we have

φ(2)(θ) = exp(θYt + θµh+

(βθ +

θ2

2

)ελ(h)V (2)

t

)EQt

[exp

(∫ t+h

tψL′(s)(f

(2)(s, θ))T (λds))]

,

where f (2)(s, θ) = (βθ+ θ2/2)ελ(t+ h− s) and where ψL′(s) denotes the log–transformed Laplacetransform of L′(s). Note that L′ denotes the Levy seed associated with L. Further, T denotes therandom measure associated with τ .

13

5 CHANGE OF MEASURE AND VARIANCE RISK PREMIA

Note that in the case when ω is deterministic, then we have an analytic formula for the Laplacetransform of the conditional distribution of Y . In the stochastic case, the integral has to be evaluatedusing Monte Carlo methods. Finally, we can use the Fourier inversion formula presented above forcomputing the option price based on our new stochastic volatility model.

Remark In the formulae above, we see that Model 1 and Model 2 are fundamentally different fromeach other. While the volatility of volatility component enters as an argument in the log–transformedLaplace transform of the driving Levy process, this is not the case in Model 2. Here, the additionalvolatility component enters as a multiplicative factor to the log–transformed Laplace transform.

5 Change of measure and variance risk premia

After we have studied the new stochastic volatility model based on a volatility modulated non–Gaussian OU process both under the physical and under the risk neutral probability measure, wenext focus on the link between the physical and the risk neutral properties.

So far, we have seen that the additional stochastic variability of the variance component can bemotivated both from an empirical point of view when studying asset price data under the physicalmeasure, since the additional component introduces much more flexibility in the behaviour of theintegrated variance, and also when studying option prices since the integrated variance also entersdirectly in the option pricing formula.

In this section we will show that the additional stochastic volatility of volatility component alsoplays a key role in determining the dynamics of the variance risk premium (VRP). We will demonstratethat, under fairly mild assumptions, the stochastic dynamics of the VRP are determined solely by thevolatility of volatility component.

In order to understand the relationship between the risk–neutral world and the physical world, wehave to study how to change the corresponding probability measures. Hence, in the following, wefirst discuss measure changes, where we are particularly interested in structure preserving changes ofmeasure, before we turn to one of the key results of this paper: The role of the stochastic variabilityof variance in determining the variance risk premium.

5.1 Structure preserving change of measure

Nicolato and Venardos (2003) derived the class of structure preserving change of measures for theBNS model. Since the model we study in this paper is a direct generalisation of the BNS model,where an additional volatility of volatility component is included, we can proceed similarly to obtainthe class of structure preserving changes of measure.

Clearly, we have to specify first which kind of structure we wish to preserve, meaning that weshould specify a model for the volatility of volatility ω in Model 1 or the time change τ in Model 2.Here we focus on one particularly model choice, but other (related) models can be dealt with similarly.An analytically tractable choice for the volatility of volatility would be to model ω by a square rootdiffusion and to model the time change τ as absolutely continuous, i.e. τt =

∫ t0 ξsds, where (ξs)s≥0 is

itself a square root diffusion. (Alternatively, we could use a non–Gaussian OU process instead of thesquare root diffusion). We know from Veraart and Veraart (Forthcoming) how to construct structurepreserving changes of measures for such processes, and combining them with the structure preservingmeasure changes from Nicolato and Venardos (2003) we obtain a class of structure preserving changesof measure for the model studied in this paper.

14


5.2 Variance risk premia

In order to understand the influence of the volatility of volatility term even better, we study the vari-ance risk premium (VRP), which is described by the wedge between the physical and the risk neutralvolatility. The variance risk premium has been studied extensively in the very recent literature, seee.g. Drechsler and Yaron (2008), Carr and Wu (2009), Bollerslev et al. (2009), Todorov (2010), Wu(2010).

We denote by P the physical probability measure on the filtered probability space (Ω,F , Ft,P).Let Z denote a positive martingale with mean 1. Then we define a new probability measure fort, h ≥ 0 by

Qt(A) := E(IAZt)) =∫AZtdP, for all A ∈ Ft.

Clearly, Qt is a probability measure on Ft with

dQt

dP= Zt.

Furthermore, since Z is a martingale, we have Qt+h(A) = Qt(A) for all A ∈ Ft. Clearly, the riskneutral measure will be obtained from the physical measure by a measure transformation like the onedefined above. Hence, in the following we will refer to Q as the risk neutral probability measure.

Note that variance swaps are written on the (annualised) realised variance. For [t, t + h] witht, h ≥ 0, we define a partition t = t0 < · · · < tn = t+ h with ti − ti−1 = 1/n for i = 1, . . . , n,

RVt,t+h :=n∑i=1

log(StiSti−1

).

Hence, the variance risk premium over the interval [t, t+ h] is given by

V RPt,t+h :=1h

[Et(RVt,t+h)− EQ

t (RVt,t+h)],

where Et(·) := E(·|Ft).We approximate the realised variance by the quadratic variation and, hence, the approximated

variance risk premium (AVRP) is given by

AV RPt,t+h :=1h

[Et([Y ][t,t+h])− EQ

t ([Y ][t,t+h])],

where the approximation error is given by

V RPt,t+h −AV RPt,t+h =1h

[Et(RVt,t+h − [Y ][t,t+h])− EQ

t (RVt,t+h − [Y ][t,t+h])].

We know from the central limit theorem for realised variance, see Jacod (2008), that the error in theapproximation is of order O(

√n−1) (both under Q and under P) and is hence negligible to this order.

Since, we have an explicit formula for the integrated squared volatility process, we can formulatea fairly explicit general formula for the variance risk premium in the following proposition, which isproved in the Appendix.

15


Proposition 4 The approximated variance risk premium for Model i, for i = 1, 2, is given by

AV RP(i)t,t+h =

1h

[Et(ζ(i)(t, t+ h)

)− EQ

t

(ζ(i)(t, t+ h)

)], (12)

where

ζ(i)(t, t+ h) :=∫ t+h


s +∑

t≤s≤t+h(∆Js)2.

Equivalently, we have

AV RP(i)t,t+h =

1h

Et(ζ(i)(t, t+ h)

(1− Zt+h

Zt

)).

Formula (12) contains a fairly explicit formula for the variance risk premium associated with thestochastic volatility model given by a volatility modulated non–Gaussian OU process. In particular,we see that the volatility of volatility component influences the dynamics of the variance risk premiumin quite a direct fashion through the process L(i).

5.3 Variance risk premia under structure preserving changes of measures

It turns out that we can obtain an even more explicit result if we add the following two assumptions.

(A1) The jump process in the asset price, denoted by J , is a pure jump Levy process.

(A2) The risk neutral probability measure is obtained by a structure preserving change of measure.

Note that we can easily show that structure preserving changes of measure for our new model classexist, where we extend the corresponding results in Nicolato and Venardos (2003), Veraart and Veraart(Forthcoming).

In the following, we will denote by AV RP(i)ct,t+h the part of the AVRP due to the continuous

component in the price and by AV RP dt,t+h the part due to the jumps J , i.e.

AV RP(i)ct,t+h =

1h

[Et(ζ(i)c(t, t+ h)

)− EQ

t

(ζ(i)c(t, t+ h)

)],

AV RP dt,t+h =1h

[Et(ζd(t, t+ h)

)− EQ

t

(ζd(t, t+ h)

)],

where

ζ(i)c(t, t+ h) :=∫ t+h


s , ζd(t, t+ h) :=∑

t≤s≤t+h(∆Js)2.

Proposition 5 Under assumptions (A1) and (A2), we obtain the following result for the jump part ofthe AVRP:

AV RP dt,t+h =∫

Rx2νJ(dx)−

∫Rx2νQ

J (dx),

where νJ(dx)dt and νQJ (dx)dt are the predictable compensators of the Poisson random measure

associated with J under P and under Q, respectively.

16


So, the jump part of the variance risk premium is only given by a constant.Next, let us study the continuous part of AVRP.

Proposition 6 Assume that assumptions (A1), (A2), (T) hold. Let v(1)t := ωλt and v(2)

t := ξλt. Fori = 1, 2, we have

AV RP(i)ct,t+h =

1h

[v

(i)t

(E(L1)− EQ(L1)

)(h− ελ(h)) (13)

+ λ

∫ t+h

tελ(t+ h− s)

(E(L1)Et

(v(i)s − v

(i)t

)− EQ(L1)EQ

t

(v(i)s − v

(i)t

))ds

].

Remark Note that L is a subordinator and, hence, E(L1) > 0. Also, under the structure preservingchange of measure, the predictable compensator of L changes and, hence, E(L1)− EQ(L1) 6= 0.

Remark Note that we obtain exactly the same results in Model 2 as in Model 1, where ω is replacedby the density process ξ of the time change τ .

The above propositions show that the stochastic dynamics of the variance risk premium are solely de-termined by the stochastic volatility of volatility component ω and τ , respectively. If these terms werenot stochastic, then the variance risk premium (assuming a structure preserving change of measure)would be deterministic, which would contradict recent empirical findings e.g. by Drechsler and Yaron(2008), Bollerslev et al. (2009).

The formula (13) in Proposition 6 can be computed explicitly in the case where ω and ξ, respec-tively, are given by a stationary square root diffusion, see the following Corollary.

Corollary 7 Assume that assumptions (A1), (A2), (T) hold. Let v(1)t := ωλt and v(2)

t := ξλt. Also, fori = 1, 2, let

dv(i)t = a

(b− v(i)

t

)dt+ g

√v

(i)t dBv

t ,

where a, b, g are positive constants satisfying the Feller condition 2ab > g2 (and v(i)0 > 0) and Bv is

a standard Brownian motion. Then we get for i = 1, 2 that

AV RP(i)ct,t+h = v

(i)t F (1)(t, t+ h) + F (2)(t, t+ h),

where F (1)(t, t + h) and F (2)(t, t + h) are explicitly known deterministic functions, given in theAppendix.

Also in the case of a non–Gaussian OU process, we get explicit results, see the following Corollary.

Corollary 8 Assume that assumptions (A1), (A2), (T) hold. Let v(1)t := ωλt and v(2)

t := ξλt. Also, fori = 1, 2, let

dv(i)t = −av(i)

t dt+ dLvat,

where a > 0 and Lv is a Levy subordinator. Then we get for i = 1, 2 that

AV RP(i)ct,t+h = v

(i)t G(1)(t, t+ h) +G(2)(t, t+ h),

where G(1)(t, t + h) and G(2)(t, t + h) are explicitly known deterministic functions, given in theAppendix.

17

6 CONCLUDING REMARKS

So, we have obtained an explicit formula for the AVRP which depends on the physical andrisk neutral parameters of the underlying model. Further, we see that the stochastic dynamics ofAV RPt,t+h (as a stochastic process in t with fixed h > 0) are determined by the volatility of volatil-ity component vt.

Note that the classical BNS model (assuming a structure preserving change of measure) impliesthat the variance risk premium is deterministic. This fact can be easily derived from our results above.However, by including a stochastic variance of variance term, we allow for stochastic dynamics of thevariance risk premium.

Remark Note that both Drechsler and Yaron (2008) and Bollerslev et al. (2009) also established alink between a volatility of volatility component and the VRP. However, they study a self–containedgeneral equilibrium model and show that the variance risk premium is solely driven by the volatilityof (consumption growth) volatility, where this explicit formula has been derived using a log–linearapproximation.

Here, we do not work with an equilibrium model, but extend one of the popular asset price models,the BNS model, to allow for an additional volatility of volatility component. In order to derive ourexplicit formula for VRP and to establish the link to the volatility of volatility component, we didnot need any approximation due to the high analytic tractability of our new model. However, themain assumption we made was that the physical and the risk neutral probability measures are relatedthrough a structure preserving change of measure, which seems to be a strong, but nevertheless rathernatural assumption from a modelling point of view.

6 Concluding remarks

This paper has introduced a new class of stochastic volatility models which is given by volatility mod-ulated non–Gaussian Ornstein Uhlenbeck (VMOU) processes. We have shown that the new modelclass is highly analytical tractable and, in particular, we have derived an explicit formula for the in-tegrated squared volatility process, which plays a key role in determining an explicit formula for thevariance risk premium.

Next, we have shown that the additional volatility of volatility component can be used to introducethe leverage effect in a new way. Also, we have developed a new methodology for allowing for longmemory and volatility of volatility of volatility simultaneously: This can be done by combining theconcepts of extended subordination of Levy bases (or of more general random measures) and of ran-domisation of the memory parameter of the OU process through a suitable choice of the characteristicquadruplet.

Another key result we have established in this paper is the fact that the stochastic volatility ofvolatility component solely determines the stochastic dynamics of the variance risk premium (as-suming a structure preserving change of measure). Given the empirical evidence that the variancerisk premium has stochastic dynamics, including a stochastic volatility of volatility component into astochastic volatility model is hence not just a modelling choice but a necessity and can be easily dealtwith by our new class of VMOU processes.

Clearly, there are various natural extensions of our new model. For instance, we will address mul-tivariate extensions of VMOU processes and, also, superpositions of such multivariate processes infuture research. Multivariate OU processes and their superpositions have recently been introduced byBarndorff-Nielsen and Stelzer (2011) and have been applied as multivariate stochastic volatility mod-els by Barndorff-Nielsen and Stelzer (Forthcoming); Muhle-Karbe et al. (2010); Pigorsch and Stelzer

18

REFERENCES

(2009). Furthermore, we will address multivariate extensions of the VMOU of type 2, i.e. the timechange case, in future research, which will be based on the new concept of multivariate subordinationintroduced by Barndorff-Nielsen (2010) and Barndorff-Nielsen and Pedersen (2010).

Funding

This work was supported by the Center for Research in Econometric Analysis of Time Series, CRE-ATES, funded by the Danish National Research Foundation.

References

Aıt-Sahalia, Y. and Jacod, J. (2009), ‘Testing for jumps in a discretely observed process’, Annals ofStatistics 37(1), 184–222.

Bandi, F. M. and Reno, R. (2008), Nonparametric stochastic volatility. Working paper (June 2008).Available at SSRN: http://ssrn.com/abstract=1158438.

Barndorff-Nielsen, O. E. (2001), ‘Superposition of Ornstein–Uhlenbeck type processes’, Theory ofProbability and its Applications 45, 175–194.

Barndorff-Nielsen, O. E. (2010), Levy bases and extended subordination. Thiele Centre ResearchReport No. 12, Aarhus University.

Barndorff-Nielsen, O. E., Benth, F. E. and Veraart, A. E. D. (2010), Modelling energy spot pricesby Levy semistationary processes. CREATES Research Paper 2010–18. Available at SSRN:http://ssrn.com/abstract=1597700.

Barndorff-Nielsen, O. E., Benth, F. and Veraart, A. (Forthcoming), Ambit processes and stochasticpartial differential equations, in G. Di Nunno and B. Ø ksendal, eds, ‘Advanced MathematicalMethods for Finance’, Springer, Berlin.

Barndorff-Nielsen, O. E. and Leonenko, N. (2005), ‘Spectral properties of superpositions of Ornstein–Uhlenbeck type processes’, Methodology and computing in applied probability 7, 335–352.

Barndorff-Nielsen, O. E. and Pedersen, J. (2010), Meta–times and extended subordination. ThieleResearch Report No. 15, December 2010.

Barndorff-Nielsen, O. E. and Shephard, N. (2001), ‘Non–Gaussian Ornstein–Uhlenbeck–based mod-els and some of their uses in financial economics (with discussion)’, Journal of the Royal StatisticalSociety Series B 63, 167–241.

Barndorff-Nielsen, O. E. and Shephard, N. (2002), ‘Econometric analysis of realised volatility and itsuse in estimating stochastic volatility models’, Journal of the Royal Statistical Society B 64, 253–280.

Barndorff-Nielsen, O. E. and Shephard, N. (2003), ‘Integrated OU processes’, Scandinavian Journalof Statistics 30, 277–295.

19

REFERENCES

Barndorff-Nielsen, O. E. and Shephard, N. (2007), Variation, jumps, market frictions and high fre-quency data in financial econometrics, in R. Blundell, P. Torsten and W. K. Newey, eds, ‘Advancesin Economics and Econometrics. Theory and Applications, Ninth World Congress’, EconometricSociety Monographs, Cambridge University Press.

Barndorff-Nielsen, O. E. and Shephard, N. (Forthcoming), Financial Volatility in Continuous Time,Cambridge University Press, Cambridge. Forthcoming.

Barndorff-Nielsen, O. E. and Stelzer, R. (2011), ‘Multivariate supOU processes’, Annals of AppliedProbability 21, 140–182.

Barndorff-Nielsen, O. E. and Stelzer, R. (Forthcoming), ‘The multivariate supOU stochastic volatilitymodel’, Mathematical Finance . Forthcoming.

Behme, A., Lindner, A. and Maller, R. (2011), ‘Stationary solutions of the stochastic differentialequation dVt = Vt−dUt + dLt with Levy noise’, Stochastic Processes and their Applications121, 91–108.

Black, F. (1976), ‘Studies of stock market volatility changes’, Proceedings of the Business and Eco-nomic Statistics Section, American Statistical Association pp. 177–181.

Bollerslev, T., Litvinova, J. and Tauchen, G. (2006), ‘Leverage and volatility feedback effects in high-frequency data’, Journal of Financial Econometrics 4(3), 353–384.

Bollerslev, T., Tauchen, G. and Zhou, H. (2009), ‘Expected stock returns and variance risk premia’,Review of Financial Studies 22(11), 4463–4492.

Bouchaud, J.-P., Matacz, A. and Potters, M. (2001), ‘Leverage effect in financial markets: The re-tarded volatility model’, Physical Review Letters 87(228701).

Carr, P. and Madan, D. (1999), ‘Option valuation and the fast Fourier transform’, Journal of Compu-tational Finance 2.

Carr, P. and Wu, L. (2009), ‘Variance risk premiums’, Review of Financial Studies 22(3), 1311–1341.

Christie, A. A. (1982), ‘The stochastic behavior of common stock variances’, Journal of FinancialEconomics 10, 407–432.

Christoffersen, P., Heston, S. and Jacobs, K. (2009), ‘The shape and term structure of the indexoption smirk: why multifactor stochastic volatility models work so well’, Management Science55(12), 1914–1932.

Christoffersen, P., Jacobs, K., Ornthanalai, C. and Wang, Y. (2008), ‘Option valuation with long-runand short-run volatility components’, Journal of Financial Economics 90(3), 272 – 297.

Comte, F. and Renault, E. (1998), ‘Long memory in continuous–time stochastic volatility models’,Mathematical Finance 8, 291–257.

Cox, J. C., Ingersoll, J. E. J. and Ross, S. A. (1985), ‘A theory of the term structure of interest rates’,Econometrica 53(2), 385–408.

20

REFERENCES

Dette, H., Podolskij, M. and Vetter, M. (2006), ‘Estimation of integrated volatility in continuous timefinancial models with applications to goodness–of–fit testing’, Scandinavian Journal of Statistics33, 259–278.

Drechsler, I. and Yaron, A. (2008), What’s vol got to do with it. AFA 2009 San Francisco MeetingsPaper.

Duffie, D., Filipovic, D. and Schachermayer, W. (2003), ‘Affine processes and applications in finance’,Annals of Applied Probability 13, 984–1053.

Fasen, V. and Kluppelberg, C. (2007), Extremes of supou processes, in F. Benth, G. Di Nunno,T. Lindstrom, B. Ø ksendal and T. Zhang, eds, ‘Stochastic Analysis and Applications. The AbelSymposium’, Vol. 2, Springer, Berlin, pp. 340–359.

Ghysels, E., Harvey, A. and Renault, E. (1996), Stochastic volatility, in C. R. Maddala, G. S. and Rao,ed., ‘Handbook of Statistics’, Vol. 14 of Statistical Methods in Finance, North Holland, pp. 119–183.

Granger, C. J. (1981), ‘Some properties of time series data and their use in econometric model speci-fication’, Journal of Econometrics 16, 121–130.

Harvey, A. and Shephard, N. (1996), ‘Estimation of an asymmetric stochastic volatility model forasset returns’, Journal of Business and Economic Statistics 14, 429–434.

Heston, S. L. (1993), ‘A closed–form solution for options with stochastic volatility with applicationsto bond and currency options’, Review of Financial Studies 6, 327–343.

Jacod, J. (2008), ‘Asymptotic properties of realized power variations and related functionals of semi-martingales’, Stochastic Processes and their Applications 118(4), 517–559.

Kallsen, J. (2006), A didactic note on affine stochastic volatility models, in Y. Kabanov, R. Liptser andJ. Stoyanov, eds, ‘From Stochastic Calculus to Mathematical Finance’, Springer, Berlin, pp. 343–368.

Karatzas, I. and Shreve, S. E. (1991), Brownian Motion and Stochastic Calculus, second edn, Springer,New York.

Keller-Ressel, M. (2008), Affine Processes — Theory and Applications in Finance, PhD thesis, Tech-nische Universitat Wien.

Kristensen, D. (2010), ‘Nonparametric filtering of the realized spot volatility: A kernel–based ap-proach’, Econometric Theory 26, 60–93.

Lee, R. W. (2004), ‘Option pricing by transform methods: Extensions, unification, and error control’,Journal of Computational Finance 7, 51–86.

Lee, S. S. and Mykland, P. A. (2008), ‘Jumps in financial markets: A new nonparametric test andjump dynamics’, Review of Financial Studies 21(6), 2535–2563.

Lewis, A. L. (2001), A simple option formula for general jump–diffusion and other exponential Levyprocesses. Working paper, Envision Financial Systems.

21

REFERENCES

Meddahi, N. and Renault, E. (2004), ‘Temporal aggregation of volatility models’, Journal of Econo-metrics 119, 355–379.

Muhle-Karbe, J., Pfaffel, O. and Stelzer, R. (2010), Option pricing in multivariate stochastic volatilitymodels of OU type. Preprint.

Nelson, D. B. (1991), ‘Conditional heteroscedasticity in asset returns: a new approach’, Econometrica59(2), 347–370.

Nicolato, E. and Venardos, E. (2003), ‘Option pricing in stochastic volatility models of the Ornstein–Uhlenbeck type’, Mathematical Finance 13, 445–466.

Pigorsch, C. and Stelzer, R. (2009), A multivariate Ornstein–Uhlenbeck type stochastic volatilitymodel. Preprint.

Raible, S. (2000), Levy Processes in Finance: Theory, Numerics, and Empirical Facts, PhD thesis,University of Freiburg.

Rajput, B. and Rosinski, J. (1989), ‘Spectral representation of infinitely divisible distributions’, Prob-ability Theory and Related Fields 82, 451–487.

Shephard, N. and Andersen, T. G. (Forthcoming), Stochastic volatility: Origins and overview, inT. G. Andersen, R. Davis, J.-P. Kreiss and T. Mikosch, eds, ‘Handbook of Financial Time Series’,Springer. Forthcoming.

Shephard, N., ed. (2005), Stochastic Volatility: Selected Readings, Oxford University Press, Oxford.

Tauchen, G. (2004), Remarks on recent developments in stochastic volatility: statistical modellingand general equilibrium. Working paper, Department of Economics, Duke University.

Tauchen, G. (2005), Stochastic volatility and general equilibrium. Working paper, Department ofEconomics, Duke University.

Todorov, V. (2010), ‘Variance risk premium dynamics: The role of jumps’, The Review of FinancialStudies 23(1), 345–383.

Veraart, A. E. D. (2010), ‘Inference for the jump part of quadratic variation of Ito semimartingales’,Econometric Theory 26, 331 –368.

Veraart, A. E. D. and Veraart, L. A. M. (Forthcoming), ‘Stochastic volatility and stochastic leverage’,Annals of Finance .

Veraart, A. and Winkel, M. (2010), Time change, in R. Cont, ed., ‘Encyclopedia of QuantitativeFinance’, Vol. 4, Wiley, pp. 1812–1816.

Wu, L. (2010), ‘Variance dynamics: Joint evidence from options and high–frequency returns’, Journalof Econometrics . In press.

Yu, J. (2005), ‘On leverage in a stochastic volatility model’, Journal of Econometrics 127(2), 165–178.

22

A CUMULANTS

A Cumulants

A.1 Cumulants of V (i)

Here, we study various conditional and unconditional moments of V (i) and we derive its autocorre-lation function. We only present the results and omit the rather lengthy proofs (except for one case)since they consist of straightforward computations.

In order to simplify the exposition, we fix the following notation. For i ∈ N, we denote theith cumulant of the Levy subordinator Lt by κi(Lt). Clearly, we have κi(Lt) = tκi(L1) and, inparticular, we write κi := 1

λκi(Lλ) = κi(L1).

Remark In order to ensure that our model (4) is uniquely identified, we have to set the variance (orthe mean) of L1 to a fixed value. Otherwise one could always multiply ω by a constant and rescaleL correspondingly, which results in an identification problem. For convenience, we will later setκ2(L1) = V ar(L1) = 1. Similarly, we impose the same identifiability condition for model (5).

Furthermore, we will write γ(h) = Cov(ωt, ωt+h) for the covariance function of ω. We will carry outall computations for a general stationary volatility process ω, which is independent of the subordinatorL.

First, we compute the moments of V (1), when we condition on ω. Clearly, if ω was deterministic,these results would also hold unconditionally.

Proposition 9 The conditional mean, variance and covariance are given by

E(V

(1)t

∣∣∣ω) = e−λtEω(V

(1)0

)+ λκ1(L1)

∫ t

0e−λ(t−s)ωλsds,

V ar(V

(1)t

∣∣∣ω) = e−2λtV arω(V

(1)0

)+ λκ2(L1)

∫ t

0e−2λ(t−s)ω2

λsds,

Cov(V

(1)t , V

(1)t+h

∣∣∣ω) = e−λhV ar(V

(1)t

∣∣∣ω) ,Cor

(V

(1)t , V

(1)t+h

∣∣∣ω) = e−λh.

Proof of Proposition 9 Note that throughout the proof, we skip the superscript in the volatility pro-cess, i.e. we write V = V (1).

For the mean, we have

Eω(Vt) = e−λtEω(V0) + λκ1(L1)∫ t

0e−λ(t−s)ωλsds.

For the second moment, we get from Ito’s formula

V 2t − V 2

0 = −2λ∫ t

0V 2s ds+ 2

∫ t

0VsωλsdLλs +

∫ t

0ω2λsd[L]λs.

Taking the conditional expectation Eω(·) := E(·|ω), we get

Eω(V 2t )− Eω(V 2

0 ) = −2λ∫ t

0Eω(V 2

s )ds+ 2λκ1(L1)∫ t

0Eω(Vsωλs)ds+ λκ2(L1)

∫ t

0Eω(ω2

λs)ds

= −2λ∫ t

0Eω(V 2

s )ds+ 2λκ1(L1)∫ t

0Eω(Vs)ωλsds+ λκ2(L1)

∫ t

0ω2λsds.

23

A CUMULANTS

Hence,

d

dtEω(V 2

t ) = −2λEω(V 2t ) + 2λκ1(L1)Eω(Vt)ωλt + λκ2(L1)ω2

λt.

We solve the stochastic differential equation and obtain

Eω(V 2t ) = e−2λtEω(V 2

0 ) +∫ t

0e−2λ(t−s)Ξsds,

where

Ξs := 2λκ1(L1)Eω(Vs)ωλs + λκ2(L1)ω2λs

= 2λκ1(L1)e−λsEω(V0)ωλs + 2λ2κ21(L1)ωλs

∫ s

0e−λ(s−u)ωλudu+ λκ2(L1)ω2

λs.

Consequently, we have

Eω(V 2t ) = e−2λtEω(V 2

0 ) +∫ t

0e−2λ(t−s)Ξsds

= e−2λtEω(V 20 ) + 2λκ1(L1)Eω(V0)

∫ t

0e−2λ(t−s)e−λs︸︷︷︸=e−λte−λ(t−s)

ωλsds

+ 2λ2κ21(L1)

∫ t

0e−2λ(t−s)ωλs

∫ s

0e−λ(s−u)ωλududs︸︷︷︸

=e−2λt∫ t0 e

λsωλs∫ s0 e

λuωλududs

+λκ2(L1)∫ t

0e−2λ(t−s)ω2

λsds.

Hence,

V arω(Vt) = e−2λtV arω(V0) + λκ2(L1)∫ t

0e−2λ(t−s)ω2

λsds.

Similar computations lead to the result for the covariance, where we used that

Eω(VtVt+h) = e−λhEω(V 2t ) + λκ1(L1)Eω(Vt)

∫ t+h

te−λ(t+h−s)ωλsds.

Next, we compute the unconditional mean, variance and covariance of the VMOU process V (1).

Proposition 10 Let V (1)t = V

(1)0 e−λt +

∫ t0 e−λ(t−s)ωλsdLs. Then, for h > 0,

E(V

(1)t

)= e−λtE(V (1)

0 ) + κ1(L1)E(ω0)(1− e−λt),

V ar(V

(1)t

)= e−2λtV ar

(V

(1)0

)+

12κ2(L1)E

(ω2

0

)(1− e−2λt)

+ κ21(L1)

∫ 0

−λt

∫ 0

−λtexeyγ(|x− y|)dxdy,

Cov(V

(1)t , V

(1)t+h

)= e−λhV ar

(V

(1)t

).

24

B PROOFS

Proof The results follow essentially from Proposition 9.

Corollary 11 The mean, variance and autocovariance of the stationary process V (1) are given by

E(V

(1)t

)= κ1(L1)E(ω0),

V ar(V

(1)t

)=

12κ2(L1)E

(ω2

0

)+ κ2

1(L1)∫ 0

−∞

∫ 0

−∞exeyγ(|x− y|)dxdy,

Cor(V

(1)t , V

(1)t+h

)= e−λhV ar

(V

(1)t

).

Remark When we compute the corresponding formulae for the time change case, i.e. for V (2), weget almost the same results as for V (1) in the case where the time change is given by τt =

∫ t0 ξsds

for a positive, stationary process ξ: In the first moment, we need to replace E(ω0) by E(ξ0). For thevariances and covariances, we need to replace E(ω2

0) by E(ξ0). To see that note that

E(L(1)t ) = E(ω0)λE(L1)t,

E(L(2)t ) = E(ξ0)λE(L1)t.

However, for the quadratic variation, we have

E([L(1)]t) = E(ω20)λV ar(L1)t,

E([L(2)]t) = E(ξ0)λV ar(L1)t.

Also, we will use γ(h) = Cov(ξt, ξt+h).

B Proofs

Proof of Proposition 2 Note that P0 = 0 and E(Pt) = 0. From Ito’s product rule, we get that

PtV(1)t =

∫ t

0V (1)s dPs +

∫ t

0PsdV

(1)s + [P, V (1)]t

=∫ t

0V (1)s dPs − λ

∫ t

0PsV

(1)s ds+

∫ t

0PsωλsdLλs + [P, V (1)]t.

Taking expectations, we get

Cov(Pt, V(1)t ) = E(PtV

(1)t ) = −λ

∫ t

0E(PsV (1)

s )ds+ λE(L1)∫ t

0E (Psωλs) ds.

The above equation is an integral equation, which can be solved as soon as we have computed thesecond term on the right hand side. We do that by applying Ito’s product rule again and obtain

E (Puωλu) = E(∫ u

0Psdωλs

)+ E

(∫ u

0

√Vs−d[W,ωλ]s

)= −αλ

∫ u

0E(Psωλs)ds+ ρλγ

∫ u

0E(√

V(1)s√ωλs

)ds,

25

B PROOFS

which is yet another integral equation. Since V (1) and ωλ are strictly positive (provided the Feller

condition 2αβ > γ2 and ω0 > 0 holds) , we have that g(s) := E(√

V(1)s√ωλs

), for a strictly

positive function g. Solving the differential equation

d

duE (Puωλu) = −αλE(Puωλu) + ρλγg(u),

we get

E (Puωλu) = exp(−αλu)(∫ u

0ρλγg(s) exp(αλs)ds

)=: ρg(u) 6≡ 0,

for a strictly positive function g. Next, we define G(u) := λE(L1)E (Puωλu). Solving

d

dtE(PtV

(1)t ) = −λE(PtV

(1)t ) +G(t),

we obtain

E(PtV(1)t ) = exp(−λt)

∫ t

0G(x)eλxdx 6≡ 0.

Further, Ito’s formula leads to the following result for higher moments of order n ∈ R, n ≥ 2(provided they exist):

E (Pnt ) =n(n− 1)

2

∫ t

0E(Pn−2s V (1)

s

)ds.

In particular, we have

Cov(Pt, P 2t ) = E

(P 3t

)= 3

∫ t

0E(PsV (1)

s )ds = 3λE(L1)∫ t

0

∫ s

0eλuE (Puωλu) duds 6≡ 0.

Proof of Proposition 3 Since the integrated variance appears in the asset price formula, we computethe Laplace transform of the integrated variance first. In particular, we focus on the conditionalintegrated variance over the interval [t, t + h] given Ft for t, h ≥ 0. We use the notation EQ

t (·) :=EQ(·|Ft). Further, using (7), we get

EQt

[exp

(θ

∫ t+h

tV (i)s ds

)]= EQ

t

[exp

(θ

(ελ(h)V (i)

t +∫ t+h


s

))]= exp

(θελ(h)V (i)

t

)EQt

[exp

(θ

(∫ t+h


s

))].

We define the following σ–algebras G(1) := σ(ωλs : s ≤ t + h) ∪ Ft and G(2) := σ(τλs : s ≤t+ h) ∪ Ft.

26

B PROOFS

Laplace transforms for Model 1

Then

EQt

[exp

(θ

(∫ t+h

tελ(t+ h− s)dL(1)

s

))]= EQ

t

[exp

(θ

(∫ t+h

tελ(t+ h− s)ωλsdLλs

))]= EQ

t

[exp

(θ

(∫ t+h

tελ(t+ h− s)ωλsdLλs

))∣∣∣∣G(1)

]= EQ

t

[exp

(∫ t+h

tψLλ(θελ(t+ h− s)ωλs)ds

)]= EQ

t

[exp

(λ

∫ T

tψL(θελ(t+ h− s)ωλs)ds

)],

where ψLλ and ψL denote the log–transformed Laplace transform of Lλ and L, respectively:For the conditional distribution of Y (1), we get

φ(1)(θ) := EQt (exp(θY (1)

t+h)) = exp(θY (1)t )EQ

t (exp(θ(Y (1)t+h − Y

(1)t )))

= exp(θY (1)t )EQ

t

(exp

(θ

(µh+ β

∫ t+h

tV (1)s ds+

∫ t+h

t

√V

(1)s dWs

))).

Let G(1) := σ(Lλs|s ≤ t+ h) ∪ Ft. Then

φ(1)(θ) = exp(θYt)EQt

[EQ(

exp(θ

(µh+ β

∫ t+h

tV (i)s ds+

∫ t+h

t

√V

(i)s dWs

)))| G(i) ∪ G(1)

]= exp(θYt)EQ

t

[(exp

(θ

(µh+ β

∫ t+h

tV (1)s ds

)))

EQ(

exp(θ

∫ t+h

t

√V

(1)s dWs

)∣∣∣∣G(1) ∪ G(1)

)︸︷︷︸

=exp(θ2

2

∫ t+ht V

(1)s ds

)

= exp(θYt)EQ

t

[(exp

((θµh+

(θβ +

θ2

2

)∫ t+h

tV (1)s ds

)))]= exp

(θYt + θµh+

(βθ +

θ2

2

)ελ(h)V (1)

t

)EQt

[(exp

(∫ t+h

tf (1)(s, θ)dLλs

))]= exp

(θYt + θµh+

(βθ +

θ2

2

)ελ(h)V (1)

t

)EQt

[exp

(λ

∫ t+h

tψL(f (1)(s, θ))ds

)],

where f (1)(s, θ) := (βθ + θ2/2)ελ(t+ h− s)ωλs.

Laplace transforms for Model 2

In Model 2, we proceed differently. As in Section 3.6, we denote by T the random measure associatedwith τ and by L′ the Levy seed associated with L. Then we deduce from Barndorff-Nielsen (2010)that the Laplace transformation is given by

EQt

[exp

(θ

(∫ t+h

tελ(t+ h− s)dL(2)

s

))]= EQ

t

[exp

(θ

(∫ t+h

tελ(t+ h− s)dLτλs

))]27

B PROOFS

= EQt

[exp

(∫ t+h

tψL′(s)(θελ(T − s))T (λds)

)],

where ψL′(s) denotes the log–transformed Laplace transform of L′(s). Altogether, we get

φ(2)(θ) = exp(θY (2)t )EQ

t

[(exp

((θµh+

(θβ +

θ2

2

)∫ t+h

tV (2)s ds

)))]= exp

(θY

(2)t + θµh+

(βθ +

θ2

2

)ελ(h)V (2)

t

)EQt

[exp

(∫ t+h

tψL′(s)(f

(2)(s, θ))T (λds))]

,

where f (2)(s, θ) = (βθ + θ2/2)ελ(t+ h− s).

Proof of Proposition 4 We apply the following Bayes rule, see Karatzas and Shreve (1991, p.193).For any Ft measurable random variable ζ with EQt+h |ζ| <∞ and for 0 ≤ s ≤ t ≤ t+ h, we have

EQt+hs (ζ) =

1Zs

Es(ζZt),

and, hence

Es(ζ)− EQt+hs (ζ) = Es

(ζ

(1− Zt

Zs

)).

As a particular case, we get for the approximated variance risk premium, since [Y ][t,t+h] is Ft+hmeasurable,

AV RPt,t+h = Et(

[Y ][t,t+h]

(1− Zt+h

Zt

)).

Recall that

[Y ][t,t+h] =∫ t+h

tσ2sds+

∑t≤s≤t+h

(∆Js)2 = [Y ]c[t,t+h] + [Y ]d[t,t+h],

where [Y ]c[t,t+h] denotes the continuous part of the quadratic variation and [Y ]d[t,t+h] the jump part.

Next we plug in the explicit formula for V (i)+, see (7), and we obtain

[Y ]c[t,t+h] = V(i)+t+h − V

(i)+t = ελ(h)V (i)

t +∫ t+h


s .

Hence

Et(

[Y ]c[t,t+h]

(1− Zt+h

Zt

))= ελ(h)Vt Et

((1− Zt+h

Zt

))︸︷︷︸

=0

+ Et(∫ t+h


s

(1− Zt+h

Zt

))= Et

(∫ t+h


s

(1− Zt+h

Zt

)).

28

B PROOFS

For the jump part of the quadratic variation, we get

Et(

[Y ]d[t,t+h]

(1− Zt+h

Zt

))= Et

∑t≤s≤t+h

(∆Js)2

(1− Zt+h

Zt

) .

Proof of Proposition 5 Since the jumps come from a Levy process, the conditional expectation Etequals the unconditional one and we get

AV RP dt,t+h =1h

Et

∑t≤s≤t+h

(∆Js)2

− EQt

∑t≤s≤t+h

(∆Js)2

=∫

Rx2νJ(dx)−

∫Rx2νQ

J (dx),

where νJ(dx)dt and νQJ (dx)dt are the predictable compensators of the Poisson random measure as-

sociated with J under P and under Q, respectively.

Proof of Proposition 6 For v(1)t := ωλt and v(2)

t := ξt, we get for i = 1, 2,

Et(∫ t+h


s

)= λE(L1)

∫ t+h

tελ(t+ h− s)Et

(v(i)s

)ds.

Under a structure preserving measure change, we have for i = 1, 2,

EQt

(∫ t+h


s

)= λEQ(L1)

∫ t+h

tελ(t+ h− s)EQ

t

(v(i)s

)ds.

Hence

Et(∫ t+h


s

)− EQ

t

(∫ t+h


s

)= λ

∫ t+h

tελ(t+ h− s)

(E(L1)Et

(v(i)s

)− EQ(L1)EQ

t

(v(i)s

))ds

= v(i)t

(E(L1)− EQ(L1)

)(h− ελ(h))

+ λ

∫ t+h

tελ(t+ h− s)

(E(L1)Et

(v(i)s − v

(i)t

)− EQ(L1)EQ

t

(v(i)s − v

(i)t

))ds.

Note that L is a subordinator and, hence, E(L1) > 0. Also, under the structure preserving change ofmeasure, the predictable compensator of L changes and, hence, E(L1)−EQ(L1) 6= 0. This concludesthe proof.

29

B PROOFS

Proof of Corollary 7 Throughout the proof we assume that s ≥ t. Further, we skip the superscriptand write vt := v

(i)t for ease of exposition. Then we have

Et(vs − vt) = Et(∫ s

tdvu

)= ab(s− t)− a

∫ s

tEt(vu)du.

Next, we define the random variable Zu := Et(vu). Then, we have

dZs = a(b− Zs)ds, Zt = vt.

Hence, we get

Zs = Et(vs) = Zte−a(s−t) + b(1− e−a(s−t)) = vte

−a(s−t) + b(1− e−a(s−t)),

and

Et(vs)− vt = vt(e−a(s−t) − 1) + b(1− e−a(s−t)) = −aεa(s− t)vt + abεa(s− t).

Consequently, we obtain

λ

∫ t+h

tελ(t+ h− s) (E(L1)Et (vs − vt)) ds = E(L1)G(t, t+ h)(b− vt),

where

G(t, t+ h) = G(t, t+ h, a, λ) = λa

∫ t+h

tελ(t+ h− s)εa(s− t)ds

= −a+ λ

aλ+ h− a

λ(λ− a)e−λh +

λ

a(λ− a)e−ah.

(14)

The results under the risk neutral measure are essentially the same using the risk neutral parametersaQ, bQ. Also, we denote byGQ(t, t+h) := G(t, t+h, aQ, λ) the functionG defined in (14) evaluatedat the risk neutral parameter. Altogether, we have

AV RP(i)ct,t+h = v

(i)t F (1)(t, t+ h) + F (2)(t, t+ h),

where for κ1 = E(L1) and κQ1 = E(L1)

F (1)(t, t+ h) := F (1)(t, t+ h, λ, a, aQ, κ1, κQ1 )

=(κ1 − κQ

1

)(1− ελ(h)

h

)− 1h

(κ1G(t, t+ h)− κQ

1 GQ(t, t+ h)

),

F (2)(t, t+ h) := F (2)(t, t+ h, λ, a, aQ, b, bQ, κ1, κQ1 )

=1h

[κ1G(t, t+ h)b− κQ

1 GQ(t, t+ h)bQ

].

30

B PROOFS

Proof of Corollary 8 Throughout the proof we assume that s ≥ t. Again, we skip the superscriptand write vt := v

(i)t for ease of exposition. Then we have

Et(vs − vt) = Et(∫ s

tdvu

)= −a

∫ s

tEt(vu)du+ a(s− t)E(Lv1).

So, when we define b := E(Lv1), we get exactly the same results as in the previous Corollary and wecan define

G(1)(t, t+ h) := F (1)(t, t+ h), and G(2)(t, t+ h) := F (2)(t, t+ h).

31

stochastic volatility of volatility and variance risk premia

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