arXiv 1902.043672019-02-12q-fin.CP

参数期权定价中Chebyshev插值的低秩张量逼近

Low-rank tensor approximation for Chebyshev interpolation in parametric option pricing

Kathrin Glau, Daniel Kressner, Francesco Statti

摘要：

处理高维是解决金融中出现的问题的计算方法开发的主要挑战之一，其中需要准确和实时地执行定价，校准和风险评估等任务。在解决这个问题的越来越多的文献中，Gass等人。[14]提出了一种基于Chebyshev插值的参数期权定价的复杂性降低技术。然而，随着参数数量的增加，这种方法受到维数灾难的影响。在这篇文章中，我们将这种方法扩展到处理高维问题：此外，利用低秩结构使我们能够考虑高维参数空间。我们的方法的核心是张量插值表示张量训练（TT）格式和开发一种有效的方法，基于张量完成，近似插值系数。我们将新方法应用于两个模型问题：Heston模型中的美式期权定价和多维布莱克-斯科尔斯模型中的欧式一篮子期权定价。在这些例子中，我们处理维数高达25的参数空间。数值结果证实了这些问题的低秩结构和我们的方法相比，先进的技术的有效性。

英文摘要：

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. [14] 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 tensor train (TT) 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 multi-dimensional 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.

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摘要：

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