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

Kathrin Beatrice Glau, Daniel Kressner, Francesco Statti

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.

2020

A New Approach For American Option Pricing: The Dynamic Chebyshev Method

Kathrin Beatrice Glau

We introduce a new method to price American options based on Chebyshev interpolation. In each step of a dynamic programming time-stepping we approximate the value function with Chebyshev polynomials. The key advantage of this approach is that it allows us to shift the model-dependent computations into an offline phase prior to the time-stepping. In the offline part a family of generalized (conditional) moments is computed by an appropriate numerical technique such as a Monte Carlo, PDE, or Fourier transform based method. Thanks to this methodological flexibility the approach applies to a large variety of models. Online, the backward induction is solved on a discrete Chebyshev grid, and no (conditional) expectations need to be computed. For each time step the method delivers a closed form approximation of the price function along with the options' delta and gamma. Moreover, the same family of (conditional) moments yield multiple outputs including the option prices for different strikes, maturities, and different payoff profiles. We provide a theoretical error analysis and find conditions that imply explicit error bounds for a variety of stock price models. Numerical experiments confirm the fast convergence of prices and sensitivities. An empirical investigation of accuracy and runtime also shows an efficiency gain compared with the least-squares Monte Carlo method introduced by Longstaff and Schwartz [Rev. Financ. Stud., 14 (2001), pp. 113-147]. Moreover, we show that the proposed algorithm is flexible enough to price barrier and multivariate barrier options.

SIAM PUBLICATIONS2019