# A Hull and White Formula for a General Stochastic Volatility Jump-Diffusion Model with Applications to the Study of the Short-Time Behavior of the Implied Volatility

Authors: Elisa Alòs; Jorge A. León; Monique Pontier; Josep Vives
Journal: Journal of Applied Mathematics and Stochastic Analysis
 (2008)

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## Abstract

We obtain a Hull and White type formula for a
general jump-diffusion stochastic volatility model, where the
involved stochastic volatility process is correlated not only with
the Brownian motion driving the asset price but also with the
asset price jumps.
Towards this end, we establish an anticipative Itô's formula, using Malliavin calculus techniques for Lévy processes on the canonical space. As an application, we show that the dependence of the volatility process on the asset price jumps has no effect on the short-time behavior of the
at-the-money implied volatility skew.


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## Body

## 1. Introduction

It iswell known that classical stochastic volatility
models, where the volatility is allowed to be a diffusion process, are able to
capture the dependence of the implied volatility as a function of the strike
(thesmile or the skew). Nevertheless, they can not explain its
dependence with respect to time to maturity (term
structure). For example, empirical observations indicate that the
at-the-money skew slope is approximately O((T−t)−1/2) (here T−t denotes the time to maturity), while the rate
for these stochastic volatility models is O(1) (e.g., Lewis [1], Lee [2], or Medvedev and Scaillet [3]). The introduction of jumps in the asset price dynamics is
a natural extension of classical stochastic volatility models proposed with the
aim to capture this short-time behavior. Although the rate of the skew slope
for models with jumps is still O(1) (see Medvedev and Scaillet [3]), they allow
flexible modelling, and generate skews and smiles similar to those observed
empirically (see Bates [4], Barndorff-Nielsen and Shephard [5, 6], or
Carr and Wu [7]).In Alòs et al. [8], the authors
considered general jump-diffusion stochastic volatility models where the
volatility is not necessarily a diffusion. They proved that for a volatility process
independent of price jumps (as in the Bates [4] case) the at-the-money skew
slope explodes if and only if the Malliavin derivative of the volatility
process Dsσt also does when s↑t.
This allows us to consider new models—where the volatility is not required to
be Markovian nor to be a diffusion process—that capture the short-time
explosion of skew slopes. The basic idea in that work was to expand option
prices around the classical Hull and White expression by means of Malliavin
calculus, following similar ideas as in Alòs [9]. This gives us a
decomposition of option prices that allows us to identify the effect of
correlation (between the volatility and the Brownian motion driving stock
prices) and the effect of jumps in the at-the-money skew slope when time to
maturity tends to zero. This can be interpreted as an answer of a demand in
Fouque et al. [10, end of page 54].In this paper, we study jump-diffusion stochastic
volatility models allowing the volatility process to be correlated also with
the price jumps (see Bakshi et al. [11] and Duffie et al. [12], among others). Our main goal in this work is to describe
analytically the effect of this extension in the at-the-money short-time
behavior of the stochastic volatility. The idea will be again to decompose
option prices around the Hull and White term, now by using Malliavin calculus
for Lévy processes (see Solé et al. [13], Løkka [14] and Petrou
[15]). In comparison with the formula obtained in Alòs et al. [8], here we obtain an extra term because the volatility depends now on the
jump price. This representation allows us to show that the existence of
correlation between the volatility process and the price jumps does not have
any influence on the at-the-money skew of the implied volatility as time runs
to expiry, confirming an heuristic idea explained in Gatheral [16, page 70].The paper is organized as follows. In Section2, we give
the main hypotheses and notations. Section 3 is devoted to introduce the
Malliavin calculus framework needed in the remaining of the paper. In Section
4, we obtain the Hull and White formula. In Section 5, we apply it to the
problem of describing the at-the-money short time skew of the implied
volatility. Finally, Section 6 is devoted to the conclusions.


## 2. Main Hypotheses and Notations

We consider a log-price process, under the market
chosen risk-neutral probability measure, given by(2.1)Xt=x+(r−λk)t−12∫0tσs2ds+∫0tσs(ρdWs+1−ρ2dBs)+Zt,where, t∈[0,T], x is the current log-price, r is the instantaneous interest rate, W and B are independent standard Brownian motions, ρ∈[−1,1],
and Z is a compound Poisson process, independent of W and B,
with intensity λ,
finite Lévy measure ν,
and with k:=(1/λ)∫ℝ(ey−1)ν(dy).We assume that the processσ is adapted to the filtration generated by W and Z. So, in this paper, generalizing Alòs et al. [8], we allow the volatility to have nonpredictable jump times as
advocated by Bakshi et al. [11] and Duffie et al. [12], among others.In the following, we denote byℱW,ℱB,
and ℱZ the filtrations generated by the independent
processes W,B,
and Z,
respectively. Moreover, we define ℱ:=ℱW∨ℱB∨ℱZ.It is well known that if we price a European call with
strike priceK by the formula(2.2)Vt=e−r(T−t)E[(eXT−K)+∣ℱt],where E denotes the expectation with respect to a
risk-neutral measure, there is no arbitrage opportunity. Thus Vt is a possible price for this derivative.In the sequel, we use the following notation:(i)
The processvt:=(Yt/(T−t))1/2, with Yt:=∫tTσs2ds, denotes the future average volatility.(ii)
WithBS(t,x,σ) we represent the classical Black-Scholes
function with constant volatility σ,
current log stock price x,
time to maturity T−t, strike price K,
and interest rate r. This function can be written
as(2.3)BS(t,x,σ)=exΦ(x−xt*σT−t+σ2T−t)−ext*Φ(x−xt*σT−t−σ2T−t),where xt*=logK−r(T−t) is the future log-price at t and Φ is the cumulative probability function of the
standard normal law.(iii)
WithN we denote the Poisson random measure on [0,T]×ℝ such that Zt=∫[0,t]×ℝxN(ds,dx). Moreover, N(ds,dx):=N(ds,dx)−dsν(dx) is the compensated Poisson random measure.(iv)
We consider the operatorℒBS(σ):=∂t+(1/2)σ2∂xx2+(r−(1/2)σ2)∂x−r which satisfies ℒBS(σ)BS(⋅,⋅,σ)=0.


## 3. Required Tools of Malliavin Calculus for Lévy Processes

### 3.1. Introduction

In this section, we introduce the tools of Malliavin
calculus for Lévy processes that we need in the rest of the paper.Consider a complete probability space(Ω,ℱ,ℙ) and let L={Lt,t∈[0,T]} be a càdlàg Lévy process with triplet (γ,σ,ν). See for example the book of Sato [17] for a
general theory of Lévy processes.It is well known thatL can be represented as(3.1)Lt=γt+σWt+∬(0,t]×{|x|>1}xN(ds,dx)+limϵ↓0∬(0,t]×{ϵ<|x|≤1}xN(ds,dx),where W is a Brownian motion and N is the Poisson random measure associated to ν. It is also known that ℱL=ℱW∨ℱN. See for example, Solé et al. [13].In general, the construction of a Malliavin calculus,
based on a chaos expansion, for a certain process follows three main steps.
First of all, to prove a chaotic representation property, secondly, to define
formally the gradient and divergence operators, and finally, to give their
probabilistic interpretations.In the last years, several approaches to the Malliavin
calculus for Lévy processes have been developed, with different probabilistic
interpretations of gradient operators. Between them, we mention the approach of
Nualart and Schoutens [18], the approach of Solé et al. [13], based
on Itô [19], and the approach of Løkka [14] and Petrou [15]. In this
paper, we follow the last one, because in it, the form of the gradient operator
simplifies strongly our computations. As a tool for our results we develop in
Sections 3.3 and 3.4 two transfer formulas between the second and the third
methods. Finally, let us remark that we are under the conditions of Løkka's
approach because our Lévy measure ν is finite.


### 3.2. The Chaotic Representation Property

Consider the space[0,T]×ℝ with its Borel family of sets ℬ([0,T]×ℝ). We can introduce the centered independent
random measure given by(3.2)M(E):=σ∫E(0)dWt+∬E′N(dt,dx),E∈ℬ([0,T]×ℝ),where E(0):={t∈[0,T]:(t,0)∈E} and E′:=E−E(0)×{0}. Its variance is given by(3.3)μ(E)=σ2∫E(0)dt+∬E′dt⁢ν(dx).Note thatW can be seen as a centered independent Gaussian
random measure on [0,T],
and N(ds,dx) can be seen as a centered independent random
measure on [0,T]×ℝ0 where ℝ0=ℝ−{0}. Thus we can write(3.4)M(ds,dx)=σ(W⊗δ0)(ds,dx)+N(ds,dx),where δ0 is the Dirac's delta, that is, a unitary mass
on the point {0}.In this context, we can define stochastic multiple
integralsIn with respect to M with kernels in the Hilbert spaces(3.5)Ln2:=L2(([0,T]×ℝ)n,⁢ℬ([0,T]×ℝ)n,μ⊗n),in the usual way, and to prove
that if {ℱtX,t∈[0,T]} is the completed natural filtration of X, for any random variable F∈L2(Ω,ℱTX,ℙ) we have the chaotic
representation(3.6)F=∑n=0∞In(fn),where the kernels are unique if
we take them symmetric.


### 3.3. The Malliavin-Type Derivative

Let us denote by𝔻1,2 the set of random variables in L2(Ω),
of the form (3.6), such that ∑n=1∞nn!∥fn∥Ln22<∞. The Malliavin derivative DMF of a random variable F∈𝔻1,2 is the process {Dt,xMF,(t,x)∈[0,T]×ℝ} defined by Dt,xMF:=∑n=1∞nIn−1(fn((t,x),⋅)).In order to give the probabilistic interpretation of
this operator, we assume, in the remaining, as in Solé et al. [13], that the underlying probability space is the canonical Lévy space (ΩW×ΩN,ℱW⊗ℱN,ℙW⊗ℙN). That is, (ΩW,ℱW,ℙW) is the canonical Wiener space and (ΩN,ℱN,ℙN) is the canonical Lévy space of the compound
Poisson process with Lévy measure ν. Moreover, we assume that W and Z are the canonical processes.Letω=(ωW,ωN) be an element of this space. So, ωW is a continuous trajectory null at the origin
and ωN is a sequence of (jump instant, jump size) pairs(3.7)ωN:=((t1,x1),(t2,x2),(t3,x3),…).From Petrou [15], we have(3.8)Dt,0M=1σDtW1{σ>0},where DtW denotes the classical Malliavin derivative
with respect to the Brownian motion W (e.g., Nualart [20]). Denote by 𝔻W its domain.In order to obtain the probabilistic interpretation ofDt,xM for x≠0,
we consider the following transformation.Given(t,x)∈[0,T]×ℝ0,
we can add to any ωN a jump of size x at instant t,
denote the new element(3.9)ωt,xN:=((t,x),(t1,x1),(t2,x2),…),and write ωt,x:=(ωW,ωt,xN). So, for any (t,x)∈[0,T]×ℝ0,
we can define the operator Tt,xF:=F(ωt,x). As it is shown in Solé et al. [13, Proposition 4.8] this is a well-defined operator.For allF∈𝔻1,2, define(3.10)Dt,xNF=Tt,xF−F,x≠0,and denote by 𝔻N its domain.This operator is related toDM. Indeed, in Solé et al. [13] is
considered the random measure(3.11)M¯(E):=σ∫E(0)dWt+∬E′xdN(t,x),E∈B([0,T]×ℝ).Observe that(3.12)M¯(E)=I1(h(x)1E(t,x)),where (3.13)h(x):=x1{x≠0}+1{x=0}.Therefore, we obtain the transfer
principle(3.14)Dt,xMF=h(x)Dt,xM¯F,for F∈𝔻1,2 and (t,x)∈[0,T]×ℝ. Here DM¯ is the annihilation
operator used in Solé et al. [13].Thus, combining results from Solé et al. [13] and Alòs et al. [21] it is easy to show that for F∈L2(Ω),(3.15)DNF∈L2([0,T]×ℝ0,⁢μ),F∈DW⟺F∈D1,2,and in this case Dt,xMF=Dt,xNF,x≠0.Observe that we have proved(3.16)Dt,xM=1{σ>0}1{0}(x)1σDtW+1ℝ0(x)Dt,xN,which follows from (3.8) and (3.10).Observe also that it is immediate from (3.10), to see
that(3.17)Dt,xN(FG)=FDt,xNG+GDt,xNF+Dt,xNFDt,xNG.Finally, we have𝔻1,2=𝔻W∩𝔻N,
due to Solé et al. [13, Section 2] and the equalities (3.12) and
(3.14).


### 3.4. The Skorohod-Type Integral

Letu∈L2(Ω×[0,T]×ℝ,ℙ⊗μ).
For almost all (t,x) we have the chaotic decomposition (see Section
3.2)(3.18)ut,x=∑n=0∞In(fn(t,x,⋅)),where fn∈𝕃n+12 is symmetric in the n last variables.Letf^n be the symmetrization in all n+1 variables. Then, we define the Skorohod
integral of u by(3.19)δ(u)=∑n=0∞In+1(f^n),in L2(Ω), provided u∈DomδM, that means(3.20)∑n=0∞(n+1)!∥f^n∥Ln+122<∞.Moreover, ifF∈𝔻1,2 and u∈DomδM we have the duality relation(3.21)E(FδM(u))=E∫0T∫ℝu(t,x)Dt,xMFμ(dt,dx).So, in this sense,δM is the dual of the operator DM. Sometimes we will write δt(u) instead of δ(u𝟙[0,t]).Note, also, that the transfer principle (3.14) allows us
to establish a transfer principle between δM and the divergence operator δM¯ given in Solé et al. [13].
Namely,(3.22)δM(u)=δM¯(uh),u∈DomδM.The following lemma is useful for our purposes. A
version of this lemma in the pure jump case is given in Di Nunno et al. [22, Theorem 3.13].Lemma 3.1.
LetF∈𝔻1,2 and u∈DomδM such that u⋅(F+DMF⋅𝟙ℝ0)∈L2(Ω×[0,T]×ℝ,ℙ⊗μ). Then,
(3.23)u⁢⋅(F+DMF⁢⋅1ℝ0)∈DomδM⟺FδM(u)−∫[0,T]×ℝu(t,x)Dt,xMFμ(dt,dx)∈L2(Ω),and in this case(3.24)δM(u⁢⋅F)=FδM(u)−δM(u⁢⋅DMF⁢⋅1ℝ0)−∫[0,T]×ℝu(t,x)Dt,xMFμ(dt,dx).Proof.
This result follows using relations
(3.16) and (3.21); Alòs et al. [21, Lemma 2.4 and Proposition 2.5]
and the transfer formulas (3.14) and (3.22). Note that, in our case, F does not need to be bounded. This is a
consequence of the fact that if G is a bounded random variable of L2(ΩN) and ν is finite, we have that G∈𝔻N and DNG is also bounded.In order to give the relation betweenδM and the pathwise integral with respect to N, we consider the following two
sets.Definition 3.2.
We define𝕃1,2:=L2([0,T]×ℝ;𝔻1,2).Observe that ifu={u(s,y):(s,y)∈[0,T]×ℝ} is a random field of 𝕃1,2 we have, in particular, that u and DMu are in L2(ℙ⊗μ) and L2(ℙ⊗μ⊗μ),
respectively. Moreover 𝕃1,2⊆DomδM.Definition 3.3.
We
define𝕃−1,2 as the subset of 𝕃1,2 of random fields u such that the following ℙ⊗μ -a.s. left-limits exist and belong to L2(ℙ⊗μ)(3.25)u−(s,y)=limr↑s,x↑yu(r,x),D−u(s,y)=limr↑s,x↑yDs,yMu(r,x).Proposition 3.4.
Assume thatu is a random field belonging to 𝕃−1,2. Assume(3.26)∫0T∫ℝ0|u−(s,x)|N(ds,dx)∈L2(Ω),where ∫0T∫ℝ0u(s,x)N(ds,dx) is the classical path-by-path integral defined
by ∑ΔZt≠0u(t,ΔZt).
Then,T−u:=u−+D−u∈DomδM, and in this case,
(3.27)δM((u−+D−u)⋅1ℝ0)=∫0T∫ℝ0u−(s,x)N(ds,dx)−∫0T∫ℝ0D−u(s,x)ν(dx)ds,or equivalently, (3.28)δM(T−u⁢⋅1ℝ0)=∫0T∫ℝ0u−(s,x)N(ds,dx)−∫0T∫ℝ0T−u(s,x)ν(dx)ds.Proof.
Assume as a first step thatu∈𝕃−1,2 is bounded. Then, DMu, u− and D−u are also bounded on ℝ0.
In particular (3.26) is true.
We begin considering the following partition of[0,∞)×ℝ:(3.29)0=s0<s1<⋯<sn<∞=sn+1,−∞=x0<x1<⋯<xm<∞=xm+1.
Then, we can define(3.30)un,m(s,x)=∑i=0n∑j=0mu(si,xj)1(si,si+1](s)1(xj,xj+1](x).
Using Lemma3.1, we have that for all n and m(3.31)δM(un,m⁢⋅1ℝ0)+δM(Dun,m⁢⋅1ℝ0)=∑i=0n∑j=0mu(si,xj)δM(1(si,si+1]1(xj,xj+1]1ℝ0)−∑i=0n∑j=0m∫0T∫ℝ01(si,si+1](s)1(xj,xj+1](x)Ds,xMu(si,xj)ν(dx)ds.
First of all, observe that ifr,s∈]si,si+1] and x,y∈]xj,xj+1], then (Ds,yMun,m)(r,x)=u(si,xj,ωs,y)−un,m(r,x) and (Ds,yMu)(r,x)=u(r,x,ωs,y)−u(r,x) almost surely go to the same limit whatever n and m go to infinity or r↑s and x↑y. By the theorem hypothesis this limit is D−u.
Observe now that beingu bounded, and having un,m the same bound, D−u and u− are also L2-limits. So, using that δM is a closed operator, the left-hand side in
(3.31) goes to δM((u−+D−u)⋅𝟙ℝ0) in L2 if we prove that the terms on the right-hand
side converge in L2 to the limits defined by the proposition.
For the first term in the right-hand side, observe
thatδM coincides with the path-by-path integral
because the integrand is deterministic. Then, using u is bounded and the dominated convergence
theorem we obtain the expected L2-limit. For the second term, we have also a
direct application of dominated convergence theorem.
In order to prove the nonbounded case observe that we
can assume thatu is positive, because the formula that we want
to prove is linear. Then, for the general case, we simply have to apply the
result separately to the positive and negative parts.
So, letu≥0 and uK=u⋀K. Of course, uK≤u and uK converges increasingly to u. We have, as a consequence of the first step,
that
(3.32)δM((T−uK)⋅1ℝ0)=∫0T∫ℝ0uK−(s,x)N(ds,dx)−∫0T∫ℝ0T−uK(s,x)ν(dx)ds.
Beingu− and T−u in L2, we have that uK− and T−uK go up to u− and T−u in L2, respectively. So, hypothesis (3.26), the
monotone convergence theorem and the closeness of the operator δM yield the result.Remark 3.5.
Notice from the proof of Proposition3.4 that we can change the space 𝕃−1,2 by a similar space with left limits with
respect to the time variable s but with right limits with respect the space
variable y.Remark 3.6.
Observe
thatT−u=u− when u is adapted to the filtration generated by N.
Therefore, in such case(3.33)∫0t∫ℝ0u−(s,y)N(ds,dy)=δM(u−(⋅,⋅)1[0,t]×ℝ0(⋅,⋅)).That is, in this case, the
pathwise and Skorohod integrals with respect to N are the same.Observe that in the last two results there is no
contribution ofW because on ℝ0 the operator δM coincides with the Skorohod-type integral with
respect to N,
as the following result shows.Lemma 3.7.
LetδW and δN be the adjoint operators of DW and DN,
respectively, and u∈DomδM. Then u also belongs to DomδW∩DomδN and(3.34)δM(u)=σδW(u⋅,0)+δN(u1ℝ0).Proof.
This result is implied by (3.21) and
(3.16).


### 3.5. The Anticipating Itô's Formula

The basic tool for our results is the following
anticipative Itô formula. Recall that the processX is introduced in (2.1) and Y is the future average volatility, which is an
anticipative process, even σ is adapted.Theorem 3.8.
Letσ2∈𝕃1,2 and F:[0,T]×ℝ×[0,∞)→ℝ be a bounded function in C1,2,2([0,T]×ℝ×[0,∞)) with bounded derivatives.
Then,(3.35)F(t,Xt,Yt)−F(0,X0,Y0)=∫0t∂sF(s,Xs,Ys)ds+∫0t∂xF(s,Xs,Ys)(r−σs22−λk)ds+δtW,B(∂xF(⋅,X.,Y.)σ.)−∫0t∂yF(s,Xs,Ys)σs2ds+ρ∫0t∂xy2F(s,Xs,Ys)Λsds+12∫0t∂xx2F(s,Xs,Ys)σs2ds+δtN(T−u⁢⋅1ℝ0)+∫0t∫ℝ0T−u(s,x)ν(dx)ds,where δW,B is the Skorohod integral with respect to the
Brownian motion ρWs+1−ρ2Bs,Λs:=(∫sTDsWσr2dr)σs and u(s,x):=F(s,Xs−+x,Ys)−F(s,Xs−,Ys).Proof.
The
proof is as in Alòs et al. [8] combined by Proposition 3.4 to treat
the sum of jump terms.
We apply it to the random fieldu(s,x)=F(s,Xs−+x,Ys)−F(s,Xs−,Ys). Here, the independence between Z,B,
and W,
the fact that Y is a continuous process and the fact that Z is a compound Poisson process with a finite
number of jumps on every compact time interval play a key role.
Indeed, letTi denote these jump instants. Then, (3.36)F(Ti+1,XTi+1,YTi+1)−F(Ti,XTi,YTi)=∫TiTi+1−dF(s,Xs,Ys)+F(Ti+1,XTi+1,YTi+1)−F(Ti+1,XTi+1−,YTi+1).
The first term yields a standard Itô formula
concerning continuous process, so Alòs et al. [8] results apply
and we get the six first terms in the right-hand side of the Theorem 3.8 formula.
On other hand, the sum of second terms is the path by path integral(3.37)∫0t∫ℝ0u(s,x)N(ds,dx).Note here that F,X−,Y are left continuous so u=u−.
Then, using Proposition 3.4 we get the last sum is equal to: (3.38)δtN(T−u)+∫0t∫ℝ0T−u(s,x)ν(dx)ds.


## 3.1. Introduction

In this section, we introduce the tools of Malliavin
calculus for Lévy processes that we need in the rest of the paper.Consider a complete probability space(Ω,ℱ,ℙ) and let L={Lt,t∈[0,T]} be a càdlàg Lévy process with triplet (γ,σ,ν). See for example the book of Sato [17] for a
general theory of Lévy processes.It is well known thatL can be represented as(3.1)Lt=γt+σWt+∬(0,t]×{|x|>1}xN(ds,dx)+limϵ↓0∬(0,t]×{ϵ<|x|≤1}xN(ds,dx),where W is a Brownian motion and N is the Poisson random measure associated to ν. It is also known that ℱL=ℱW∨ℱN. See for example, Solé et al. [13].In general, the construction of a Malliavin calculus,
based on a chaos expansion, for a certain process follows three main steps.
First of all, to prove a chaotic representation property, secondly, to define
formally the gradient and divergence operators, and finally, to give their
probabilistic interpretations.In the last years, several approaches to the Malliavin
calculus for Lévy processes have been developed, with different probabilistic
interpretations of gradient operators. Between them, we mention the approach of
Nualart and Schoutens [18], the approach of Solé et al. [13], based
on Itô [19], and the approach of Løkka [14] and Petrou [15]. In this
paper, we follow the last one, because in it, the form of the gradient operator
simplifies strongly our computations. As a tool for our results we develop in
Sections 3.3 and 3.4 two transfer formulas between the second and the third
methods. Finally, let us remark that we are under the conditions of Løkka's
approach because our Lévy measure ν is finite.


## 3.2. The Chaotic Representation Property

Consider the space[0,T]×ℝ with its Borel family of sets ℬ([0,T]×ℝ). We can introduce the centered independent
random measure given by(3.2)M(E):=σ∫E(0)dWt+∬E′N(dt,dx),E∈ℬ([0,T]×ℝ),where E(0):={t∈[0,T]:(t,0)∈E} and E′:=E−E(0)×{0}. Its variance is given by(3.3)μ(E)=σ2∫E(0)dt+∬E′dt⁢ν(dx).Note thatW can be seen as a centered independent Gaussian
random measure on [0,T],
and N(ds,dx) can be seen as a centered independent random
measure on [0,T]×ℝ0 where ℝ0=ℝ−{0}. Thus we can write(3.4)M(ds,dx)=σ(W⊗δ0)(ds,dx)+N(ds,dx),where δ0 is the Dirac's delta, that is, a unitary mass
on the point {0}.In this context, we can define stochastic multiple
integralsIn with respect to M with kernels in the Hilbert spaces(3.5)Ln2:=L2(([0,T]×ℝ)n,⁢ℬ([0,T]×ℝ)n,μ⊗n),in the usual way, and to prove
that if {ℱtX,t∈[0,T]} is the completed natural filtration of X, for any random variable F∈L2(Ω,ℱTX,ℙ) we have the chaotic
representation(3.6)F=∑n=0∞In(fn),where the kernels are unique if
we take them symmetric.


## 3.3. The Malliavin-Type Derivative

Let us denote by𝔻1,2 the set of random variables in L2(Ω),
of the form (3.6), such that ∑n=1∞nn!∥fn∥Ln22<∞. The Malliavin derivative DMF of a random variable F∈𝔻1,2 is the process {Dt,xMF,(t,x)∈[0,T]×ℝ} defined by Dt,xMF:=∑n=1∞nIn−1(fn((t,x),⋅)).In order to give the probabilistic interpretation of
this operator, we assume, in the remaining, as in Solé et al. [13], that the underlying probability space is the canonical Lévy space (ΩW×ΩN,ℱW⊗ℱN,ℙW⊗ℙN). That is, (ΩW,ℱW,ℙW) is the canonical Wiener space and (ΩN,ℱN,ℙN) is the canonical Lévy space of the compound
Poisson process with Lévy measure ν. Moreover, we assume that W and Z are the canonical processes.Letω=(ωW,ωN) be an element of this space. So, ωW is a continuous trajectory null at the origin
and ωN is a sequence of (jump instant, jump size) pairs(3.7)ωN:=((t1,x1),(t2,x2),(t3,x3),…).From Petrou [15], we have(3.8)Dt,0M=1σDtW1{σ>0},where DtW denotes the classical Malliavin derivative
with respect to the Brownian motion W (e.g., Nualart [20]). Denote by 𝔻W its domain.In order to obtain the probabilistic interpretation ofDt,xM for x≠0,
we consider the following transformation.Given(t,x)∈[0,T]×ℝ0,
we can add to any ωN a jump of size x at instant t,
denote the new element(3.9)ωt,xN:=((t,x),(t1,x1),(t2,x2),…),and write ωt,x:=(ωW,ωt,xN). So, for any (t,x)∈[0,T]×ℝ0,
we can define the operator Tt,xF:=F(ωt,x). As it is shown in Solé et al. [13, Proposition 4.8] this is a well-defined operator.For allF∈𝔻1,2, define(3.10)Dt,xNF=Tt,xF−F,x≠0,and denote by 𝔻N its domain.This operator is related toDM. Indeed, in Solé et al. [13] is
considered the random measure(3.11)M¯(E):=σ∫E(0)dWt+∬E′xdN(t,x),E∈B([0,T]×ℝ).Observe that(3.12)M¯(E)=I1(h(x)1E(t,x)),where (3.13)h(x):=x1{x≠0}+1{x=0}.Therefore, we obtain the transfer
principle(3.14)Dt,xMF=h(x)Dt,xM¯F,for F∈𝔻1,2 and (t,x)∈[0,T]×ℝ. Here DM¯ is the annihilation
operator used in Solé et al. [13].Thus, combining results from Solé et al. [13] and Alòs et al. [21] it is easy to show that for F∈L2(Ω),(3.15)DNF∈L2([0,T]×ℝ0,⁢μ),F∈DW⟺F∈D1,2,and in this case Dt,xMF=Dt,xNF,x≠0.Observe that we have proved(3.16)Dt,xM=1{σ>0}1{0}(x)1σDtW+1ℝ0(x)Dt,xN,which follows from (3.8) and (3.10).Observe also that it is immediate from (3.10), to see
that(3.17)Dt,xN(FG)=FDt,xNG+GDt,xNF+Dt,xNFDt,xNG.Finally, we have𝔻1,2=𝔻W∩𝔻N,
due to Solé et al. [13, Section 2] and the equalities (3.12) and
(3.14).


## 3.4. The Skorohod-Type Integral

Letu∈L2(Ω×[0,T]×ℝ,ℙ⊗μ).
For almost all (t,x) we have the chaotic decomposition (see Section
3.2)(3.18)ut,x=∑n=0∞In(fn(t,x,⋅)),where fn∈𝕃n+12 is symmetric in the n last variables.Letf^n be the symmetrization in all n+1 variables. Then, we define the Skorohod
integral of u by(3.19)δ(u)=∑n=0∞In+1(f^n),in L2(Ω), provided u∈DomδM, that means(3.20)∑n=0∞(n+1)!∥f^n∥Ln+122<∞.Moreover, ifF∈𝔻1,2 and u∈DomδM we have the duality relation(3.21)E(FδM(u))=E∫0T∫ℝu(t,x)Dt,xMFμ(dt,dx).So, in this sense,δM is the dual of the operator DM. Sometimes we will write δt(u) instead of δ(u𝟙[0,t]).Note, also, that the transfer principle (3.14) allows us
to establish a transfer principle between δM and the divergence operator δM¯ given in Solé et al. [13].
Namely,(3.22)δM(u)=δM¯(uh),u∈DomδM.The following lemma is useful for our purposes. A
version of this lemma in the pure jump case is given in Di Nunno et al. [22, Theorem 3.13].Lemma 3.1.
LetF∈𝔻1,2 and u∈DomδM such that u⋅(F+DMF⋅𝟙ℝ0)∈L2(Ω×[0,T]×ℝ,ℙ⊗μ). Then,
(3.23)u⁢⋅(F+DMF⁢⋅1ℝ0)∈DomδM⟺FδM(u)−∫[0,T]×ℝu(t,x)Dt,xMFμ(dt,dx)∈L2(Ω),and in this case(3.24)δM(u⁢⋅F)=FδM(u)−δM(u⁢⋅DMF⁢⋅1ℝ0)−∫[0,T]×ℝu(t,x)Dt,xMFμ(dt,dx).Proof.
This result follows using relations
(3.16) and (3.21); Alòs et al. [21, Lemma 2.4 and Proposition 2.5]
and the transfer formulas (3.14) and (3.22). Note that, in our case, F does not need to be bounded. This is a
consequence of the fact that if G is a bounded random variable of L2(ΩN) and ν is finite, we have that G∈𝔻N and DNG is also bounded.In order to give the relation betweenδM and the pathwise integral with respect to N, we consider the following two
sets.Definition 3.2.
We define𝕃1,2:=L2([0,T]×ℝ;𝔻1,2).Observe that ifu={u(s,y):(s,y)∈[0,T]×ℝ} is a random field of 𝕃1,2 we have, in particular, that u and DMu are in L2(ℙ⊗μ) and L2(ℙ⊗μ⊗μ),
respectively. Moreover 𝕃1,2⊆DomδM.Definition 3.3.
We
define𝕃−1,2 as the subset of 𝕃1,2 of random fields u such that the following ℙ⊗μ -a.s. left-limits exist and belong to L2(ℙ⊗μ)(3.25)u−(s,y)=limr↑s,x↑yu(r,x),D−u(s,y)=limr↑s,x↑yDs,yMu(r,x).Proposition 3.4.
Assume thatu is a random field belonging to 𝕃−1,2. Assume(3.26)∫0T∫ℝ0|u−(s,x)|N(ds,dx)∈L2(Ω),where ∫0T∫ℝ0u(s,x)N(ds,dx) is the classical path-by-path integral defined
by ∑ΔZt≠0u(t,ΔZt).
Then,T−u:=u−+D−u∈DomδM, and in this case,
(3.27)δM((u−+D−u)⋅1ℝ0)=∫0T∫ℝ0u−(s,x)N(ds,dx)−∫0T∫ℝ0D−u(s,x)ν(dx)ds,or equivalently, (3.28)δM(T−u⁢⋅1ℝ0)=∫0T∫ℝ0u−(s,x)N(ds,dx)−∫0T∫ℝ0T−u(s,x)ν(dx)ds.Proof.
Assume as a first step thatu∈𝕃−1,2 is bounded. Then, DMu, u− and D−u are also bounded on ℝ0.
In particular (3.26) is true.
We begin considering the following partition of[0,∞)×ℝ:(3.29)0=s0<s1<⋯<sn<∞=sn+1,−∞=x0<x1<⋯<xm<∞=xm+1.
Then, we can define(3.30)un,m(s,x)=∑i=0n∑j=0mu(si,xj)1(si,si+1](s)1(xj,xj+1](x).
Using Lemma3.1, we have that for all n and m(3.31)δM(un,m⁢⋅1ℝ0)+δM(Dun,m⁢⋅1ℝ0)=∑i=0n∑j=0mu(si,xj)δM(1(si,si+1]1(xj,xj+1]1ℝ0)−∑i=0n∑j=0m∫0T∫ℝ01(si,si+1](s)1(xj,xj+1](x)Ds,xMu(si,xj)ν(dx)ds.
First of all, observe that ifr,s∈]si,si+1] and x,y∈]xj,xj+1], then (Ds,yMun,m)(r,x)=u(si,xj,ωs,y)−un,m(r,x) and (Ds,yMu)(r,x)=u(r,x,ωs,y)−u(r,x) almost surely go to the same limit whatever n and m go to infinity or r↑s and x↑y. By the theorem hypothesis this limit is D−u.
Observe now that beingu bounded, and having un,m the same bound, D−u and u− are also L2-limits. So, using that δM is a closed operator, the left-hand side in
(3.31) goes to δM((u−+D−u)⋅𝟙ℝ0) in L2 if we prove that the terms on the right-hand
side converge in L2 to the limits defined by the proposition.
For the first term in the right-hand side, observe
thatδM coincides with the path-by-path integral
because the integrand is deterministic. Then, using u is bounded and the dominated convergence
theorem we obtain the expected L2-limit. For the second term, we have also a
direct application of dominated convergence theorem.
In order to prove the nonbounded case observe that we
can assume thatu is positive, because the formula that we want
to prove is linear. Then, for the general case, we simply have to apply the
result separately to the positive and negative parts.
So, letu≥0 and uK=u⋀K. Of course, uK≤u and uK converges increasingly to u. We have, as a consequence of the first step,
that
(3.32)δM((T−uK)⋅1ℝ0)=∫0T∫ℝ0uK−(s,x)N(ds,dx)−∫0T∫ℝ0T−uK(s,x)ν(dx)ds.
Beingu− and T−u in L2, we have that uK− and T−uK go up to u− and T−u in L2, respectively. So, hypothesis (3.26), the
monotone convergence theorem and the closeness of the operator δM yield the result.Remark 3.5.
Notice from the proof of Proposition3.4 that we can change the space 𝕃−1,2 by a similar space with left limits with
respect to the time variable s but with right limits with respect the space
variable y.Remark 3.6.
Observe
thatT−u=u− when u is adapted to the filtration generated by N.
Therefore, in such case(3.33)∫0t∫ℝ0u−(s,y)N(ds,dy)=δM(u−(⋅,⋅)1[0,t]×ℝ0(⋅,⋅)).That is, in this case, the
pathwise and Skorohod integrals with respect to N are the same.Observe that in the last two results there is no
contribution ofW because on ℝ0 the operator δM coincides with the Skorohod-type integral with
respect to N,
as the following result shows.Lemma 3.7.
LetδW and δN be the adjoint operators of DW and DN,
respectively, and u∈DomδM. Then u also belongs to DomδW∩DomδN and(3.34)δM(u)=σδW(u⋅,0)+δN(u1ℝ0).Proof.
This result is implied by (3.21) and
(3.16).


## 3.5. The Anticipating Itô's Formula

The basic tool for our results is the following
anticipative Itô formula. Recall that the processX is introduced in (2.1) and Y is the future average volatility, which is an
anticipative process, even σ is adapted.Theorem 3.8.
Letσ2∈𝕃1,2 and F:[0,T]×ℝ×[0,∞)→ℝ be a bounded function in C1,2,2([0,T]×ℝ×[0,∞)) with bounded derivatives.
Then,(3.35)F(t,Xt,Yt)−F(0,X0,Y0)=∫0t∂sF(s,Xs,Ys)ds+∫0t∂xF(s,Xs,Ys)(r−σs22−λk)ds+δtW,B(∂xF(⋅,X.,Y.)σ.)−∫0t∂yF(s,Xs,Ys)σs2ds+ρ∫0t∂xy2F(s,Xs,Ys)Λsds+12∫0t∂xx2F(s,Xs,Ys)σs2ds+δtN(T−u⁢⋅1ℝ0)+∫0t∫ℝ0T−u(s,x)ν(dx)ds,where δW,B is the Skorohod integral with respect to the
Brownian motion ρWs+1−ρ2Bs,Λs:=(∫sTDsWσr2dr)σs and u(s,x):=F(s,Xs−+x,Ys)−F(s,Xs−,Ys).Proof.
The
proof is as in Alòs et al. [8] combined by Proposition 3.4 to treat
the sum of jump terms.
We apply it to the random fieldu(s,x)=F(s,Xs−+x,Ys)−F(s,Xs−,Ys). Here, the independence between Z,B,
and W,
the fact that Y is a continuous process and the fact that Z is a compound Poisson process with a finite
number of jumps on every compact time interval play a key role.
Indeed, letTi denote these jump instants. Then, (3.36)F(Ti+1,XTi+1,YTi+1)−F(Ti,XTi,YTi)=∫TiTi+1−dF(s,Xs,Ys)+F(Ti+1,XTi+1,YTi+1)−F(Ti+1,XTi+1−,YTi+1).
The first term yields a standard Itô formula
concerning continuous process, so Alòs et al. [8] results apply
and we get the six first terms in the right-hand side of the Theorem 3.8 formula.
On other hand, the sum of second terms is the path by path integral(3.37)∫0t∫ℝ0u(s,x)N(ds,dx).Note here that F,X−,Y are left continuous so u=u−.
Then, using Proposition 3.4 we get the last sum is equal to: (3.38)δtN(T−u)+∫0t∫ℝ0T−u(s,x)ν(dx)ds.


## 4. The Hull and White Formula

Now we have the following extension of the Hull and
White formula.Theorem 4.1.
Letσ and X be as in Theorem 3.8. Then,(4.1)Vt=E(BS(t,Xt,vt)∣ℱt)+ρ2E(∫tTe−r(s−t)∂xG(s,Xs,vs)Λsds∣ℱt)+E(∫tT∫ℝ0e−r(s−t)(T−BS(s,Xs−+y,vs)−T−BS(s,Xs−,vs)ds⁢ν(dy)∣ℱt)−λkE(∫tTe−r(s−t)∂xBS(s,Xs,vs)ds∣ℱt),where G=(∂xx2−∂x)BS.Remark 4.2.
Note that in the case thatσ only depends on the filtration generated by W, we have T−=Id. Consequently, in this case, we obtain the Hull
and White formula given in Alòs et al. [8].Proof.
This proof is similar to the one of
the Theorem 4.2. in Alòs et al. [8]. Notice that BS(T,XT,vT)=VT. Then, from (2.2) we have(4.2)e−rtVt=E(e−rTBS(T,XT,vT)∣ℱt).
Now, our idea is to apply the Itô formula (Theorem3.8)
to the process e−rtBS(t,Xt,vt). As the derivatives of BS(t,x,σ) are not bounded we use an approximating
argument, changing vt by(4.3)vtϵ:=1T−t(Yt+ϵ),and BS(t,x,σ) by BSn(t,x,σ):=BS(t,x,σ)ψn(x), where ψn(Xt):=ϕ((1/n)x), for some ϕ∈𝒞b2 such that ϕ(x)=1 for all x<1 and ϕ(x)=0 for all x>2. Now, applying Theorem 3.8 between t and T to function(4.4)F:(t,x,y)⟼e−rtBSn(t,x,y+ϵT−t),and grouping terms according to
the type of derivative we obtain(4.5)e−rTBSn(T,XT,vTϵ)=e−rtBSn(t,Xt,vtϵ)+∫tTe−rsℒBS(σs)BSn(s,Xs,vsϵ)ds−12∫tTe−rs∂σBSn(s,Xs,vsϵ)(σs2−(vsϵ)2)vsϵ(T−s)ds−λk∫tTe−rs∂xBSn(s,Xs,vsϵ)ds+δW,B(e−rs∂xBSn(s,Xs,vsϵ)σs1(t,T](s))+ρ2∫tTe−rs∂σx2BSn(s,Xs,vsϵ)1vsϵ(T−s)Λsds+δtN(e−rs(T−BSn(s,Xs−+y,vsϵ)−T−BSn(s,Xs−,vsϵ)))+∫tT∫ℝ0e−rs(T−BSn(s,Xs−+y,vsϵ)−T−BSn(s,Xs−,vsϵ))ds⁢ν(dy).
Notice thatℒBS(σs)BSn(s,Xs,vsϵ)=(ℒBS(σs)BS(s,Xs,vsϵ))ψn(Xs)+An(s), where(4.6)An(s)=12σs2[2∂xBS(s,Xs,vsϵ)ψn′(Xs)+BS(s,Xs,vsϵ)(ψn′′(Xs)−ψn′(Xs))]+rBS(s,Xs,vsϵ)ψn′(Xs).
Also note that the classical relation between theGamma,
the Vega, and the Delta gives us that(4.7)∂σBS(s,x,σ)1σ(T−s)=(∂xx2−∂x)BS(s,x,σ).
Then we can write(4.8)e−rTBSn(T,XT,vTϵ)=e−rtBSn(t,Xt,vtϵ)+∫tTe−rs[(ℒBS(σs)BS)(s,Xs,vsϵ)ψn(Xs)+An(s)]ds−12∫tTe−rs(∂xx2−∂x)BS(s,Xs,vsϵ)ψn(Xs)(σs2−(vsϵ)2)ds−λk∫tTe−rs∂xBSn(s,Xs,vsϵ)ds+δW,B(e−rs∂xBSn(s,Xs,vsϵ)σs1(t,T](s))+ρ2∫tTe−rs[(∂x(∂xx2−∂x)BS)(s,Xs,vsϵ)ψn(Xs)+(∂xx2−∂x)BS(s,Xs,vsϵ)ψn′(Xs)]Λsds+δtN(e−rs(T−BSn(s,Xs−+y,vsϵ)−T−BSn(s,Xs−,vsϵ)))+∫tT∫ℝ0e−rs(T−BSn(s,Xs−+y,vsϵ)−T−BSn(s,Xs−,vsϵ))ds⁢ν(dy).
Hence, taking into account thatℒBS(σs)=ℒBS(vsϵ)+(1/2)(σs2−(vsϵ)2)(∂xx2−∂x) it follows that (using the fact that ℒBS(vsϵ)BS(s,Xs,vsϵ)=0)(4.9)e−rTBSn(T,XT,vTϵ)=e−rtBSn(t,Xt,vtϵ)+∫tTe−rsAn(s)ds−λk∫tTe−rs∂xBSn(s,Xs,vsϵ)ds+δW,B(e−rs∂xBSn(s,Xs,vsϵ)σs1(t,T](s))+ρ2∫tTe−rs[(∂x(∂xx2−∂x)BS)(s,Xs,vsϵ)ψn(Xs)+(∂xx2−∂x)BS(s,Xs,vsϵ)ψn′(Xs)]Λsds+δtN(e−rs(T−BSn(s,Xs−+y,vsϵ)−T−BSn(s,Xs−,vsϵ)))+∫tT∫ℝ0e−rs(T−BSn(s,Xs−+y,vsϵ)−T−BSn(s,Xs−,vsϵ))ds⁢ν(dy).
Now, taking conditional expectations we obtain
that(4.10)E(e−rTBSn(T,XT,vTϵ)∣ℱt)=E{e−rtBSn(t,Xt,vtϵ)+∫tTe−rsAn(s)ds−λk∫tTe−rs∂xBSn(s,Xs,vsϵ)ds+ρ2∫tTe−rs[(∂x(∂xx2−∂x)BS)(s,Xs,vsϵ)ψn(Xs)+(∂xx2−∂x)BS(s,Xs,vsϵ)ψn′(Xs)]Λsds+∫tT∫ℝ0e−rs(T−BSn(s,Xs−+y,vsϵ)−T−BSn(s,Xs−,vsϵ))ds⁢ν(dy)∣ℱt}.
Let us remark that continuity ofBSn,vϵ and left continuity of X− imply that (T−BSn(s,Xs−+y,vsϵ)=BSn(s,Xs−+y,T−vϵ(s,y)).
Finally, we obtain the result proceeding as in the
proof of Theorem 3 in Alòs et al. [8]. That is, letting first n↑∞,
then ϵ↓0 and using the dominated convergence
theorem.Remark 4.3.
The additional term given byT−BS can be detailed as follows. Suppose that σr2=f(Wu,Zu,u≤r). Then we can define (4.11)vs2=limt↑s,y↑x1T−t∫tTTs,x(σr2)dr.
But forr>s, Ts,x(σr2)=f(Wu,Zu+x𝟙{s≤u},u≤r),(4.12)vs2=limt↑s1T−t∫tTf(Wu,Zu+x1{s≤u},⁢u≤r)dr=1T−s∫sTσr2dr,where σr2=f(Wu,Zu+x𝟙{s≤u},u≤r).
For example, consider the following pure volatility
jump case described in Álvarez [23]. See also Espinosa and Vives [24]. For i≥0,
let Ti be the jump instants and ΔTiZ be the jump sizes of process Z, with T0=0. Assume that the dynamic of σ is given by (4.13)σt2=∑i=0Ntσi21[Ti,Ti+1[(t),with σi2=σi−12+f(ΔTiZ), for a certain function f. In this case, we have σr2=σr2+f(x)𝟙{r≥s}(r) and so, the explicit computation of vs gives vs2=vs2+f(x).


## 5. Short Time Behavior of the Implied Volatility

In this section, we show that the short-time behavior
of the at-the-money implied volatility is the same as in the case where the
volatilityσ is independent of the filtration of Z, even the Hull and White formula is different
in the last case (see Remark 4.2). This is a fact that must be taken in account
for pricing and hedging.LetIt(Xt) denote the implied volatility process. By
definition it satisfies Vt=BS(t,Xt,It(Xt)). Assume that σ∈𝕃1,2 is as in model (2.1). Proceeding as in Alòs et al. [8], the derivative of the implied volatility with respect to
the log-strike k=logK is(5.1)∂It∂k(xt∗)=−E(∫tT(∂xF(s,Xs,vs)−(1/2)F(s,Xs,vs))ds∣ℱt)∂σBS(t,xt∗,It(xt∗))|Xt=xt∗,⁢a.s.,where(5.2)F(s,Xs,vs):=ρ2e−r(s−t)∂xG(s,Xs,vs)Λs−λke−r(s−t)∂xBS(s,Xs,vs)+∫ℝe−r(s−t)(BS(s,Xs−+y,vs)−BS(s,Xs−+y,vs))ν(dy).Now, in order to study the limit of(∂It/∂k)(xt∗) as T↓t we need to introduce the following hypotheses:
(H1)
σ∈𝕃W2,4=L4([0,T]×ℝ;𝔻W2,4).(H2)
There exists a constantα>−(1/2) such that, for all 0<t<s<r<T,(5.3)E((DsWσr)2∣ℱt)≤C(r−s)2α,E((DθWDsWσr)2∣ℱt)≤C(r−s)2α(r−θ)−2α.(H3)
For every fixedt>0,sups,r,θ∈[t,T]E((σsσr−σθ2)2∣ℱt)→0 as T→t.Theorem 5.1.
Under the Hypotheses (H1)–(H3) we
have(1)
assume thatα in (H2) is nonnegative and that there exists a ℱt-measurable random variable DtW,+σt such that, for every t>0,(5.4)sups,r∈[t,T]|E((DsWσr−DtW,+σt)∣ℱt)|⟶0,a.s. as T→t; then(5.5)limT→t∂It∂k(xt∗)=1σt(λk+ρDtW,+σt2),(2)
assume thatα in (H2) is negative and that there exists a ℱt-measurable random variable Ltα,+σt such that, for every t>0,(5.6)1(T−t)2+α∫tT∫sTE(DsWσr∣ℱt)dr⁢ds−Ltα,+σt⟶0,a.s. as T→t; then(5.7)limT→t(T−t)−α∂It∂k(xt∗)=ρσtLtα,+σt.Proof.
We can write(5.8)−∂σBS(t,xt∗,It(xt∗))∂It∂k(xt∗)=ρ2E(∫tTe−r(s−t)(∂x−12)∂x(∂xx2−∂x)BS(s,Xs,vs)Λsds∣ℱt)|Xt=xt∗+E(∫tT∫ℝe−r(s−t)(∂x−12)[BS(s,Xs+y,vs)−BS(s,Xs,vs)]ν(dy)ds∣ℱt)|Xt=xt∗−λkE(∫tTe−r(s−t)(∂x−12)∂xBS(s,Xs,vs)ds∣ℱt)|Xt=xt∗=T1+T2+T3.
The termT2 is O(T−t) due to the fact that the following majoration
is uniform on σ
(5.9)|BS(t,x,σ)|+|∂xBS(t,x,σ)|≤2ex+K.
Now the result follows as in Alòs et al. [8, Proposition 6 and Theorem 7].


## 6. Conclusion

The Malliavin
calculus for Lévy processes appears to be a natural tool to deal with the
future average volatility in jump-diffusion models where the volatility is
correlated with both Brownian motion and compound Poisson process driving the
stock price process. Proceeding as in Alòs et al. [8], this
powerful calculus allows us to identify the effect of both correlations. In
particular, we have seen that the correlation with the asset price jumps has no
effect on the short-time behavior of the volatility skew.



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*Source: 359142-2009-02-10.xml*