Dimension and Variance Reduction for Monte Carlo Methods for High-Dimensional Models in Finance

Abstract

One-way coupling often occurs in multi-dimensional models in finance. In this paper, we present a dimension reduction technique for Monte Carlo (MC) methods, referred to as drMC, that exploits this structure for pricing plain-vanilla European options under a N-dimensional one-way coupled model, where N is arbitrary. The dimension reduction also often produces a significant variance reduction.

The drMC method is a dimension reduction technique built upon (i) the conditional MC technique applied to one dimension and (ii) the derivation of a closed-form solution for the conditional Partial Differential Equation (PDE) that arises via Fourier transforms. In the drMC approach, the option price can be computed simply by taking the expectation of this closed-form solution. Hence, the approach results in a powerful dimension reduction from N to one, which often results in a significant variance reduction as well, since the variance associated with the other (N-1) factors in the original model are completely removed from the drMC simulation. Moreover, under the drMC framework, hedging parameters, or Greeks, can be computed in a much more efficient way than in traditional MC techniques.

A variance reduction analysis of the method is presented and numerical results illustrating the method's efficiency are provided.

Keywords: conditional Monte Carlo, variance reduction, dimension reduction, cross-currency, Fourier transform, partial differential equations

JEL Classification: E40, E43, G12, G13, C61, C63