The Convergence Analysis of the Numerical Calculation to Price the Time-Fractional Black–Scholes Model

H. Mesgarani, M. Bakhshandeh, Y. Esmaeelzade Aghdam and J. F. Gómez-Aguilar ()
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H. Mesgarani: Shahid Rajaee Teacher Training University
M. Bakhshandeh: Shahid Rajaee Teacher Training University
Y. Esmaeelzade Aghdam: Shahid Rajaee Teacher Training University
J. F. Gómez-Aguilar: CONACyT-Tecnológico Nacional de México/CENIDET

Computational Economics, 2023, vol. 62, issue 4, No 17, 1845-1856

Abstract: Abstract In this paper, the approximate solution u(x, t) of the temporal fractional Black–Scholes model involving the time derivative in the Caputo sense with initial and boundary conditions has been studied. This equation has the main part in defining the European option in the financial activities. Time discretization is performed by linear interpolation with a temporally $$\tau ^{2-\alpha }$$ τ 2 - α order accuracy, and the Chebyshev collocation is based on the orthogonal polynomials used for spatial discretization. Additionally, the convergence and stability analysis of the specified methods are considered. Finally, the numerical solutions of some examples were obtained and compared with their analytical solutions that demonstrate the high accuracy and feasibility of the proposed approach.

Keywords: Time-fractional Black–Scholes equation; Chebyshev polynomials of the third kind; Linear interpolation; Collocation method; Convergence analysis; 91G80; 34K37; 97N50 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10614-022-10322-x

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