SIAM Journal on Financial Mathematics, Volume 11, Issue 3, Page 897-927, January 2020.
Treating high dimensionality is one of the main challenges in the development of computational methods for solving problems arising in finance, where tasks such as pricing, calibration, and risk assessment need to be performed accurately and in real-time. Among the growing literature addressing this problem, Gass et al. [Finance Stoch., 22 (2018), pp. 701--731] propose a complexity reduction technique for parametric option pricing based on Chebyshev interpolation. As the number of parameters increases, however, this method is affected by the curse of dimensionality. In this article, we extend this approach to treat high-dimensional problems: Additionally, exploiting low-rank structures allows us to consider parameter spaces of high dimensions. The core of our method is to express the tensorized interpolation in the tensor train format and to develop an efficient way, based on tensor completion, to approximate the interpolation coefficients. We apply the new method to two model problems: American option pricing in the Heston model and European basket option pricing in the multidimensional Black--Scholes model. In these examples, we treat parameter spaces of dimensions up to 25. The numerical results confirm the low-rank structure of these problems and the effectiveness of our method compared to advanced techniques.

中文翻译：

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参数期权定价中Chebyshev插值的低秩张量逼近

SIAM金融数学杂志，第11卷，第3期，第897-927页，2020年1月。
在解决金融中出现的问题的计算方法的开发中，处理高维度是主要挑战之一，在该方法中，需要准确，实时地执行诸如定价，校准和风险评估之类的任务。在有关这一问题的越来越多的文献中，加斯等人。[Finance Stoch。，22（2018），pp。701--731]提出了一种基于Chebyshev插值的参数化期权定价复杂度降低技术。但是，随着参数数量的增加，此方法会受到维度诅咒的影响。在本文中，我们将这种方法扩展为处理高维问题：此外，利用低秩结构使我们可以考虑高维参数空间。我们方法的核心是以张量列格式表示张量插值，并基于张量完成度开发一种有效的方法来近似插值系数。我们将新方法应用于两个模型问题：Heston模型中的美国期权定价和多维Black-Scholes模型中的欧洲篮子期权定价。在这些示例中，我们处理的参数空间最大为25。数值结果证实了这些问题的低秩结构，以及与先进技术相比，该方法的有效性。