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arXiv 2403.02832q-fin.CPcs.NAmath.NA
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带域变换的拟蒙特卡洛方法用于多资产期权的高效傅里叶定价

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

Weierstrass Institute for Applied Analysis and Stochastics (WIAS)（魏尔斯特拉斯应用分析与随机性研究所）；Mathematical Institute, Utrecht University（乌得勒支大学数学研究所）；Delft Institute of Applied Mathematics, EEMCS, TU Delft（代尔夫特理工大学应用数学研究所）；Department of Mathematics, SAMPS, NTUA（雅典国立技术大学数学系）；Institute of Applied and Computational Mathematics, FORTH（希腊研究与技术创新基金会应用与计算数学研究所）；Chair of Mathematics for Uncertainty Quantification, RWTH Aachen University（亚琛工业大学不确定性量化数学教席）；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）

Christian Bayer, Chiheb Ben Hammouda, Antonis Papapantoleon, Michael Samet, Raul Tempone

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AI总结：

针对傅里叶方法在高维多资产期权定价中的维度灾难问题，提出使用随机化拟蒙特卡洛（RQMC）结合域变换提高可扩展性，通过边界增长条件设计变换保持被积函数正则性，数值实验验证了RQMC在傅里叶域中优于MC和TP方法。

摘要：

高效定价多资产期权是量化金融中的一个重大挑战。傅里叶方法利用傅里叶域中被积函数的正则性，能够准确且快速地估值通常在物理域中缺乏正则性的期权。然而，由于常用数值求积技术的张量积（TP）结构，大多数现有的傅里叶方法在高维设置中面临障碍。为克服这一困难，本文提倡使用随机化拟蒙特卡洛（RQMC）求积来提高傅里叶方法在高维中的可扩展性。RQMC技术受益于被积函数的平滑性，缓解了维度灾难，同时提供了实用的误差估计。然而，RQMC在无界域$\mathbb{R}^d$上的适用性需要域变换到$[0,1]^d$，这可能导致变换后的被积函数在超立方体角点处出现奇异性，从而降低RQMC的性能。为规避这一困难，我们基于变换后被积函数的边界增长条件设计了一种高效的域变换过程。所提出的变换保留了原始被积函数足够的正则性，以实现RQMC方法的快速收敛。为验证我们的分析，我们展示了在傅里叶空间中使用带有适当变换的RQMC评估各种定价模型、收益函数和维度下期权的效率。最后，我们强调了在傅里叶域中应用RQMC相对于MC或TP，以及在物理域中相对于MC对于最多15个资产的期权的计算优势。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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