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

ArXiv ID: 2403.02832 “View on arXiv”

Authors: Unknown

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.

Keywords: Multi-asset Options, Fourier Methods, Randomized Quasi-Monte Carlo (RQMC), Derivatives Pricing, High Dimensionality, Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper involves advanced mathematics including domain transformations, boundary growth conditions, and analysis of RQMC convergence, while providing empirical validation across multiple models and dimensions up to 15 assets, though without public code or specific backtesting metrics.

  flowchart TD
    A["Research Goal<br>Improve efficiency of Fourier methods<br>for multi-asset option pricing<br>in high dimensions"] --> B["Input Data<br>Multi-asset options (2-15 dimensions)<br>Various pricing models & payoffs"]
    B --> C["Key Methodology<br>Randomized Quasi-MC (RQMC)<br>in Fourier domain<br>with Domain Transformation"]
    C --> D["Computational Process<br>Apply transformation to map<br>unbounded domain R^d to [0,1"]^d<br>while preserving integrand regularity]
    D --> E["RQMC Quadrature<br>Exploit smoothness in Fourier domain<br>Alleviate curse of dimensionality<br>Provide error estimates"]
    E --> F["Key Findings<br>RQMC outperforms MC & Tensor Product<br>in Fourier domain<br>Significant speedup for 15-asset options"]
    F --> G["Outcome<br>Efficient, scalable method for<br>high-dimensional derivative pricing"]