Variance Reduction for Multivariate Monte Carlo Simulation

Abstract

As derivative securities become more complex, solution by Monte Carlo simulation is an increasingly necessary tool. But Monte Carlo methods are computationally demanding, and the size of the simulation sample required to achieve reasonable accuracy rapidly escalates beyond what is feasible given current technology when multiple stochastic factors are involved. Variance reduction techniques help considerably. One of the simplest is use of antithetic variables, which imposes on the simulated data the true constraint that the distribution from which the sample is to be drawn is symmetric. Moment matching goes further, by constraining the mean and variance of the sample to match the desired values. In this article, Wang goes further still, to show how to force simulated multivariate vectors to have the right correlations. In the first step a multivariate sample of independent variables is simulated and its sample correlation matrix is calculated. This will be close to the identity matrix but not exactly equal, due to sampling noise. Cholesky factorization of the sample correlation matrix is then used to transform the initial sample of slightly correlated factors into one whose elements are perfectly uncorrelated in the sample. A second Cholesky factorization of the target correlation matrix that is desired for the variables transforms them into a sample with exactly the correct correlations. As Wang demonstrates, this produces a sharp increase in performance for multivariate Monte Carlo problems, even when the simulation sample is constructed from nonrandom low-discrepancy sequences rather than by stochastic simulation.