Dimension Reduction Techniques in Quasi-Monte Carlo Methods for Option Pricing

Many problems in mathematical finance (e. g., the pricing of financial derivative securities) can be formulated as high-dimensional integrals, which are often tackled by quasi-Monte Carlo methods. Quasi-random points have the special property that the one-dimensional projections and the earlier dimensional projections are extremely well distributed. This property strongly motivates us to reduce the effective dimensions of the problems. This paper deals with multiasset path-dependent options. The goal of this paper is twofold: (1) to design new methods for dimension reduction and to offer further insight into dimension reduction techniques, and (2) to investigate the practical performance of various dimension reduction strategies in conjunction with different quasi-random points (digital nets and good lattice points). We propose two groups of two-stage procedures for the constructions of correlated multidimensional Brownian motions, which provide a general framework to combine and use different dimension reduction techniques and optimize the use of quasi-random points. We find the inherent weights that characterize the weighted property and the dimension structure of the underlying functions (e. g., the sensitivity indices, the effective dimensions, etc.), and show how each dimension reduction strategy affects them. Numerical experiments on multiasset path-dependent options demonstrate that significant improvements can be achieved by proper use of dimension reduction strategies and quasi-random points. It is also shown that different quasi-random point sets have different behavior with respect to dimension reduction strategies.