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.
Highlights
IntroductionIt is well 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 the smile or the skew
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
It is well 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 the smile or the skew
Summary
We introduce the tools of Malliavin calculus for Levy processes that we need in the rest of the paper. See for example the book of Sato 17 for a general theory of Levy processes. The construction of a Malliavin calculus, based on a chaos expansion, for a certain process follows three main steps. Property, secondly, to define formally the gradient and divergence operators, and to give their probabilistic interpretations. Several approaches to the Malliavin calculus for Levy processes have been developed, with different probabilistic interpretations of gradient operators. We follow the last one, because in it, the form of the gradient operator simplifies strongly our computations. Let us remark that we are under the conditions of Løkka’s approach because our Levy measure ν is finite
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