Quasi-Monte Carlo Simulation of Random Walks in Finance

Abstract

The need to numerically simulate stochastic processes arises in many fields. Frequently this is done by discretizing the process into small time steps and applying pseudo-random sequences to simulate the randomness. This paper address the question of how to use quasi-Monte Carlo methods to improve this simulation. Special techniques must be applied to avoid the problem of high dimensionality which arises when a large number of time steps are required. One such technique, the generalized Brownian bridge, is described here. The method is applied to a classical problem from finance, the valuation of a mortgage backed security portfolio. When expressed as an integral, this problem is nominally 360 dimensional. The analysis of the integrand presented here explains the effectiveness of the quasi-random sequences on this high dimensional problem.