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Publication# Low-Rank Tensor Approximation for Chebyshev Interpolation in Parametric Option Pricing

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.

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Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it - for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it - for example, both a latitude and longitude are required to locate a point on the surface of a sphere.

Finance

Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, which is the study of production, distribution, and consumption of money, assets, goods and services (the discipline of financial economics bridges the two). Finance activities take place in financial systems at various scopes, thus the field can be roughly divided into personal, corporate, and public finance.

Interpolation

In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable.

Options are some of the most traded financial instruments and computing their price is a central task in financial mathematics and in practice. Consequently, the development of numerical algorithms fo