Implicit cubic B-spline scheme for the fractional Black-Scholes model with Caputo-Hadamard derivative

Document Type : Research Paper

Author

Department of Mathematics, Payame Noor University (PNU), P.O.Box 19395-4697, Tehran, Iran

Abstract

In this study, we introduce a novel numerical scheme for solving the Black–Scholes equation endowed with a Caputo-Hadamard fractional time derivative. The temporal derivative is discretized via a finite-difference approach, ensuring both stability and high accuracy in capturing the model’s dynamics. For spatial discretization, we employ a cubic B-spline quasi-interpolation technique, which delivers a robust and efficient framework. A rigorous error analysis is carried out, establishing theoretical bounds that confirm the method’s reliability. Finally, we validate the proposed scheme through a series of numerical experiments, demonstrating its effectiveness in practical financial modeling and its ability to address the challenges posed by fractional differential equations.

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References

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Journal of Mahani Mathematical Research
Volume 15, Issue 1 - Serial Number 33
January 2026
Pages 485-508

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History

  • Receive Date: 07 April 2025
  • Revise Date: 16 September 2025
  • Accept Date: 19 October 2025

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