Quasi-Monte Carlo with Domain Transformation for Efficient Fourier Pricing of Multi-Asset Options

TL;DR

This paper introduces a domain transformation technique combined with randomized quasi-Monte Carlo (RQMC) methods to efficiently price multi-asset options in high dimensions using Fourier methods, overcoming traditional computational challenges.

Contribution

It develops a novel domain transformation approach that preserves integrand regularity, enabling RQMC to effectively scale Fourier-based option pricing to high-dimensional problems.

Findings

RQMC with domain transformation outperforms Monte Carlo in high dimensions.

The method achieves faster convergence and better accuracy for multi-asset options.

Applicable to models with up to 15 assets, demonstrating practical efficiency.

Abstract

Efficiently pricing multi-asset options poses a significant challenge in quantitative finance. Fourier methods leverage the regularity properties of the integrand in the Fourier domain to accurately and rapidly value options that typically lack regularity in the physical domain. However, most of the existing Fourier approaches face hurdles in high-dimensional settings due to the tensor product (TP) structure of the commonly employed numerical quadrature techniques. To overcome this difficulty, this work advocates using the randomized quasi-MC (RQMC) quadrature to improve the scalability of Fourier methods with high dimensions. The RQMC technique benefits from the smoothness of the integrand and alleviates the curse of dimensionality while providing practical error estimates. Nonetheless, the applicability of RQMC on the unbounded domain, $\mathbb{R}^d$, requires a domain transformation to $[0,1]^d$, which may result in singularities of the transformed integrand at the corners of the hypercube, and hence deteriorate the performance of RQMC. To circumvent this difficulty, we design an efficient domain transformation procedure based on boundary growth conditions on the transformed integrand. The proposed transformation preserves sufficient regularity of the original integrand for fast convergence of the RQMC method. To validate our analysis, we demonstrate the efficiency of employing RQMC with an appropriate transformation to evaluate options in the Fourier space for various pricing models, payoffs, and dimensions. Finally, we highlight the computational advantage of applying RQMC over MC or TP in the Fourier domain, and over MC in the physical domain for options with up to 15 assets.