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Computational modeling of the time-fractional Black–Scholes equations and its numerical solution using a fourth-order improvised cubic B-spline collocation method

Author

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  • Guangyu Fan

    (School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai 200090, P. R. China)

  • Beibei Wu

    (School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai 200090, P. R. China)

Abstract

In this paper, an improvised cubic B-spline method (ICSCM) for the time-fractional Black–Scholes model (TFBSM) governing European option pricing is presented. The numerical method utilizes the ICSCM for spatial discretization and employs a finite difference method for temporal discretization. Stability analysis of the scheme is performed using the von Neumann scheme. Also, the method is proved to be convergent of order 2−β in temporal directions, where β is order of the fractional derivative. Three test examples verify the effectiveness and accuracy of the proposed method. Numerical and graphical results show that the results obtained by this method to solve the equation are in good agreement with the analytical solution, and achieve smaller error norms and perform better in numerical accuracy.

Suggested Citation

  • Guangyu Fan & Beibei Wu, 2025. "Computational modeling of the time-fractional Black–Scholes equations and its numerical solution using a fourth-order improvised cubic B-spline collocation method," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 36(11), pages 1-28, November.
    • Handle: RePEc:wsi:ijmpcx:v:36:y:2025:i:11:n:s0129183125500263
    DOI: 10.1142/S0129183125500263
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    Keywords

    Time-fractional Black–Scholes equation; cubic B-splines; improvised collocation method; von-Neumann; L2 and L∞ error norms;
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    JEL classification:

    • L2 - Industrial Organization - - Firm Objectives, Organization, and Behavior

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