Option pricing under multifactor Black–Scholes model using orthogonal spline wavelets

Dana Černá and Kateřina Fiňková

Mathematics and Computers in Simulation (MATCOM), 2024, vol. 220, issue C, 309-340

Abstract: The paper focuses on pricing European-style options on multiple underlying assets under the Black–Scholes model represented by a nonstationary partial differential equation. The numerical solution of such equations is challenging in dimensions exceeding three, primarily due to the so-called curse of dimensionality. The main contribution of the paper is the design and analysis of the method based on combining the sparse wavelet-Galerkin method and the Crank–Nicolson scheme with Rannacher time-stepping enhanced by Richardson extrapolation, which helps overcome the curse of dimensionality. The next contribution is constructing a new orthogonal cubic spline wavelet basis on the interval and a sparse tensor product wavelet basis on the unit cube, which is suitable for the proposed method. The resulting method brings the following important advantages. The method is higher-order convergent with respect to both temporal and spatial variables, and the number of basis functions is significantly reduced compared to a full grid. Furthermore, many matrices involved in the computation are identity matrices, which results in a considerable simplification of the algorithm. Moreover, we prove that the condition numbers of discretization matrices are uniformly bounded and do not depend on the dimension, even without preconditioning, which leads to a small number of iterations when solving the resulting linear system. Numerical experiments are presented for several types of European-style options.

Keywords: Black–Scholes model; European option; Orthogonal spline wavelet; Sparse grid; Wavelet-Galerkin method; Condition number (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:220:y:2024:i:c:p:309-340

DOI: 10.1016/j.matcom.2024.01.020

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