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Black–Scholes option pricing equations described by the Caputo generalized fractional derivative

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  • Fall, Aliou Niang
  • Ndiaye, Seydou Nourou
  • Sene, Ndolane

Abstract

Fractional Black–Scholes equation is a constructive financial equation. The model is used to determine the value of the option without a transaction cost. The analytical solutions of the fractional Black–Scholes equations have been addressed. The Caputo generalized fractional derivative has been used. The homotopy perturbation method has been developed for obtaining the analytical solutions of the fractional Black–Scholes equation (BSE) and the generalized fractionalBSE. The analytical solutions of the fractionalBSE and the generalized fractionalBSE have been represented graphically. The effect of the order ρ of the generalized fractional derivative in the diffusion processes has been analyzed.

Suggested Citation

  • Fall, Aliou Niang & Ndiaye, Seydou Nourou & Sene, Ndolane, 2019. "Black–Scholes option pricing equations described by the Caputo generalized fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 108-118.
    • Handle: RePEc:eee:chsofr:v:125:y:2019:i:c:p:108-118
    DOI: 10.1016/j.chaos.2019.05.024
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    References listed on IDEAS

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    1. Abdeljawad, Thabet & Al-Mdallal, Qasem M. & Jarad, Fahd, 2019. "Fractional logistic models in the frame of fractional operators generated by conformable derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 119(C), pages 94-101.
    2. Owolabi, Kolade M. & Atangana, Abdon, 2018. "Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations," Chaos, Solitons & Fractals, Elsevier, vol. 111(C), pages 119-127.
    3. Thabet Abdeljawad & Qasem M. Al-Mdallal & Mohamed A. Hajji, 2017. "Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications," Discrete Dynamics in Nature and Society, Hindawi, vol. 2017, pages 1-8, June.
    4. Qasem Al-Mdallal & Kashif Ali Abro & Ilyas Khan, 2018. "Analytical Solutions of Fractional Walter’s B Fluid with Applications," Complexity, Hindawi, vol. 2018, pages 1-10, February.
    5. Al-Mdallal, Qasem M. & Abu Omer, Ahmed S., 2018. "Fractional-order Legendre-collocation method for solving fractional initial value problems," Applied Mathematics and Computation, Elsevier, vol. 321(C), pages 74-84.
    6. Sene, Ndolane, 2018. "Stokes’ first problem for heated flat plate with Atangana–Baleanu fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 68-75.
    7. 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.
    8. Al-Mdallal, Qasem M., 2018. "On fractional-Legendre spectral Galerkin method for fractional Sturm–Liouville problems," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 261-267.
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    Cited by:

    1. Jugal Mohapatra & Sudarshan Santra & Higinio Ramos, 2024. "Analytical and Numerical Solution for the Time Fractional Black-Scholes Model Under Jump-Diffusion," Computational Economics, Springer;Society for Computational Economics, vol. 63(5), pages 1853-1878, May.
    2. Anshima Singh & Sunil Kumar, 2024. "An Efficient Numerical Method Based on Exponential B-splines for a Time-Fractional Black–Scholes Equation Governing European Options," Computational Economics, Springer;Society for Computational Economics, vol. 64(4), pages 1965-2002, October.
    3. Sivaporn Ampun & Panumart Sawangtong, 2021. "The Approximate Analytic Solution of the Time-Fractional Black-Scholes Equation with a European Option Based on the Katugampola Fractional Derivative," Mathematics, MDPI, vol. 9(3), pages 1-15, January.
    4. Sene, Ndolane, 2020. "Second-grade fluid model with Caputo–Liouville generalized fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    5. Sene, Ndolane & Abdelmalek, Karima, 2019. "Analysis of the fractional diffusion equations described by Atangana-Baleanu-Caputo fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 158-164.
    6. Akgül, Esra Karatas & Akgül, Ali & Yavuz, Mehmet, 2021. "New Illustrative Applications of Integral Transforms to Financial Models with Different Fractional Derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    7. Zhang, Lihong & Wang, Jun & Wang, Bin, 2020. "Energy market prediction with novel long short-term memory network: Case study of energy futures index volatility," Energy, Elsevier, vol. 211(C).
    8. Agus Sugandha & Endang Rusyaman & Sukono & Ema Carnia, 2023. "A New Solution to the Fractional Black–Scholes Equation Using the Daftardar-Gejji Method," Mathematics, MDPI, vol. 11(24), pages 1-25, December.
    9. Abdi, N. & Aminikhah, H. & Sheikhani, A.H. Refahi, 2022. "High-order compact finite difference schemes for the time-fractional Black-Scholes model governing European options," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).

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