Probabilistic representation of integration by parts formulae for some stochastic volatility models with unbounded drift

Abstract

Abstract. In this paper, we establish a probabilistic representation as well as some integration by parts formulae for the marginal law at a given time maturity of some stochastic volatility model with unbounded drift. Relying on a perturbation technique for Markov semigroups, our formulae are based on a simple Markov chain evolving on a random time grid for which we develop a tailor-made Malliavin calculus. Among other applications, an unbiased Monte Carlo path simulation method stems from our formulas so that it can be used in order to numerically compute with optimal complexity option prices as well as their sensitivities with respect to the initial values or Greeks in finance, namely the Delta and Vega , for a large class of non-smooth European payoff. Numerical results are proposed to illustrate the efficiency of the method.

On Some Asymptotic Expansions of Skew Diffusions

“…For a review on these subjects, we refer the reader to [12]. It has also recently been revisited in a series of works [1,7,4] to provide some probabilistic representation and integration by parts formula of the marginal law of various processes as an alternative to standard Malliavin calculus techniques or to establish the well-posedness in the weak sense of some degenerate stochastic differential equations (SDEs for short) [8,9].…”