Decomposition of the Pricing Formula for Stochastic Volatility Models Based on Malliavin-Skorohod Type Calculus

Abstract

The goal of this survey article is to present in detail a method that, for a financial derivative under a certain stochastic volatility model, allows to obtain a decomposition of its pricing formula that distinguishes clearly the impact of correlation and jumps. This decomposed pricing formula, usually called Hull and White type formula, can be potentially useful for model selection and calibration. The method is based on the obtention of an ad-hoc anticipating Itô formula.

Supported by grant MEC MTM 2012-31192.

1 Introduction

The decomposition method presented in this paper is based on a series of works developed during the last ten years. In [1], E. Alòs obtained a decomposition of the pricing formula, usually called Hull and White type formula, for a plain vanilla call under a correlated stochastic volatility model, with minor hypothesis on the volatility process related with its Malliavin derivability. The decomposition was obtained applying an ad-hoc extension of the anticipative Itô formula given in [2]. The obtained formula showed clearly the impact on prices of adding correlation between price and volatility in stochastic volatility models.

In [3] the same type of formula was obtained adding also finite activity jumps in the price process. A new term appeared, showing the impact of jumps. In [5] the formula was extended to the case of assuming jumps also on the volatility process. Still a new term appeared in the formula. Finally, on [9], the result in [5] was extended for free to the case of infinite activity and finite variation jumps, and with a certain restriction in the interpretation of the formula, to the case of infinite activity and infinite variation jumps. The very general model considered in this last paper covers almost all stochastic volatility models with and without jumps, treated in the literature.

As we see in the paper, the presence of correlation and jumps in stochastic volatility models is relevant. Additional terms in the pricing formula appears from correlation, from jumps in the price process and from jumps in the volatility. Malliavin-Skorohod calculus and the decomposition method allow to obtain these pricing formulas that clearly distinguish the effect of correlation than the effect of jumps, for different types of jump models. If the stochastic volatility is correlated only with the continuous part of the price process, only Gaussian Malliavin-Skorohod calculus is needed. If the stochastic volatility is also correlated with price jumps, Lévy Malliavin-Skorohod calculus is needed.

Section 2 is devoted to the Brownian (no jump) case and Sect. 3 treats the Lévy case.

2 Decomposition of the Pricing Formula Under a General Brownian Stochastic Volatility Model

The main reference of the theory presented in this section is [1].

2.1 The Model

Let $T>0$ be a finite horizon, $S=\{S_t, t\in [0,T]\}$ a price process, $X_t=\log S_t$ the corresponding log price process and $r>0$ the fixed interest rate. We assume the following exponential model with stochastic volatility for the dynamics of the log-price, under a market chosen risk-neutral probability:

$$X_{t}=x+rt-\frac{1}{2}\int _{0}^{t}{\sigma _{s}^{2}}ds+\int _{0}^{t}\sigma _{s}(\rho dW_{s}+\sqrt{1-\rho ^{2}}dB_{s})$$

where x is the current log-price, W and B are independent standard Brownian motions and $\rho \in (-1,1).$

We denote by $\mathcal{F}^W$ and $\mathcal{F}^B$ the filtrations generated by the independent processes W and B. Moreover, we define $\mathcal{F}$, the filtration associated to S, by $\mathcal{F}:\,=\,\mathcal{F}^{W} \vee \mathcal{F}^{B}.$ We consider our price model defined on the product of the canonical spaces of processes W and B.

The volatility process $\sigma$ is assumed to be a square-integrable stochastic process, adapted to $\mathcal{F}^W$ and with strictly positive and càdlàg trajectories.

Note that this is a very general stochastic volatility model. In this sense, recall the following facts:

• The model is a generalization of Heston model or other classical correlated stochastic volatility models in the sense that we do not assume a concrete dynamics for the volatility process $\sigma$.

• If $\rho =0$ we have a generalization in the same sense as before of different non correlated stochastic volatility models as Hull-White, Scott, Stein-Stein or Ball-Roma.

• If $\sigma$ is deterministic or constant we have the classical Osborne-Samuelson-Black-Scholes model.

For information about correlated and non-correlated stochastic volatility models, a good reference is [8].

Stochastic volatility models pursue the goal to replicate price surfaces of plain vanilla options (depending on time to maturity and strike) given by derivative markets or vanilla desks. The stochastic volatility $\sigma$ is a process not directly observable, so it is not easy to model. This is a justification for trying to assume minimal conditions on it.

Let $H_T$ be the payoff of a financial derivative. Assume it is a $\mathcal{F}_T-$measurable functional. Its price is given by $V_t=e^{-r(T-t)}\mathbb {E}_t(H_T)$ where $\mathbb {E}_t:\,=\,\mathbb {E}(\cdot |\mathcal{F}_t).$ To fix ideas we will concentrate on the case of a plain vanilla call, that is, $H_T=(S_T-K)^+.$ So, our goal is to obtain a decomposition of

$$V_t=e^{-r(T-t)} \mathbb {E}_t((S_T-K)^+)$$

under our risk neutral model, in order to clarify the effect of correlation in the price.

2.2 Fast Summary of Brownian Malliavin-Skorohod Calculus

Here we simply recall some basic definitions and facts necessary for our purpose. See for example [10] for a complete presentation of the theory.

Let W and $(\varOmega ^W, \mathcal{F}^W, \mathbb {P}_W)$ be the canonical Wiener process and its canonical space, respectively. Recall that $\varOmega ^W:\,=\,C_0([0,T])$ is the space of continuous functions on [0, T], null at the origin. Denote by $\mathbb {E}_W$ the expectation with respect to $\mathbb {P}_W.$

Consider the family of smooth functionals of type

$$F=f(W_{t_1},\dots , W_{t_n})$$

for any $n\ge 0,$ $t_1,\dots t_n\in [0,T]$ and $f\in C^{\infty }_b (\mathbb {R}^n).$

Given a smooth functional F we define its Malliavin derivative $D^W F$ as the element of $L^2 (\varOmega ^W\times [0,T])$ given by

$$D_t F=\sum _{i=1}^n \partial _i f(W_{t_1},\dots , W_{t_n}){1\!\!1}_{[0,t_i]}(t).$$

The operator $D^W$ is closed and densely defined in $L^2 (\varOmega ^W)$, and its domain $Dom\,D^W$ is the closure of the smooth functionals with respect the norm

$$||F||_{Dom\,D^W}:\,=\,\big (\mathbb {E}_W(|F|^2)+\mathbb {E}_W\int _0^T |D^W_t F|^2 dt\big )^{\frac{1}{2}}.$$

We define $\delta ^W$ as the dual operator of $D^W.$ Given $u \in L^2 (\varOmega ^W\times [0,T])$, $\delta ^W (u)$ is the element of $L^{2}(\varOmega ^W)$ characterized by

$$\mathbb {E}_W(F\delta ^W (u))=\mathbb {E}_W\int _0^T u_t D^W_tF dt$$

for any $F \in Dom\,D^W.$ Note that taking $F\equiv 1$ we obtain

$$\mathbb {E}_W(\delta ^W (u))=0.$$

The following results will be helpful:

• If F, G and $F\cdot G$ belong to $Dom\,D^W$ we have

  $$D^W(F\cdot G)=F D^WG+G D^WF.$$

• If $F\in Dom\,D^W$, $u\in Dom\,\delta ^W$ and $F\cdot u\in Dom\,\delta ^W$ then

  $$\delta ^W (F\cdot u)=F\delta ^W (u)-\int _0^T u_t D^W_t F dt.$$

• It is well known that $D^W$ can be interpreted as a directional derivative on the Wiener space and $\delta ^W$ is an extension of the classical Itô integral.

We define the space $\mathbb {L}^{1,2}_W\ {:=}\ L^{2}([0,T]; Dom\,D^W)$, that is the space of processes $u \in L^{2}([0,T] \times \varOmega ^W)$ such that $u_t \in Dom\,D^W$ for almost all t and $Du \in L^2 (\varOmega ^W\times [0,T]^2).$ It can be proved that $\mathbb {L}^{1,2}_W\subseteq Dom\,\delta ^W$ and

$$\mathbb {E}_W(\delta ^W (u)^2)\le ||u||_{\mathbb {L}^{1,2}_W}^2\ {:=}\ \mathbb {E}_W(||u||^2_{L^2 ([0,T])})+\mathbb {E}_W(||D^W u||^2_{L^2 ([0,T]^2)}).$$

Finally, we will denote $\delta ^W_t (u)\ {:=}\ \delta ^W (u {1\!\!1}_{[0,t]}).$

2.3 The Hull and White Formula

If we assume constant volatility we have the well known geometric Brownian model. In this case, the price $V_t$ is given by the well known Black-Scholes formula:

$$V_t=BS(t,X_t,\sigma )=e^x \varPhi (d_+)-Ke^{-r(T-t)} \varPhi (d_-)$$

where

$$d_{\pm }=\frac{X_t-\log K+r(T-t)}{\sigma \sqrt{T-t}}\pm \frac{\sigma \sqrt{T-t}}{2}$$

and $\varPhi$ is the cumulative probability function of the standard normal law.

If we allow $\sigma =\sigma (t)$ be a deterministic function, it is easy to see, that

$$X_T-X_t\sim \mathcal{N}((r-\frac{1}{2}{\bar{\sigma }}^2_t)(T-t), {\bar{\sigma }}^2_t (T-t)),$$

where

$${\bar{\sigma }}_t\ {:=}\ \sqrt{\frac{1}{T-t}\int _t^T {\sigma }^2(s)ds}$$

is the so called future average volatility. Define ${\bar{\sigma }}_T$ as the limit of ${\bar{\sigma }}_t$ when $t\uparrow T.$

So, in this case, the pricing formula is exactly the Black-Scholes formula changing $\sigma$ by ${\bar{\sigma }}_t,$ that is, $V_t=BS(t,X_t,{\bar{\sigma }}_t).$ This suggests that it is the future average volatility and not the volatility the really relevant quantity in pricing. Black-Scholes formula would be nothing more than the particular case of constant future average volatility.

If $\sigma$ is a stochastic process uncorrelated with price, that is, $\rho =0$ in our model, we have, following for example [7]:

$$V_t=\mathbb {E}_t(BS(t,X_t,{\bar{\sigma }}_t)).$$

This is the classical Hull and White formula and covers non correlated stochastic volatility models as the cases of Hull-White, Scott, Stein-Stein, Ball-Roma and others. The proof is immediate, conditioning first by $\mathcal{F}_t\vee \mathcal{F}^{W}_T.$

Note that the future average volatility ${\bar{\sigma }}_t$ is an anticipative process. This suggest the use of Malliavin-Skorohod calculus as a natural tool to deal with this type of processes.

In the correlated case we have the following theorem:

Theorem 1

Assume

• (A1): $\sigma ^2\in \mathbb {L}^{1,2}_W.$

• (A2): $\sigma \in \mathbb {L}^{1,2}_W.$

Then we have,

$$V_t=\mathbb {E}_t[BS(t, X_t, {\bar{\sigma }}_t)]+\frac{\rho }{2}\mathbb {E}_t[\int _t^T e^{-r(s-t)} (\partial ^3_x-\partial ^2_x)BS(s,X_s,{\bar{\sigma }}_s)\varLambda _sds]$$

where

$$\varLambda _s\ {:=}\ (\int _s^T D^W_s \sigma _r^2 dr )\sigma _s.$$

Proof

The proof is based on the so called decomposition method.

Recall that

$$V_T=(e^{X_T}-K)^+=BS(T,X_T,{\bar{\sigma }}_T)$$

and so

$$e^{-rt}V_t=\mathbb {E}_t (e^{-rT}BS(T,X_T,{\bar{\sigma }}_T)).$$

The idea of the proof consists in applying an ad-hoc anticipative Itô formula to the process

$$e^{-rs}BS(s,X_s,{\bar{\sigma }}_s)$$

between t and T, take conditional expectations $\mathbb {E}_t$ and multiply by $e^{rt}.$ This gives the expansion for $V_t.$

The ad-hoc Itô formula is an adaptation to our case of the anticipative Itô formula proved in [2]. Define

$$Y_t\ {:=}\ (T-t){\bar{\sigma }}_t^2=\int _t^T \sigma _r^2 dr.$$

Thanks to (A1), we are under the conditions of Theorem 1 in [1], and so, for any $F\in C^{1,2,2}_b ([0,T]\times \mathbb {R}\times [0,\infty )),$ we have

$$\begin{aligned} F(t,X_{t},Y_t)= & {} F(0,X_{0},Y_0)+\int _{0}^{t}{\partial _{s}F} (s,X_{s},Y_{s})ds+\delta _t^{W,B}({\partial _{x}F}(\cdot ,X_{\cdot },Y_{\cdot })\sigma _{\cdot })\\ \nonumber&+\int _{0}^{t}{\partial _{x}F}(s,X_{s},Y_{s})(r-\frac{\sigma _{s}^{2}}{2})ds-\int _{0}^{t}{\partial _{y}F}(s,X_{s},Y_{s})\sigma _{s}^{2}ds\\ \nonumber&+\rho \int _{0}^{t}{\partial _{xy} F}(s,X_{s},Y_{s})\varLambda _{s}ds+\frac{1}{2}\int _{0}^{t}{\partial _x^2 F}(s,X_{s},Y_{s})\sigma _{s}^{2}ds. \end{aligned}$$

Now we want to apply this result to

$$F(s,x,y)\ {:=}\ e^{-rs} BS(s,x,\sqrt{\frac{y}{T-s}}),$$

but this function doesn’t satisfy the required conditions of the previous Itô formula because the derivatives are not bounded, so we need to use a mollifier argument.

For $n\ge 1$ and $\delta >0,$ we consider the approximation,

$$F_{n,\delta }(s,x,y)\ {:=}\ e^{-rs}BS(s,x,\sqrt{\frac{y+\delta }{T-s}})\phi (\frac{x}{n}),$$

where $\phi \in C^2_b (\mathbb R),$ such that $\phi (z)=1$ if $|z|\le 1$, $\phi (z)\in [0,1]$ if $|z|\in [1,2]$ and $\phi (z)=0$ if $|z|>2.$

Then, applying the previous ad-hoc Itô formula to $F_{n,\delta } (s,X_s,Y_s)$, taking the conditional expectation $\mathbb {E}_t$, using the fact that Skorohod type integrals have zero expectation and multiplying by $e^{rt}$ we obtain

$$\begin{aligned}&\mathbb {E}_t(e^{-r(T-t)}BS(T,X_T,{\bar{\sigma }}_T^{\delta })\phi (\frac{X_T}{n}))\\ =&\mathbb {E}_t(BS(t,X_t,{\bar{\sigma }}_t^{\delta })\phi (\frac{X_t}{n}))\\&+\mathbb {E}_t(\int ^{T}_{t}e^{-r(s-t)}A_{n}(s)ds)\\&+\frac{\rho }{2}\mathbb {E}_t(\int ^{T}_{t}e^{-r(s-t)}(\partial _x^3-\partial _x^2)BS(s,X_{s},{\bar{\sigma }}^{\delta }_{s})\phi _{n}(\frac{X_{s}}{n})\varLambda _s ds\\&+\frac{\rho }{2}\mathbb {E}_t(\int ^{T}_{t}e^{-r(s-t)}(\partial _x^2-\partial _x)BS(s,X_{s},{\bar{\sigma }}^{\delta }_{s})\frac{1}{n}\phi ^{'}(\frac{X_{s}}{n}))\varLambda _{s}ds \end{aligned}$$

where

$${\bar{\sigma }}^{\delta }_s:=\sqrt{\frac{Y_s+\delta }{T-s}}$$

and

$$\begin{aligned} A_{n}(s):= & {} \frac{\sigma ^{2}_{s}}{n}\partial _{x}BS(s,X_{s},{\bar{\sigma }}^{\delta }_{s})\phi ^{'}(\frac{X_{s}}{n})\\&+\frac{\sigma ^{2}_{s}}{2n}BS(s,X_{s},{\bar{\sigma }}^{\delta }_{s})(\frac{1}{n}\phi ^{''}(\frac{X_{s}}{n})-\phi ^{'}(\frac{X_{s}}{n}))\\&+\frac{r}{n} BS(s,X_{s},{\bar{\sigma }}^{\delta }_{s})\phi ^{'}(\frac{X_{s}}{n}). \end{aligned}$$

The details can be found in [9] (erasing there the terms depending on jumps, that will be treated later in this paper).

Finally, the result follows from the dominated convergence theorem taking limits first on $n\uparrow \infty$ and then on $\delta \downarrow 0.$ The dominated convergence runs thanks to the properties of Black-Scholes function and (A2). For the left hand side and the two first terms on the right hand side we use the fact that function $BS(t,x,\sigma )$ is bounded by $e^x+K$ and its derivative $(\partial _x BS)(t,x,\sigma )$ is bounded by $e^x.$ For the last two terms on the right hand side we use Lemma 2 in [3] that says that for any $n\ge 0$,

$$|\mathbb {E}[\partial _x^n (\partial _x^2-\partial _x) BS(s,X_s,{\bar{\sigma }}_s)|\mathcal{F}_t\vee \mathcal{F}^W_T]|\le C_n(\rho ) (\int _t^T \sigma _s^2 ds)^{-\frac{n+1}{2}},$$

for a certain constant $C_n(\rho )$ that depends only on n and $\rho .$

For example, for the third term on the right hand side, we have

$$\begin{aligned}&\mathbb {E}_t(\int ^{T}_{t}e^{-r(s-t)}(\partial _x^3-\partial _x^2)BS(s,X_{s},{\bar{\sigma }}^{\delta }_{s})\phi _{n}(\frac{X_{s}}{n})\varLambda _s ds)\\= & {} \mathbb {E}_t(\int ^{T}_{t}e^{-r(s-t)}\mathbb {E}[(\partial _x^3-\partial _x^2)BS(s,X_{s},{\bar{\sigma }}^{\delta }_{s})|\mathcal{F}_t\vee \mathcal{F}^W_T]\phi _{n}(\frac{X_{s}}{n})\varLambda _s ds). \end{aligned}$$

And, applying the lemma,

$$|\mathbb {E}[(\partial _x^3-\partial _x^2)BS(s,X_{s},v^{\delta }_{s})|\mathcal{F}_t\vee \mathcal{F}^W_T]\phi _{n}(\frac{X_{s}}{n})\varLambda _s|\le C_1 (\rho )\frac{|\varLambda _s|}{\int _t^T \sigma _s^2 ds}.$$

Using the chain rule for $D^W$, the problem reduces to show

$$\mathbb {E}_t(\frac{\int _t^T (\int _t^T |D_s^W \sigma _u|\sigma _u dr) \sigma _s ds}{\int _t^T \sigma _r^2 dr})<\infty ,$$

and applying Cauchy-Schwarz inequality twice, we can bound this expression by

$$\begin{aligned}&\mathbb {E}_t(\frac{(\int _t^T (\int _t^T |D_s^W \sigma _u| \sigma _u du)^2 ds)^{\frac{1}{2}}}{(\int _t^T \sigma _u^2 du)^{\frac{1}{2}}})\\&\le \mathbb {E}_t(\frac{(\int _t^T (\int _t^T |D_s^W \sigma _u|^2 dr)(\int _t^T \sigma ^2_u du) ds)^{\frac{1}{2}}}{(\int _t^T \sigma _u^2 du)^{\frac{1}{2}}})\\&\le \mathbb {E}_t((\int _t^T\int _t^T |D_s^W \sigma _u|^2 duds)^{\frac{1}{2}})\\&\le (\mathbb {E}_t\int _t^T\int _t^T |D_s^W \sigma _u|^2 duds)^{\frac{1}{2}}. \end{aligned}$$

So, (A2) proves that this expression is finite.

For the fourth term in the right hand side, applying the lemma and using C as a generic constant, we have

$$|\mathbb {E}[(\partial _x^2-\partial _x)BS(s,X_{s},{\bar{\sigma }}^{\delta }_{s})|\mathcal{F}_t\vee \mathcal{F}^W_T]\frac{1}{n}\phi ^{'}_{n}(\frac{X_{s}}{n})\varLambda _s|\le \frac{C}{n}\frac{|\varLambda _s|}{(\int _t^T \sigma _s^2 ds)^{\frac{1}{2}}}.$$

So, we have to show

$$\mathbb {E}_t(\frac{\int _t^T\sigma _s (\int _s^T |D_s^W \sigma _u| \sigma _u du) ds}{(\int _t^T \sigma _u^2 du)^{\frac{1}{2}}})<\infty ,$$

that follows applying Cauchy-Schwarz inequality, similarly to the previous case.

Remark 1

Note that hypothesis (A2) can be changed by the following alternative hypothesis of uniform ellipticity (A$2^\prime$): The process $\sigma ^2$ defined on [0, T] is uniformly bounded below by a constant $a>0.$ In fact (A1) and (A$2^\prime$), jointly, imply (A2).

3 The Lévy Case

The main references for this section are [3, 5, 9].

3.1 A Very General Stochastic Volatility Lévy Model

Assume now the following exponential Lévy model with stochastic volatility for the dynamics of the log-price, under a market chosen risk-neutral probability:

$$X_{t}=x+rt-\frac{1}{2}\int _{0}^{t}{\sigma _{s}^{2}}ds +\int _{0}^{t}\sigma _{s}(\rho dW_{s}+\sqrt{1-\rho ^{2}}dB_{s})+L^0_t$$

where $L^0$ is a pure jump Lévy process with possibly infinitely many jumps with triplet $(\gamma _0, 0, \nu )$ and independent of W and B. Now, the volatility process $\sigma$ is assumed to be adapted to the filtration generated by W and $L^0.$

Due to the well known Lévy-Itô decomposition we can write

$$L^0_t=\gamma _0 t+\int _0^t \int _{\{|y|>1\}} y N(ds,dy)+\lim _{\varepsilon \downarrow 0}\int _0^t \int _{\{\varepsilon <|y|\le 1\}} y {\widetilde{N}}(ds,dy)$$

where N denotes the Poisson measure associated to Lévy process, ${\widetilde{N}}(ds, dy):=N(ds, dy)-\nu (dy)ds$ is the compensated Poisson measure and the limit is a.s. and uniformly on compacts.

For the integers $i\ge 0$, we consider the following constants, provided they exist:

$$c_i:=\sum _{k=i}^{\infty } \int _\mathbb {R} \frac{y^k}{k!}\nu (dy).$$

Observe that in particular

$$\begin{aligned} c_0= & {} \int _\mathbb {R} e^y \nu (dy),\\ c_1= & {} \int _\mathbb {R} (e^y-1) \nu (dy),\\ c_2= & {} \int _\mathbb {R} (e^y-1-y) \nu (dy). \end{aligned}$$

In order to $e^{-rt} e^{X_t}$ be a martingale, see for example [6], we must assume

$$\int _{\left| y\right| \ge 1} e^y\nu (dy)<\infty \, \text{ and } \, \gamma _0=-\int _\mathbb { R}(e^{y}-1-y{1\!\!1}_{\{\left| y\right| <1\}})\nu (dy).$$

These conditions guarantee that $\nu$ has moments of all orders greater or equal than 2 and that we can write

$$L^0_t=\int _0^t \int _\mathbb {R} y {\widetilde{N}(ds,dy)}-c_2 t.$$

So, in the following we will assume, without loosing generality, the model

$$X_t=x+(r-c_2)t-\frac{1}{2}\int ^t_0\sigma ^2_s ds+\int ^t_0\sigma _s(\rho dW_s+\sqrt{1-\rho ^2}dB_s)+J_t$$

with

$$J_t:=\int _0^t \int _\mathbb {R} y {\widetilde{N}}(ds,dy).$$

Recall that if $\int _\mathbb {R} |y|\nu (dy)=\infty$ we say that the process has infinite activity and infinite variation. In this case $c_1:=\int _\mathbb {R} (e^y-1)\nu (dy)$ and $c_0:=\int _\mathbb {R} e^y \nu (dy)$ are infinite or not defined. If $\nu$ has first order moment, that is $\int _\mathbb {R} |y|\nu (dy)<\infty ,$ we say the model has infinite activity but finite variation and $c_1$ is finite. In this case we can consider $c_2=c_1-\int _\mathbb {R}y\nu (dy)$ and rewrite

$$\int _0^t \int _\mathbb {R} y {\widetilde{N}(ds,dy)}-c_2 t=\int _0^t \int _\mathbb {R} y N(ds,dy)-c_1 t.$$

Finally, if $\nu$ is finite, the model has finite activity and so, it is a Compound Poisson process with $\nu =\lambda Q$ for a certain probability measure Q and a certain constant $\lambda =\nu (\mathbb {R})>0.$ In this case,

$$c_1=\int _\mathbb {R}(e^y-1)\nu (dy)=c_0-\lambda $$

and

$$\int _0^t \int _\mathbb {R} y N(ds,dy)=\sum _{i=1}^{N_t} V_i,$$

where N is a $\lambda -$Poisson process and $V_i$ are i.i.d. random variables with law Q, the law that produce the jumps.

Let $\mathcal{F}^J$ be the filtration generated by J. Note that this filtration is the same as the filtration generated by $L^0$ because the difference of these two processes is deterministic. Define now $\mathcal{F}$, the filtration associated to S, by

$$\mathcal{F}:= \mathcal{F}^{W} \vee \mathcal{F}^{B}\vee \mathcal{F}^{J}.$$

We will consider our price model defined on the product of the canonical spaces of processes W, B and J. This means that

$$\omega :=(\omega ^W,\omega ^B,\omega ^J)\in \varOmega :=\varOmega ^W\times \varOmega ^B\times \varOmega ^J$$

and in the rest of the paper, any hypothesis on one of the spaces will mean that the property is true almost surely with respect to the other spaces.

Note that this is a very general stochastic volatility model because, being $\sigma$ adapted to $\mathcal{F}^{W}\vee \mathcal{F}^{J}$, we are allowing jumps both in price and volatility. Recall the following facts:

• If we assume no jumps, that is $\nu =0$, we have a generalization of correlated and non correlated stochastic volatility models in the sense that we do not assume a concrete dynamics for the volatility. This is the case treated in Sect. 2.

• If we restrict our model to the case $\sigma$ adapted only to $\mathcal{F}^{W}$ we have a generalization of the Bates model in a double sense; on one hand we do not assume any concrete dynamics for the stochastic volatility and on other hand we are not assuming finite activity nor finite variation on $\nu .$

• If we assume no correlation but presence of jumps we cover for example Heston-Kou model or any uncorrelated model with the addition of Lévy jumps in the price process with any Lévy measure $\nu .$

• If $\sigma =0$ but we have jumps, we cover the so called exponential Lévy models.

3.2 Malliavin-Skorohod Type Calculus for Lévy Processes

The literature on Malliavin calculus for Lévy processes is more recent and less extended. Here we follow closely [11] and [4]. A survey of this results can be found in [12]. We refer to these references for proofs of next results.

Let us denote $\mathbb {R}_0:=\mathbb {R}-\{0\}.$ Consider the canonical version of the pure jump Lévy process J. It is defined on the space $\varOmega ^N$ given by the finite or infinite sequences of pairs $(t_i,x_i)\in (0,T]\times \mathbb {R}_0$ such that for every $\varepsilon >0$ there is only a finite number of $(t_i,x_i)$ with $\left| x_i\right| >\varepsilon .$ Of course, $t_i$ denotes a jump instant and $x_i$ a jump size.

Consider $\omega ^N\in \varOmega ^N.$ Given $(t,x) \in [0,T] \times \mathbb {R}_{0}$ we can introduce a jump of size x at instant t to $\omega ^N$ and call the new element

$$\omega ^{N}_{t,x}:=((t,x),(t_1,x_1) (t_2,x_s),...).$$

For a random variable $F\in L^2 (\varOmega ^N)$, we define

$$T_{t,x}F(\omega ^N)=F(\omega _{t,x}^N)$$

and

$$D^N_{t,x} F=\frac{T_{t,x} F (\omega ^N)-F(\omega ^N)}{x}.$$

The operator $D^N$ is closed and densely defined in $L^2 (\varOmega ^N)$ and its domain $Dom\,D^N$ can be characterized by the fact that

$$F \in Dom\,D^N \Longleftrightarrow D^N F \in L^{2}(\varOmega \times [0,T]\times \mathbb {R}_{0}, P \otimes ds\otimes x^2\nu (dx)).$$

On other hand we define $\delta ^N$ as the dual operator of $D^N.$

Given $u \in L^2 (\varOmega ^W\times [0,T]\times \mathbb {R}, P \otimes ds \otimes x^2\nu (dx))$, $\delta ^N (u)$ is the element of $L^{2}(\varOmega ^N)$ characterized by

$$\mathbb {E}_N(F\delta ^N (u))=\mathbb {E}_N(\int _0^T \int _\mathbb {R}u_{t,x} D^N_{t,x}F x^2\nu (dx)dt)$$

for any $F \in Dom\,D^N.$ In particular $\mathbb {E}_N(\delta ^N (u))=0.$

Let us denote $\delta ^N_t (u):=\delta ^N (u {1\!\!1}_{[0,t]}).$

As we have seen, $D^N$ is an increment quotient operator and it is also known that $\delta ^N_t$ is an extension of Itô integral in the sense that

$$\delta ^N_t (u {1\!\!1}_{\mathbb {R}_0})=\int _0^t \int _\mathbb {R} u(s,x)x{\tilde{N}}(ds,dx)$$

for predictable integrands u.

The following formulas will be helpful:

• If F, G and $F\cdot G$ belong to $Dom\,D^N$ we have

  $$D^N(F\cdot G)=F D^NG+G D^NF+x D^NF D^NG.$$

• If $F\in Dom\,D^N$, $u\in Dom\,\delta ^N$ and $u\cdot T_{t,x} F\in Dom\,\delta ^N$ then

  $$\delta ^N (F\cdot u)=F\delta ^N (u)-\int _0^T \int _\mathbb {R} u_{t,x} D^N_{t,x} F x^2 \nu (dx)dt-\delta ^N(x\cdot u\cdot D^NF).$$

As in the Wiener case we define the space

$$\mathbb {L}^{1,2}_N:=L^{2}([0,T] \times \mathbb {R}, Dom\,D^N),$$

that is the space of processes $u \in L^{2}([0,T]\times \mathbb {R}\times \varOmega ^N)$ such that $u_{t,x} \in Dom\,D^N$ for almost all (t, x) and $Du \in L^2 (\varOmega ^N\times ([0,T]\times \mathbb {R})^2).$

It can be proved that $\mathbb {L}^{1,2}_N\subseteq Dom\,\delta ^N$ and

$$E_N(\delta ^N (u)^2)\le ||u||_{\mathbb {L}^{1,2}_N}^2:=E_N(||u||^2_{L^2 ([0,T]\times \mathbb { R})})+E_N(||D^N u||^2_{L^2 (([0,T]\times \mathbb {R})^2)}).$$

Moreover we introduce the space $\mathbb {L}^{1,2}_{N,-}$ as the subspace of $\mathbb {L}^{1,2}_N$ of processes u such that the left-limits

$$u(s-,y):=\lim _{r\uparrow s, x\uparrow y} u(r,x)$$

and

$$D_{s,y}^{N,-} u(s-,y):=\lim _{r\uparrow s, x\uparrow y} D^N_{s,y} u(r,x)$$

exists $\mathbb {P}_N\otimes ds\otimes x^2\nu (dx)-$a.s. and belong to $L^2 (\varOmega ^N\times [0,T]\times \mathbb {R}).$

Assume $u\in \mathbb {L}^{1,2}_{N,-}$ and $\int _0^T\int _{\mathbb {R}_0} |u(s-,y)||y|N(ds,dy)\in L^2(\varOmega ^N).$ Then, for any $t\in [0,T],$

$$T^{-}_{s,y}u(s-,y):=u(s-,y)+yD^{N,-}_{s,y} u(s-,y)\in Dom\,\delta ^N_t$$

and

$$\begin{aligned} \int _0^t \int _\mathbb {R} u(s-,y)y {\tilde{N}}(ds, dy)&={\delta }^N_t (T^{-}_{s,y}u(s-,y){1\!\!1}_{\mathbb { R}_0})\\&\quad +\int _0^t \int _\mathbb {R} D^{N,-}_{s,y} u(s-,y) y^2\nu (dy)ds. \end{aligned}$$

If u is predictable we have $D_{s,y}^{N,-} u(s-,y)=0.$ Hence, in this case,

$$\int _0^t \int _\mathbb {R} u(s-,y)y {\tilde{N}}(ds, dy)={\delta }^N_t (u(s-,y) {1\!\!1}_{\mathbb {R}_0}).$$

3.3 The Hull and White Formula in the Lévy Case

Consider the following definitions in order to shorten the notation, for a suitable function F : 

• $\varDelta _{x} F(s,X_s,Y_s):=F(s,X_s+x,Y_s)-F(s,X_s,Y_s).$

• $\varDelta _{xx} F(s,X_s, Y_s):=F(s,X_s+x,Y_s)-F(s,X_s, Y_s)-x(\partial _x F)(s,X_s,Y_s).$

• $\varDelta F(s,X_s,Y_s)=F(s,X_s+x,Y_s)-F(s,X_s,Y_s)-(e^x-1)(\partial _{x} F)(s,X_{s},Y_s).$

Then, we have the following decomposition of the price formula:

Theorem 2

Assume

• (B1): $\sigma ^2\in \mathbb {L}^{1,2}_{N,-}\cap \mathbb {L}^{1,2}_W.$

• (B2): $\sigma \in \mathbb {L}^{1,2}_W.$

• (B3): For any $t\in [0,T]$, $\int _t^T \mathbb {E}_t ((\int _t^s \sigma _u^2 du)^{-2}) ds<\infty .$

Then we have

$$\begin{aligned} V_t= & {} \mathbb {E}_t(BS(t,X_{t},v_{t}))\\&+\frac{\rho }{2}\mathbb {E}_t(\int ^{T}_{t}e^{-r(s-t)}(\partial _{x}^3-\partial ^2_x) BS(s,X_{s},{\bar{\sigma }}_{s}) \varLambda _{s}ds)\\&+\mathbb {E}_t(\int ^{T}_{t}\int _\mathbb {R} e^{-r(s-t)}\varDelta BS(s, X_{s-}, {\bar{\sigma }}_s)\nu (dy)ds)\\&+\mathbb {E}_t(\int ^{T}_{t}\int _\mathbb {R}e^{-r(s-t)} D^{N,-}_{s,y} \varDelta _{y} BS(s, X_{s-}, {\bar{\sigma }}_s) y\nu (dy)ds). \end{aligned}$$

Remark 2

We can consider the following particular cases:

• 1. Observe that we cannot split the third term in two terms because in the general case

  $$\mathbb {E}_t(\int ^{T}_{t}\int _\mathbb {R} e^{-r(s-t)}\varDelta _{y}BS(s,X_{s^{-}},{\bar{\sigma }}_{s})\nu (dy)ds)$$

  and

  $$\mathbb {E}_t(\int ^{T}_{t}\int _\mathbb {R} e^{-r(s-t)}(e^y-1)\partial _{x}BS(s,X_{s^{-}},{\bar{\sigma }}_s)\nu (dy)ds)$$

  are not necessarily convergent.

• 2. Observe that if in the previous theorem we assume $\int _\mathbb {R} |y|\nu (dy)<\infty$, that is, finite variation, we obtain

  $$\begin{aligned} V_t= & {} \mathbb {E}_t(BS(t,X_{t},v_{t}))\\&+\frac{\rho }{2}\mathbb {E}_t(\int ^{T}_{t}e^{-r(s-t)}(\partial _x^3-\partial _x^2) BS(s,X_{s},{\bar{\sigma }}_{s}) \varLambda _{s}ds)\\&-\mathbb {E}_t(\int ^{T}_{t}\int _\mathbb {R} e^{-r(s-t)}(e^y-1)\partial _{x} BS(s,X_{s},{\bar{\sigma }}_{s})\nu (dy)ds)\\&+\mathbb {E}_t(\int ^{T}_{t}\int _\mathbb {R}e^{-r(s-t)} T^{-}_{s,y} \varDelta _{y}BS(s,X_{s^{-}},{\bar{\sigma }}_{s})\nu (dy)ds), \end{aligned}$$

  that is exactly the formula obtained in [5] for the finite activity case, that in fact is valid in the finite variation case.

• 3. If the volatility process is independent from price jumps, we have

  $$D^{N,-}_{s,y} \varDelta _{y} BS(s, X_{s-}, {\bar{\sigma }}_s)=0$$

  and we obtain

  $$\begin{aligned} V_t= & {} \mathbb {E}_t(BS(t,X_{t},{\bar{\sigma }}_{t}))\\&+\frac{\rho }{2}\mathbb {E}_t(\int ^{T}_{t}e^{-r(s-t)}(\partial _{x}^3-\partial _x^2) BS(s,X_{s},{\bar{\sigma }}_{s}) \varLambda _{s}ds)\\&+\mathbb {E}_t \int ^{T}_{t}\int _\mathbb {R} e^{-r(s-t)} \varDelta BS(s,X_{s^{-}},{\bar{\sigma }}_{s}) \nu (dy)ds, \end{aligned}$$

  that generalizes the formula in [3]. As in the previous remark, only in the finite variation case we recuperate exactly the formula in [3]. This formula covers Bates model and any correlated model with any type of Lévy jumps in the price process.

• 4. If moreover, the volatility process is independent from the price process, that is, $\rho =0$, we obtain

  $$V_t=\mathbb {E}_t(BS(t,X_{t},v_{t}))+\mathbb {E}_t\int ^{T}_{t}\int _\mathbb {R}e^{-r(s-t)}\varDelta BS(s,X_{s^{-}},{\bar{\sigma }}_{s})\nu (dy)ds.$$

  This covers all the so called uncorrelated models plus jumps (Heston-Kou model for example) and in the particular case of constant volatility, the so called exponential Lévy models. In the jump part we can consider infinite activity jumps as CGMY model (for $Y\ge 0$) or Meixner model for example.

Proof

We follow the same general idea of Theorem 1. The necessary ad-hoc Itô formula, see [9], is now

$$\begin{aligned} F(t,X_{t},Y_{t})= & {} F(0,X_{0},Y_{0})+\int _{0}^{t}{\partial _{s}F} (s,X_{s},Y_{s})ds+\delta _t^{W,B}({\partial _{x}F}(\cdot ,X_{\cdot },Y_{\cdot })\sigma _{\cdot })\\&+\int _{0}^{t}{\partial _{x}F}(s,X_{s},Y_{s})(r-c_2-\frac{\sigma _{s}^{2}}{2})ds-\int _{0}^{t}{\partial _{y}F}(s,X_{s},Y_{s})\sigma _{s}^{2}ds\\&+\rho \int _{0}^{t}{\partial _{xy} F}(s,X_{s},Y_{s})\varLambda _{s}ds+\frac{1}{2}\int _{0}^{t}{\partial _x^2 F}(s,X_{s},Y_{s})\sigma _{s}^{2}ds\\&+\int _0^t\int _{\mathbb {R}_0} (\varDelta _y F(s,X_{s-},Y_s)-y(\partial _yF)(s, X_{s-}, Y_{s}))\nu (dy)ds\\&+\delta _t^N (\frac{\varDelta _y F(s,X_{s-},Y_s)}{y}{1\!\!1}_{\mathbb {R}_0} (y))+\delta _t^N (D^{N,-}_{s,y}\varDelta _y (s,X_{s-},Y_s))\\&+\int _0^t \int _{\mathbb {R}} D^{N,-}_{s,y}\varDelta _y F(s,X_{s-},Y_s)y\nu (dy)ds. \end{aligned}$$

To prove it, fix first $\varepsilon >0$, and consider the process

$$\begin{aligned} X^{\varepsilon }_{t}:= & {} x+(r-c_2)t-\frac{1}{2}\int ^t_0\sigma ^2_s ds+\int ^t_0\sigma _{s}(\rho dW_s+\sqrt{1-\rho ^{2}}dB_s)\\&+\int ^t_0\int _{|x|>\varepsilon } x{\tilde{N}}(ds,dx). \end{aligned}$$

This process has a finite number of jumps and converges a.s. and in $L^2$ sense to $X_t$.

Denote by $T^{\varepsilon }_{i}$ the jump instants, and write $T^{\varepsilon }_{0}:=0$. Then

$$\begin{aligned}&F(T^{\varepsilon }_{i+1},X^{\varepsilon }_{T^{\varepsilon }_{i+1}}, Y_{T^{\varepsilon }_{i+1}})-F(T^{\varepsilon }_{i}, X^{\varepsilon }_{T^{\varepsilon }_{i}}, Y_{T^{\varepsilon }_{i}})=\int ^{T^{\varepsilon }_{i+1}-}_{T^{\varepsilon }_{i}}dF(s,X^{\varepsilon }_s,Y_s)\\&+F(T^{\varepsilon }_{i+1}, X^{\varepsilon }_{T^{\varepsilon }_{i+1}}, Y_{T^{\varepsilon }_{i+1}})-F(T^{\varepsilon }_{i+1}, X^{\varepsilon }_{T^{\varepsilon }_{i+1}-}, Y_{T^{\varepsilon }_{i+1}}). \end{aligned}$$

On the stochastic interval $[T^{\varepsilon }_{j},T^{\varepsilon }_{j+1}[$ we can apply the anticipative Itô formula for continuous process presented in Sect. 2. Then we have that

$$\partial _{x}F(s,X_{s^{-}},Y_s)\sigma _s {1\!\!1}_{[0,t]}(s) \in Dom\,\delta ^{W,B}$$

and

$$\begin{aligned} F(t,X^{\varepsilon }_t,Y_t)&=F(0,X_0,Y_0)+\int ^t_0\partial _s F(s,X^{\varepsilon }_s,Y_s)ds\\&\quad +\int ^{t}_{0}\partial _xF(s,X^{\varepsilon }_{s},Y_s)(r-\frac{\sigma ^{2}_{s}}{2}-c_2)ds+\delta ^{W,B}_t\left( \partial _xF(s,X^{\varepsilon }_{s^{-}},Y_{s})\sigma _s\right) \\&\quad -\int ^{t}_{0}\int _{\{|x|>\varepsilon \}}\partial _xF(s,X^{\varepsilon }_{s},Y_s)x\nu (dx)ds-\int ^{t}_{0}\partial _{y}F(s,X^{\varepsilon }_{s},Y_s)\sigma ^{2}_{s}ds\\&\quad +\rho \int ^{t}_{0}\partial _{xy} F(s,X^{\varepsilon }_{s},Y_{s})\varLambda _{s}ds+\frac{1}{2}\int ^t_0\partial ^2_x F(s, X^{\varepsilon }_{s},Y_s)\sigma ^2_s ds\\&\quad +\sum _i [F(T^{\varepsilon }_{i}, X^{\varepsilon }_{T^{\varepsilon }_{i}}, Y_{T^{\varepsilon }_{i}})-F(T^{\varepsilon }_{i},X^{\varepsilon }_{T^{\varepsilon }_{i}-},Y_{T^{\varepsilon }_{i}})]. \end{aligned}$$

Of course we can write

$$\sum _i [F(T^{\varepsilon }_{i}, X^{\varepsilon }_{T^{\varepsilon }_{i}}, Y_{T^{\varepsilon }_{i}})-F(T^{\varepsilon }_{i}, X^{\varepsilon }_{T^{\varepsilon }_{i}-},Y_{T^{\varepsilon }_{i}})]=\int _0^t \int _{|x|>\varepsilon } \varDelta _x F(s, X_{s-}, Y_s) N(ds, dx).$$

Then,

$$\sum _i [F(T^{\varepsilon }_{i}, X^{\varepsilon }_{T^{\varepsilon }_{i}}, Y_{T^{\varepsilon }_{i}})-F(T^{\varepsilon }_{i}, X^{\varepsilon }_{T^{\varepsilon }_{i}-},Y_{T^{\varepsilon }_{i}})]-\int ^{t}_{0}\int _{|x|>\varepsilon }\partial _x F(s,X^{\varepsilon }_{s-},Y_s)x\nu (dx)ds$$

$$=\int _0^t \int _{|x|>\varepsilon }\varDelta _{x} F(s, X_{s-}^{\varepsilon }, Y_s) {\tilde{N}}(ds, dx)+\int ^{t}_{0}\int _{|x|>\varepsilon }\varDelta _{xx} F(s, X_{s-}^{\varepsilon }, Y_s)\nu (dx)ds.$$

Observe that this equality is the crucial step of the proof. Only introducing $\varDelta _{xx} F(s, X_{s-}^{\varepsilon }, Y_s)$ we become able to apply succesfully the dominated convergence theorem, even if Y has no jumps.

Using the relation between $\delta ^N$ and the integral with respect to ${\tilde{N}}$ we have

$$\begin{aligned}&\int _0^t \int _{|x|>\varepsilon } \varDelta _x F(s, X_{s-}^{\varepsilon }, Y_s) {\tilde{N}}(ds, dx)\\= & {} {\delta }^N_t (T^{-}_{s,x} \frac{\varDelta _x F(s, X_{s-}^{\varepsilon }, Y_s)}{x} {1\!\!1}_{\{|x|>\varepsilon \}})\\&+\int _0^t \int _{|x|>\varepsilon } D^{N,-}_{s,x}\frac{\varDelta _x F(s, X_{s-}^{\varepsilon }, Y_s)}{x} x^2\nu (dx)ds. \end{aligned}$$

And using mean value theorem and the fact that first and second derivatives of F are bounded we have

$$|T^{-}_{s,x} \frac{\varDelta _x F(s, X_{s-}^{\varepsilon }, Y_s)}{x}|=|\frac{\varDelta _x F(s, X_{s-}^{\varepsilon }, T^{-}_{s,x}Y_s)}{x}|\le C,$$

$$|D^{N,-}_{r,y}\frac{\varDelta _x F(s, X_{s-}^{\varepsilon }, T^{-}_{s,x}Y_s)}{x}|\le C|D^{N,-}_{r,y}T^{-}_{s,x}Y_s|=C\int _s^T |D^{N,-}_{r,y}T^{-}_{s,x}\sigma ^2_u|du$$

and

$$|D^{N,-}_{s,x}\frac{\varDelta _x F(s, X_{s-}^{\varepsilon }, Y_s)}{x}|\le C |D^{N,-}_{s,x} Y_s|=C \int _s^T |D^{N,-}_{s,x} \sigma _r^2| dr,$$

for a generic constant C.

Finally, using (B1) and the dominated convergence theorem the right hand side of the previous equality converges when $\varepsilon$ goes to 0. The other terms converge also by the dominated convergence theorem, and the Itô formula follows.

Then, following the same steps of the proof of Theorem 1, after applying this last ad-hoc Itô formula, taking conditional expectations, using the fact that Skorohod type integrals have zero expectation and multiplying by $e^{rt}$ we obtain

$$\begin{aligned}&\mathbb {E}_t(e^{-r(T-t)} BS(T,X_{T},{\bar{\sigma }}^{\delta }_{T}) \phi (\frac{X_T}{n}))\\= & {} \mathbb {E}_t(BS(t,X_{t},{\bar{\sigma }}^{\delta }_{t})\phi (\frac{X_t}{n}))\nonumber \\&+\mathbb {E}_t(\int ^{T}_{t}e^{-r(s-t)}A_{n}(s)ds)\nonumber \\&+\frac{\rho }{2}\mathbb {E}_t(\int ^{T}_{t}e^{-r(s-t)}(\partial _x^3-\partial _x^2)BS(s,X_{s},{\bar{\sigma }}^{\delta }_{s})\phi _{n}(\frac{X_{s}}{n})\varLambda _{s}ds)\nonumber \\&+\frac{\rho }{2}\mathbb {E}_t(\int ^{T}_{t}e^{-r(s-t)}(\partial _x^2-\partial _x)BS(s,X_{s},{\bar{\sigma }}^{\delta }_{s})\frac{1}{n}\phi ^{'}(\frac{X_{s}}{n})\varLambda _{s}ds)\nonumber \\&-c_2\mathbb {E}_t(\int ^{T}_{t}e^{-r(s-t)}\partial _{x} BS(s,X_{s},{\bar{\sigma }}^{\delta }_{s})\phi (\frac{X_s}{n})ds)\nonumber \\&+\mathbb {E}_t(\int ^{T}_{t}\int _\mathbb {R} e^{-r(s-t)}\varDelta _{yy}BS(s,X_{s^{-}},{\bar{\sigma }}^{\delta }_{s})\phi (\frac{X_{s-}}{n})\nu (dy)ds)\nonumber \\&+\mathbb {E}_t(\int ^{T}_{t}\int _\mathbb {R}e^{-r(s-t)} D^{N,-}_{s,y}\frac{\varDelta _{y}BS(s,X_{s^{-}},{\bar{\sigma }}^{\delta }_{s})\phi (\frac{X_{s-}}{n})}{y} y^2\nu (dy)ds).\nonumber \end{aligned}$$

(1)

And as in Theorem 1, applying the dominated convergence theorem, letting first $n\uparrow \infty$ and then $\delta \downarrow 0$ we obtain the result.

To assure the dominated convergence, we have to treat the last three terms of (1) as a unique term and separate it in two integrals, one on $|y|\le 1$ and the other on $|y|>1.$

In the case $|y|>1$, things simplify and we obtain

$$\begin{aligned}&-\mathbb {E}_t(\int ^{T}_{t}\int _{|y|>1}e^{-r(s-t)}\partial _{x} BS(s,X_{s},{\bar{\sigma }}^{\delta }_{s})\phi (\frac{X_s}{n}) (e^y-1)\nu (dy)ds)\\&+\mathbb {E}_t(\int ^{T}_{t}\int _{|y|>1} e^{-r(s-t)}T_{s,y}^{N,-}\varDelta _{y}BS(s,X_{s^{-}},{\bar{\sigma }}^{\delta }_{s})\phi (\frac{X_{s-}}{n})\nu (dy)ds).\\ \end{aligned}$$

For the first term we use that $\partial _{x} BS(s,X_{s},{\bar{\sigma }}^{\delta }_{s})$ is bounded by $e^{X_s}$ and for the second term we use the fact that

$$|T_{s,y}^{N,-}\varDelta _{y}BS(s,X_{s^{-}},{\bar{\sigma }}^{\delta }_{s})|\le 2K+e^{X_{s-}+y}+e^{X_{s-}}$$

and that

$$\int _{\{|y|>1\}} e^y \nu (dy)<\infty .$$

In the case $|y|\le 1,$ the fifth term in the right hand side of (1) is bounded because $\partial _x BS$ is bounded. The sixth term can be written as

$$\begin{aligned}&\frac{1}{2}\mathbb {E}_t(\int ^{T}_{t}\int _{|y|\le 1}e^{-r(s-t)}\partial _x^2 BS(s,X_{s}+\alpha ,{\bar{\sigma }}^{\delta }_{s})\phi (\frac{X_{s-}}{n})y^2\nu (dy)ds)\\= & {} \frac{1}{2}\mathbb {E}_t(\int ^{T}_{t}\int _{|y|\le 1}e^{-r(s-t)}(\partial ^2_x-\partial _x)BS(s,X_{s}+\alpha ,{\bar{\sigma }}^{\delta }_{s})\phi (\frac{X_{s-}}{n})y^2\nu (dy)ds)\\&+\frac{1}{2}\mathbb {E}_t(\int ^{T}_{t}\int _{|y|\le 1}e^{-r(s-t)}\partial _{x} BS(s,X_{s}+\alpha ,{\bar{\sigma }}^{\delta }_{s})\phi (\frac{X_{s-}}{n})y^2\nu (dy)ds) \end{aligned}$$

where $|\alpha |\le |y|.$

The second term on the right hand side of this last expression is bounded because $\partial _x BS$ is bounded. For the first term we use Lemma 2 in [3] as in Theorem 1 and we bound it by

$$C \mathbb {E}_t(\int ^{T}_{t}\int _{|y|\le 1} \sqrt{\frac{1}{Y_t}} y^2\nu (dy)ds),$$

for a certain constant C. Hypothesis (B3) guarantees the convergence of this integral, because

$$\mathbb {E}_t(\int ^{T}_{t}\int _{|y|\le 1} \sqrt{\frac{1}{Y_t}} y^2\nu (dy)ds)= C (T-t) \mathbb { E}_t(\sqrt{\frac{1}{Y_t}})$$

and

$$\mathbb {E}_t(\sqrt{\frac{1}{Y_t}})\le (\mathbb {E}_t(\frac{1}{Y_t^2}))^{\frac{1}{4}}\le (\mathbb {E}_t ((\int _t^s \sigma _u^2 du)^{-2}))^{\frac{1}{4}}$$

and so, the term is bounded by (B3).

Finally, the last term of (1) can be bounded by

$$\begin{aligned}&\mathbb {E}_t(\int ^{T}_{t}\int _{|y|\le 1} |D^{N,-}_{s,y}(\partial _x BS)(s,X_{s^{-}}+\alpha ,{\bar{\sigma }}^{\delta }_{s})| y^2\nu (dy)ds)\\= & {} \mathbb {E}_t(\int ^{T}_{t}\int _{|y|\le 1} |D^{N,-}_{s,y}(\partial _x BS)(s,X_{s^{-}}+\alpha ,\sqrt{\frac{Y_s+\delta }{T-s}})| y^2\nu (dy)ds)\\= & {} \mathbb {E}_t(\int ^{T}_{t}\int _{|y|\le 1} |(\partial _{x\sigma } BS)(s,X_{s^{-}}+\alpha ,\sqrt{\frac{\theta _{s,y}+\delta }{T-s}})|\frac{|D^{N,-}_{s,y} Y_s|}{2(T-s)\sqrt{\frac{\theta _{s,y}+\delta }{T-s}}} y^2\nu (dy)ds) \end{aligned}$$

$$\begin{aligned} \quad =\frac{1}{2} \mathbb {E}_t(\int ^{T}_{t}\int _{|y|\le 1} |(\partial _x (\partial ^2_x-\partial _x)BS)(s,X_{s^{-}}+\alpha ,\sqrt{\frac{\theta _{s,y}+\delta }{T-s}})||D^{N,-}_{s,y} Y_s|y^2\nu (dy)ds), \end{aligned}$$

(2)

where $|\alpha |\le |y|$ and $\theta _{s,y}$ is a quantity between $Y_s$ and $T_{s,y}^{N,-}Y_s.$

Now, we cannot apply directly Lemma 2 in [3], but mimicking the proof we have that the last integral is less or equal than

$$C \mathbb {E}_t(\int ^{T}_{t}\int _{|y|\le 1} \frac{|D^{N,-}_{s,y} Y_s|}{\int _t^s \sigma _u^2 du} y^2\nu (dy)ds).$$

Then, applying Cauchy-Schwarz inequality, this expression is bounded by

$$C (\mathbb {E}_t(\int ^{T}_{t}\int _{|y|\le 1} \frac{1}{(\int _t^s \sigma _u^2 du)^2}y^2\nu (dy)ds))^{\frac{1}{2}} (\mathbb {E}_t(\int ^{T}_{t}\int _{|y|\le 1}|D^{N,-}_{s,y} Y_s|^2 y^2\nu (dy)ds))^{\frac{1}{2}}.$$

The first term of this product is bounded by (B3) and the second one by (B1)

Remark 3

• 1. As in the case of Theorem 1, (B2) can be changed by (A$2^\prime$).

• 2. If $\sigma$ not depends on jumps, (B3) reduces to $\mathbb {E}_t((\int _t^T \sigma _u^2 du)^{-\frac{1}{2}})<\infty ,$ that it is weaker than (A$2^\prime$).

• 3. In the case of finite variation, (B3) is not necessary.

• 4. In the complete general case, but only in this case, (B1) and (A$2^\prime$) are not enough. An alternative treatment of (2), using (A$2^\prime$), is to bound directly

  $$|(\partial _x (\partial ^2_x-\partial _x)BS)(s,X_{s^{-}}+\alpha ,\sqrt{\frac{\theta _{s,y}+\delta }{T-s}})|\le e^{X_{s-}+\alpha }(\frac{1}{\sqrt{Y_s}}+\frac{1}{Y_s}).$$

  So, we can decompose this term in two new terms. The term with $Y_s^{-\frac{1}{2}}$ can be treated easily and it is bounded with no other requirements than (B1) and (A$2^\prime$). But the term with $Y_s^{-1}$ requires to assume, alternatively to (B3), the following hypothesis,

  $$\mathbb {E}_t(\int ^{T}_{t}\int _{|y|\le 1} \frac{|D^{N,-}_{s,y} Y_s|}{Y_s} y^2\nu (dy)ds)<\infty ,$$

  that using (A$2^\prime$) is equivalent to assume

  $$(B4): \mathbb {E}_t(\int ^{T}_{t}\int _{|y|\le 1} \frac{|D^{N,-}_{s,y} Y_s|}{T-s} y^2\nu (dy)ds)<\infty .$$

  Note that this last hypothesis is stronger than (B1). So, we need (B1), (A$2^\prime$) and (B4) to guarantee the complete general case.