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Using a Mix of Finite Difference Methods and Fractional Differential Transformations to Solve Modified Black–Scholes Fractional Equations

Author

Listed:
  • Agus Sugandha

    (Doctoral Program of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Jatinangor 40132, Indonesia)

  • Endang Rusyaman

    (Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Jatinangor 40132, Indonesia)

  • Sukono

    (Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Jatinangor 40132, Indonesia)

  • Ema Carnia

    (Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Jatinangor 40132, Indonesia)

Abstract

This paper discusses finding solutions to the modified Fractional Black–Scholes equation. As is well known, the options theory is beneficial in the stock market. Using call-and-pull options, investors can theoretically decide when to sell, hold, or buy shares for maximum profits. However, the process of forming the Black–Scholes model uses a normal distribution, where, in reality, the call option formula obtained is less realistic in the stock market. Therefore, it is necessary to modify the model to make the option values obtained more realistic. In this paper, the method used to determine the solution to the modified Fractional Black–Scholes equation is a combination of the finite difference method and the fractional differential transformation method. The results show that the combined method of finite difference and fractional differential transformation is a very good approximation for the solution of the Fractional Black–Scholes equation.

Suggested Citation

  • Agus Sugandha & Endang Rusyaman & Sukono & Ema Carnia, 2024. "Using a Mix of Finite Difference Methods and Fractional Differential Transformations to Solve Modified Black–Scholes Fractional Equations," Mathematics, MDPI, vol. 12(7), pages 1-15, April.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:7:p:1077-:d:1369268
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    References listed on IDEAS

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    1. Asma Ali Elbeleze & Adem Kılıçman & Bachok M. Taib, 2013. "Homotopy Perturbation Method for Fractional Black-Scholes European Option Pricing Equations Using Sumudu Transform," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-7, May.
    2. Fadugba, Sunday Emmanuel, 2020. "Homotopy analysis method and its applications in the valuation of European call options with time-fractional Black-Scholes equation," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    3. Shengwu Zhou & Wei Li & Yu Wei & Cui Wen, 2012. "A Positivity-Preserving Numerical Scheme for Nonlinear Option Pricing Models," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-20, December.
    4. Abdon Atangana & Necdet Bildik, 2013. "Existence and Numerical Solution of the Volterra Fractional Integral Equations of the Second Kind," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-11, November.
    Full references (including those not matched with items on IDEAS)

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