A Bilinear Pseudo-spectral Method for Solving Two-asset European and American Pricing Options

M. Khasi () and J. Rashidinia ()
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M. Khasi: Iran University of Science and Technology
J. Rashidinia: Iran University of Science and Technology

Computational Economics, 2024, vol. 63, issue 2, No 16, 893-918

Abstract: Abstract This paper presents a bilinear Chebyshev pseudo-spectral method to compute European and American option prices under the two-asset Black–Scholes and Heston models. We expand a function and its derivatives into their Chebyshev series, so the differentiation matrices that act on the Chebyshev coefficients are sparse and better conditioned. First, the equation is spatially discretized using a bilinear pseudo-spectral method created by Chebyshev polynomials and a first–order matrix differential equation (MDE) is obtained. Then, using the Kronecker product, this equation is converted to a system of first–order ODEs. For solving the European options, by using an eigenvalue decomposition method, the arising system will be analytically solved. Therefore, the arising errors are because of spatial discretization and quadrature errors. For solving the American options, the approach is combined with the operator splitting method or penalty method to obtain the temporal discretization. By avoiding some transforms to convert the equation to a constant coefficient equation without mixed derivatives, we will obtain more accurate solutions. Also, transforming the European option into a set of separated ODEs by an eigenvalue decomposition causes to reduce the computational complexity. We also consider the hedge ratios which show the sensitivity of an option to the stock prices. Several numerical examples are included to show the accuracy and efficiency of the proposed approach. The results show that the spectral convergence can be achieved for models with smooth functions.

Keywords: Two-dimensional Black–Scholes equation; Heston’s model; Options pricing; Bilinear pseudo-spectral method; Kronecker product (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10614-023-10364-9

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