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 an 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 of the factors which does not depend on any other factors in the model, and (ii) the derivation of a closed-form solution to 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.

Key Words:

• Conditional Monte Carlo
• variance reduction
• dimension reduction
• cross-currency
• Fourier transform
• partial differential equations

Notes

¹ Affine PDEs are those with coefficients that are linear functions of the state variables.

² Note that, due to the quanto term, $\nu$ is in fact coupled to $r_f$, and hence we should not condition on $r_f$.

Additional information

Funding

This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada and a University of Toronto Excellence Award (UTEA). This work was also supported in part by a University of Queensland Early Career Researcher (ECR) Grant [Grant number 1006301-01-298-21-609775].