Dimension Reduction for Pricing Options Under Multidimensional Lévy Processes

The aim of this study was to develop efficient quasi-Monte Carlo algorithms for pricing European derivative securities under multidimensional Lévy models. In the paper, we first introduce the multidimensional generalized hyperbolic distribution as a normal variance–mean mixture. Using this distribution, we can model a multidimensional generalized Lévy process as a subordinated Brownian motion. Under this process, we develop practically efficient dimension reduction methods that can enhance the numerical efficiency of the quasi-Monte Carlo method. The algorithms extend the generalized linear transformation method that was originally proposed for a univariate Lévy process. We also propose hybrid types of dimension reduction methods in which the dimension reduction techniques are applied separately to the subordinator and the Brownian motion. Through numerical examples we demonstrate that the proposed method realizes a substantial gain in efficiency, relative to the naive Monte Carlo and quasi-Monte Carlo methods in the context of pricing average options.