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

2024-03-05

AI Summary

In high-dimensional multi-asset option pricing, conventional Fourier methods suffer from the curse of dimensionality in tensor-product numerical integration, while standard Monte Carlo (MC) and quasi-Monte Carlo (QMC) methods exhibit degraded convergence on unbounded domains due to integrand singularities. This paper proposes a novel adaptive regularization transformation—mapping unbounded domains to the unit hypercube—based on boundary-growth conditions. For the first time, it embeds randomized quasi-Monte Carlo (RQMC) into the Fourier pricing framework, preserving sufficient integrand regularity to achieve superlinear convergence in Fourier space for dimensions up to 15. Numerical experiments demonstrate speedups of over two orders of magnitude versus physical-domain MC, and 10–50× improvement over both Fourier-domain tensor-product quadrature and standard MC. Moreover, the method delivers verifiable, computable error estimates.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Transfer, Domain Adaptation, Multi-Task Learning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Advertising auctions, pricing, markets, and exchanges
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising

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.

Problem

• Efficiently price multi-asset options in high dimensions
• Overcome dimensionality curse in Fourier methods using RQMC
• Design domain transformation to preserve integrand regularity

Innovation

• Randomized quasi-MC for high-dimensional Fourier pricing
• Domain transformation to preserve integrand regularity
• Efficient boundary growth-based transformation design