Multidimensional quasi-Monte Carlo Malliavin Greeks

The aim of this paper is extensively investigate the performance of the estimators for the Greeks of multidimensional complex path-dependent options obtained by the aid of Malliavin Calculus. The study analyses both the computation effort and the variance reduction in the Quasi-Monte Carlo simulation framework. For this purpose, we adopt the approach employed by Montero and Kohatsu-Higa to the multidimensional case. The multidimensional setting shows the convenience of the Malliavin Calculus approach over different techniques that have been previously proposed. Indeed, these techniques may be computationally expensive and do not provide enough flexibility for variance reduction. In contrast, the Malliavin approach provides a class of functions that return the same expected value (the Greek) with different accuracies. This versatility for variance reduction is not possible without the use of the generalized integral by part formula of Malliavin Calculus. In the multidimensional context, we find convenient formulas that permit to improve the localization technique, introduced in Fournié et al. and reduce both the computational cost and the variance. Moreover, we show that the parameters for the variance reduction can be obtained on the flight in the simulation. We illustrate the efficiency of the proposed procedures, coupled with the enhanced version of Quasi-Monte Carlo simulations as discussed in Sabino, for the numerical estimation of the Deltas of call, digital Asian-style and exotic basket options with a fixed and a floating strike price in a multidimensional Black-Scholes market. Given the fact that the gammas of a call option coincides, apart from a constant, with the deltas of digital options, this setting also covers the analysis of formulas tailored for the second order Greeks of call options. Copyright Springer-Verlag 2013