Fast Orthogonal transforms for pricing derivatives with quasi-Monte Carlo

TL;DR

This paper introduces fast orthogonal transforms, including a regression-based Householder reflection, to improve the efficiency of quasi-Monte Carlo methods in high-dimensional derivative pricing.

Contribution

It proposes a new regression-based approach for approximating orthogonal transforms, enabling faster computation in QMC derivative pricing.

Findings

Transforms can be computed in $O(n\log(n))$ time.

The Householder reflection method is highly efficient.

Applications to high-dimensional financial problems show significant improvements.

Abstract

There are a number of situations where, when computing prices of financial derivatives using quasi-Monte Carlo (QMC), it turns out to be beneficial to apply an orthogonal transform to the standard normal input variables. Sometimes those transforms can be computed in time $O(n\log(n))$ for problems depending on $n$ input variables. Among those are classical methods like the Brownian bridge construction and principal component analysis (PCA) construction for Brownian paths. Building on preliminary work by Imai and Tan [3] as well as Wang and Sloan [13], where the authors try to find optimal orthogonal transform for given problems, we present how those transforms can be approximated by others that are fast to compute. We further present a new regression-based method for finding a Householder reflection which turns out to be very efficient for a wide range of problems. We apply these methods to several very high-dimensional examples from finance.