Efficiently pricing multi-asset options poses a significant challenge in quantitative finance. The Monte Carlo (MC) method remains the prevalent choice for pricing engines; however, its slow convergence rate impedes its practical application. Fourier methods leverage the knowledge of the characteristic function to accurately and rapidly value options with up to two assets. Nevertheless, they face hurdles in the high-dimensional settings due to the tensor product (TP) structure of commonly employed quadrature techniques. 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 deteriorate the rate of convergence of RQMC. To circumvent this difficulty, we design an efficient domain transformation procedure based on the derived boundary growth conditions of the integrand. This transformation preserves the sufficient regularity of the integrand and hence improves the rate of convergence of RQMC. To validate this 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.

翻译：在量化金融中，高效定价多资产期权是一项重大挑战。蒙特卡洛（MC）方法仍是定价引擎的主流选择，但其收敛速度慢，阻碍了实际应用。傅里叶方法利用特征函数知识，可精确、快速地对最多两种资产的期权进行定价。然而，由于常用求积技术的张量积（TP）结构，这类方法在高维场景中面临困难。本文主张采用随机拟蒙特卡洛（RQMC）求积法提升傅里叶方法在高维下的可扩展性。RQMC技术受益于被积函数的平滑性，可缓解维数灾难，同时提供实用的误差估计。不过，RQMC在无界域$\mathbb{R}^d$上的适用性要求将其变换到$[0,1]^d$，这可能导致变换后被积函数在超立方体角点处出现奇异性，从而降低RQMC的收敛速度。为克服这一难题，我们基于推导出的被积函数边界增长条件，设计了一种高效的域变换过程。该变换保留了被积函数的充分正则性，从而提升了RQMC的收敛速率。为验证这一分析，我们展示了采用适当变换的RQMC在傅里叶空间中对不同定价模型、收益函数及维度下期权定价的效率。最后，我们强调了在傅里叶域中应用RQMC相对于MC或TP的计算优势，以及在物理域中相对于MC对最多包含15种资产的期权定价的计算优势。