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A New Solution to the Fractional Black–Scholes Equation Using the Daftardar-Gejji Method

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  • 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

The main objective of this study is to determine the existence and uniqueness of solutions to the fractional Black–Scholes equation. The solution to the fractional Black–Scholes equation is expressed as an infinite series of converging Mittag-Leffler functions. The method used to discover the new solution to the fractional Black–Scholes equation was the Daftardar-Geiji method. Additionally, the Picard–Lindelöf theorem was utilized for the existence and uniqueness of its solution. The fractional derivative employed was the Caputo operator. The search for a solution to the fractional Black–Scholes equation was essential due to the Black–Scholes equation’s assumptions, which imposed relatively tight constraints. These included assumptions of a perfect market, a constant value of the risk-free interest rate and volatility, the absence of dividends, and a normal log distribution of stock price dynamics. However, these assumptions did not accurately reflect market realities. Therefore, it was necessary to formulate a model, particularly regarding the fractional Black–Scholes equation, which represented more market realities. The results obtained in this paper guaranteed the existence and uniqueness of solutions to the fractional Black–Scholes equation, approximate solutions to the fractional Black–Scholes equation, and very small solution errors when compared to the Black–Scholes equation. The novelty of this article is the use of the Daftardar-Geiji method to solve the fractional Black–Scholes equation, guaranteeing the existence and uniqueness of the solution to the fractional Black–Scholes equation, which has not been discussed by other researchers. So, based on this novelty, the Daftardar-Geiji method is a simple and effective method for solving the fractional Black–Scholes equation. This article presents some examples to demonstrate the application of the Daftardar-Gejji method in solving specific problems.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:24:p:4887-:d:1294980
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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. 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.
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