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

TL;DR

The paper tackles scalable pricing of high-dimensional multi-asset options by applying randomized quasi-Monte Carlo (RQMC) in the Fourier pricing framework. It introduces a model-aware domain transformation that maps $\mathbb{R}^d$ to $[0,1]^d$ using a normal-variance-mean mixture (and related constructions) to preserve integrand regularity and satisfy boundary-growth conditions. The main contributions are the first demonstration of RQMC in Fourier space for high-dimensional option pricing, practical domain-transformations tailored to GBM, GH, and VG models (including dependent assets), and extensive numerical evidence showing substantial speedups over MC in physical space and TP in Fourier space, up to 15 assets. The approach generalizes to deterministic high-dimensional integrals beyond option pricing and offers a path to efficient, accurate high-dimensional computations in finance. Overall, the work provides a principled framework to exploit analyticity in the Fourier domain with RQMC, enabling fast and reliable pricing for complex multi-asset derivatives.

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.

Figures (12)

• Figure 3.1: Effect of the domain transformation on the smoothness of the transformed integrand \ref{['eq:transformed_fourier_pricing_integral']} for a 1D put option under GBM with volatility $\sigma = 0.2$. Gaussian density: $\psi(\cdot;\tilde{\mu}, \tilde{\sigma})$, mean: $\tilde{\mu} = 0$, scale: $\tilde{\sigma}$. The used parameters are: $K = S_0 = 100, r = 0, T = 1, R = 6.58$,
• Figure 3.2: Effect of the correlation parameter, $\rho$, on the convergence of RQMC for a two-dimensional call on the minimum option under the GBM model with $S_0^j = 100$, $K = 100$, $r = 0$, $T= 1$, $\sigma_j = 0.2$, with $\boldsymbol{\Sigma}_{ij} = \rho\sigma_i \sigma_j$ for $i,j= 1,2$, $i \neq j$, $\boldsymbol{\Sigma}_{ii} = \sigma_i^2$. For the domain transformation, $\tilde{\sigma}_j = \frac{1}{\sqrt{T} \sigma_j} = 5$, where $j =1,2$. $N$: number of QMC points; $S = 30$: number of digital shifts.
• Figure 3.3: Convergence of the RQMC error for multivariate and univariate domain transormations in the case of a 2D call on the minimum option under the GBM model with $S_0^j = 100$, $K = 100$, $r = 0$, $T= 1$, $\sigma_j = 0.2, \rho = 0.7$, with $\boldsymbol{\Sigma}_{ij} = \rho\sigma_i \sigma_j$ for $i,j= 1,2$, $i \neq j$, $\boldsymbol{\Sigma}_{ii} = \sigma_i^2$. $N$: number of QMC points; $S = 30$: number of digital shifts.
• Figure 4.1: Effect of the parameter $\tilde{\sigma}$ on (a) the shape of the transformed integrand $\tilde{g}(u)$ and (b) convergence of the relative statistical error of RQMC with $S_0 = 100$, $K = 100$, $r = 0$, $T= 1$, and $\sigma = 0.2$ for a one-dimensional call option under the GBM model. $N$: number of QMC points; $S = 32$: number of digital shifts. Boundary growth condition limit: $\overline{\sigma} = \frac{1}{ \sqrt{T}\sigma} = 5$.
• Figure 4.2: Effect of the parameter $\tilde{\sigma}$ on (a) the shape of the transformed integrand $\tilde{g}(u)$ and (b) convergence of the relative statistical error of RQMC with $S_0 = 100$, $K = 100$, $r = 0$, $T= 1$, $\alpha = 20$, $\beta = -3, \delta = 0.2$ and $\lambda = 1$ for a one-dimensional call option under the GH model. $N$: number of QMC points; $S = 32$: number of digital shifts. Boundary growth condition limit: $\overline{\sigma} = \frac{1}{ T \delta } = 5$.

Theorems & Definitions (18)

• Proposition 2.4: Multivariate Fourier Pricing Valuation Formula
• proof
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Proposition 3.5
• proof
• Proposition 3.6
• proof