Monte Carlo estimation of a joint density using Malliavin calculus, and application to American options

We use the Malliavin integration by parts formula in order to provide a family of representations of the joint density (which does not involve Dirac measures) of (X_θ, X_θ+δ), where X is a d-dimensional Markov diffusion (d ≥ 1), θ > 0 and δ > 0. Following Bouchard et al. (2004), the different representations are determined by a pair of localizing functions. We discuss the problem of variance reduction within the family of separable localizing functions: We characterize a pair of exponential functions as the unique integrated-variance minimizer among this class of separable localizing functions. We test our method on the d-dimensional Brownian motion and provide an application to the problem of American options valuation by the quantization tree method introduced by Bally et al. (2002).