Abstract

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.

Keywords

  1. Chebyshev interpolation
  2. parametric option pricing
  3. high-dimensional problem
  4. tensor train format
  5. low-rank tensor approximation
  6. tensor completion

MSC codes

  1. 91G60
  2. 41A10
  3. 65DO5
  4. 15A69
  5. 65K05

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Information & Authors

Information

Published In

cover image SIAM Journal on Financial Mathematics
SIAM Journal on Financial Mathematics
Pages: 897 - 927
ISSN (online): 1945-497X

History

Submitted: 11 February 2019
Accepted: 26 May 2020
Published online: 15 September 2020

Keywords

  1. Chebyshev interpolation
  2. parametric option pricing
  3. high-dimensional problem
  4. tensor train format
  5. low-rank tensor approximation
  6. tensor completion

MSC codes

  1. 91G60
  2. 41A10
  3. 65DO5
  4. 15A69
  5. 65K05

Authors

Affiliations

Funding Information

Horizon 2020 Framework Programme https://doi.org/10.13039/100010661 : 665667
Seventh Framework Programme https://doi.org/10.13039/100011102 : 307465-POLYTE

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