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A fast implicit difference scheme with nonuniform discretized grids for the time-fractional Black–Scholes model

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  • Xin, Qi
  • Gu, Xian-Ming
  • Liu, Li-Bin

Abstract

The solution of the time-fractional Black–Scholes (TFBS) equation often exhibits a weak singularity at initial time and possible non-physical oscillations in the computed solution due to the degeneracy of the BS differential operator. To address this issue, we combine a modified graded mesh and a piecewise uniform mesh for temporal and spatial discretizations, respectively. Then we use the fast approximation (rather than the direct approximation) of the L1 scheme for the Caputo derivative to establish an implicit difference method for the TFBS model. Our analysis shows the stability and convergence of the proposed scheme, as well as the α-nonrobust error bounds. Finally, numerical results are presented to show the effectiveness of the proposed method.

Suggested Citation

  • Xin, Qi & Gu, Xian-Ming & Liu, Li-Bin, 2025. "A fast implicit difference scheme with nonuniform discretized grids for the time-fractional Black–Scholes model," Applied Mathematics and Computation, Elsevier, vol. 500(C).
    • Handle: RePEc:eee:apmaco:v:500:y:2025:i:c:s0096300325001687
    DOI: 10.1016/j.amc.2025.129441
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    References listed on IDEAS

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    1. Cartea, Álvaro & del-Castillo-Negrete, Diego, 2007. "Fractional diffusion models of option prices in markets with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 374(2), pages 749-763.
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    3. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    4. Liu, Li-Bin & Xu, Lei & Zhang, Yong, 2023. "Error analysis of a finite difference scheme on a modified graded mesh for a time-fractional diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 209(C), pages 87-101.
    5. Jumarie, Guy, 2008. "Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 271-287, February.
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