We present an asymptotic expansion of the distribution of a random variable which admits a stochastic expansion around a continuous martingale. The emphasis is put on the use of the Malliavin calculus; the uniform nondegeneracy of the Malliavin covariance under certain truncation plays an essential role as the Crame r condition did in the case of independent observations. Applications to stati...

Acknowledgements I enjoyed the help and support from numerous friends and faculty members during whilst writing this thesis. My greatest debt however, goes to my supervisor, Professor Tony Dooley, who initiated me to the fantastic field of Malliavin calculus. With his incredible breadth and depth of knowledge and intuition, he guided me to structure and clarify my thought and suggested valuable...

Using the techniques of the Malliavin calculus and standard Itô calculus methods, we give an Euler scheme to approximate the solution of a class of anticipating stochastic differential equations.

We show that the Heston volatility or equivalently the Cox-IngersollRoss process satisfying dvt = κ (θ − vt) dt+ ν √ vtdWt is Malliavin differentiable and give an explicit expression for the derivative. This result assures the applicability of Malliavin calculus in the framework of the Heston stochastic volatility model and the CoxIngersoll-Ross model for interest rates.

Recent work by Nualart and Schoutens (2000), where a kind of chaotic property for Lévy processes has been proved, has enabled us to develop a Malliavin calculus for Lévy processes. For simple Lévy processes some useful formulas for computing Malliavin derivatives are deduced. Applications for option hedging in a jump–diffusion model are given.

This paper is the sequel of Part I [1], where we showed how to use the so-called Malliavin calculus in order to devise efficient Monte-Carlo (numerical) methods for Finance. First, we return to the formulas developed in [1] concerning the “greeks” used in European options, and we answer to the question of optimal weight functional in the sense of minimal variance. Then, we investigate the use o...

The aim of this note is to provide a short and self-contained proof of Hörmander’s theorem about the smoothness of transition probabilities for a diffusion under Hörmander’s “brackets condition”. While both the result and the technique of proof are well-known, the exposition given here is novel in two aspects. First, we introduce Malliavin calculus in an “intuitive” way, without using Wiener’s ...

This paper derives asymptotic expansion formulas for option prices and implied volatilities as well as the density of the underlying asset price in multi-dimensional stochastic volatility models. In particular, the integration-byparts formula in Malliavin calculus and the push-down of Malliavin weights are effectively applied. We provide an expansion formula for generalized Wiener functionals a...

The fourth moment theorem provides error bounds in the central limit theorem for elements of Wiener chaos of any order. It was proved by Nourdin and Peccati [31] using Stein’s method and the Malliavin calculus. It was also proved by Azmoodeh, Campese and Poly [3] using Stein’s method and Dirichlet forms. This paper is an exposition on the connections between Stein’s method and the Malliavin cal...