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A spectral approach using fractional Jaiswal functions to solve the mixed time-fractional Black-Scholes European option pricing model with error analysis

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Abstract

In this paper, we propose a collocation scheme for efficiently solving the mixed time-fractional Black-Scholes (MTF-BS) model and obtaining the option price. Our approach involves deriving the mixed fractional Black-Scholes (MF-BS) partial differential equation (PDE) considering the delta hedging strategy and the mixed fractional Geometric Brownian motion (MFGBM) model. To simplify the problem, we transform the MTF-BS PDE into a modified Riemann-Liouville derivative form. Subsequently, a collocation method is employed to numerically solve the transformed equation, where the solution is represented as a series of fractional Jaiswal functions with unknown coefficients. By utilizing operational matrices and collocation points, we convert the problem into a linear system of equations, allowing for the examination of convergence and stability in the Sobolev spaces. Finally, we present four examples to demonstrate the method’s effectiveness and accuracy.

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Authors and Affiliations

  1. Mathematics Department, Kuwait University, Kuwait City, Kuwait

    Fares Alazemi & Abdulaziz Alsenafi

  2. Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

    Alireza Najafi

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  1. Fares Alazemi
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  2. Abdulaziz Alsenafi
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  3. Alireza Najafi
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Contributions

Dr. Alazemi and Dr. Alsenafi designed and wrote the main idea of the paper and solved the pde and wrote the paper text and Dr Najafi wrote the program section.

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Correspondence to Fares Alazemi.

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Alazemi, F., Alsenafi, A. & Najafi, A. A spectral approach using fractional Jaiswal functions to solve the mixed time-fractional Black-Scholes European option pricing model with error analysis. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01797-w

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