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

## TL;DR

This paper introduces a low-rank tensor approximation method using tensor train format to efficiently perform high-dimensional Chebyshev interpolation in parametric option pricing, overcoming the curse of dimensionality.

## Contribution

It extends Chebyshev interpolation for high-dimensional problems by exploiting low-rank tensor structures, enabling efficient computation in up to 25-dimensional parameter spaces.

## Key findings

- The method effectively handles high-dimensional parameter spaces in option pricing.
- Numerical results show the low-rank structure and efficiency of the approach.
- Compared to existing techniques, the proposed method demonstrates superior performance.

## 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. [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.