On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility

In this paper we use Malliavin calculus techniques to obtain an expression for the short-time behavior of the at-the-money implied volatility skew for a generalization of the Bates model, where the volatility does not need to be neither a difussion, nor a Markov process as the examples in section 7 show. This expression depends on the derivative of the volatility in the sense of Malliavin calculus.Black-Scholes formula, derivative operator, Itô's formula for the Skorohod integral, jump-diffusion stochastic volatility model

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

In this paper, generalizing results in Alòs, León and Vives (2007b), we see that the dependence of jumps in the volatility under a jump-diffusion stochastic volatility model, has no effect on the short-time behaviour of the at-the-money implied volatility skew, although the corresponding Hull and White formula depends on the jumps. Towards this end, we use Malliavin calculus techniques for Lévy processes based on Løkka (2004), Petrou (2006), and Solé, Utzet and Vives (2007).Hull and White formula, Malliavin calculus, Ito’s formula for the Skorohod integral, jumpdiffusion stochastic volatility models

Sur certaines relations entre les intégrales trajectorielles et l'opérateur de translation et son dual dans l'espace de Poisson canonique

We study the relationship between the translation operator, its dual and the pathwise integral on the Poisson space with weak conditions on the processes