Implementing Quasi-Monte Carlo Simulations with Linear Transformations

TL;DR

This paper enhances Quasi-Monte Carlo simulations for pricing complex options by introducing a fast linear transformation technique that reduces computational costs and improves convergence in high-dimensional settings.

Contribution

It develops a fast QR decomposition-based linear transformation method to accelerate high-dimensional Quasi-Monte Carlo simulations for option pricing.

Findings

Linear transformation significantly reduces computational time.

Higher convergence rate achieved for selected components.

Method outperforms standard Cholesky and PCA approaches.

Abstract

Pricing exotic multi-asset path-dependent options requires extensive Monte Carlo simulations. In the recent years the interest to the Quasi-monte Carlo technique has been renewed and several results have been proposed in order to improve its efficiency with the notion of effective dimension. To this aim, Imai and Tan introduced a general variance reduction technique in order to minimize the nominal dimension of the Monte Carlo method. Taking into account these advantages, we investigate this approach in detail in order to make it faster from the computational point of view. Indeed, we realize the linear transformation decomposition relying on a fast ad hoc QR decomposition that considerably reduces the computational burden. This setting makes the linear transformation method even more convenient from the computational point of view. We implement a high-dimensional (2500) Quasi-Monte Carlo simulation combined with the linear transformation in order to price Asian basket options with same set of parameters published by Imai and Tan. For the simulation of the high-dimensional random sample, we use a 50-dimensional scrambled Sobol sequence for the first 50 components, determined by the linear transformation method, and pad the remaining ones out by the Latin Hypercube Sampling. The aim of this numerical setting is to investigate the accuracy of the estimation by giving a higher convergence rate only to those components selected by the linear transformation technique. We launch our simulation experiment also using the standard Cholesky and the principal component decomposition methods with pseudo-random and Latin Hypercube sampling generators. Finally, we compare our results and computational times, with those presented in Imai and Tan.