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Investigation of Higher Order Localized Approximations for a Fractional Pricing Model in Finance

Author

Listed:
  • Malik Zaka Ullah

    (Mathematical Modelling and Applied Computation (MMAC) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia)

  • Abdullah Khamis Alzahrani

    (Mathematical Modelling and Applied Computation (MMAC) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia)

  • Hashim Mohammed Alshehri

    (Mathematical Modelling and Applied Computation (MMAC) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia)

  • Stanford Shateyi

    (Department of Mathematics and Applied Mathematics, School of Mathematical and Natural Sciences, University of Venda, P. Bag X5050, Thohoyandou 0950, South Africa)

Abstract

In this work, by considering spatial uniform meshes and stencils having five adjacent discretization nodes, we furnish a numerical scheme to solve the time-fractional Black–Scholes (partial differential equation) PDE to price financial options under the generalized multiquadric radial basis function (RBF). The time-fractional derivative is estimated by an L1-scheme but the spatial variable is discretized using fourth-order RBF-FD methodology. As a matter of fact, the PDE problem is transformed in the form of a linear set of algebraic equations. To support analytical discussions, numerical tests are furnished and reveal the efficacy of the presented solver.

Suggested Citation

  • Malik Zaka Ullah & Abdullah Khamis Alzahrani & Hashim Mohammed Alshehri & Stanford Shateyi, 2023. "Investigation of Higher Order Localized Approximations for a Fractional Pricing Model in Finance," Mathematics, MDPI, vol. 11(12), pages 1-12, June.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:12:p:2641-:d:1167884
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    References listed on IDEAS

    as
    1. Huamin Zhang & Feng Ding, 2013. "On the Kronecker Products and Their Applications," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-8, June.
    2. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    3. Tao Liu & Malik Zaka Ullah & Stanford Shateyi & Chao Liu & Yanxiong Yang, 2023. "An Efficient Localized RBF-FD Method to Simulate the Heston–Hull–White PDE in Finance," Mathematics, MDPI, vol. 11(4), pages 1-15, February.
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