Application of Malliavin Calculus and Wiener chaos to option pricing theory
This dissertation provides a contribution to the option pricing literature by means of some recent developments in probability theory, namely the Malliavin Calculus and the Wiener chaos theory. It concentrates on the issue of faster convergence of Monte Carlo and Quasi-Monte Carlo simulations for the Greeks, on the topic of the Asian option as well as on the approximation for convexity adjustment for fixed income derivatives. The first part presents a new method to speed up the convergence of Monte- Carlo and Quasi-Monte Carlo simulations of the Greeks by means of Malliavin weighted schemes. We extend the pioneering works of Fournie et al. (1999), (2000) by deriving necessary and sufficient conditions for a function to serve as a weight function and by providing the weight function with minimum variance. To do so, we introduce its generator defined as its Skorohod integrand. On a numerical example, we find evidence of spectacular efficiency of this method for corridor options, especially for the gamma calculation. The second part brings new insights on the Asian option. We first show how to price discrete Asian options consistent with different types of underlying densities, especially non-normal returns, by means of the Fast Fourier Transform algorithm. We then extends Malliavin weighted schemes to continuous time Asian options. In the last part, we first prove that the Black Scholes convexity adjustment (Brotherton-Ratcliffe and Iben (1993)) can be consistently derived in a martingale framework. As an application, we examine the convexity bias between CMS and forward swap rates. However, for more complicated term structures assumptions, this approach does not hold any more. We offer a solution to this, thanks to an approximation formula, in the case of multi-factor lognormal zero coupon models, using Wiener chaos theory.