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Computational approach based on wavelets for financial mathematical model governed by distributed order fractional differential equation

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  • Kumar, Yashveer
  • Singh, Vineet Kumar

Abstract

In this study, for the first time, the approximate solution of Black–Scholes option pricing distributed order time-fractional partial differential equation by means of Legendre and Chebyshev wavelets is considered. The operational matrices of Legendre and Chebyshev wavelets for integer order derivative and distributed order fractional derivative are derived. Furthermore, the combination of Gauss–Legendre quadrature formula and standard Tau method along with the obtained operational matrices reduces the distributed order time-fractional Black–Scholes model (DOTFBSM) into the system of linear algebraic equations. Convergence analysis, error bounds and numerical stability of the proposed approach are discussed in detail. The presented scheme is applied on three test examples and numerical experiments confirm the theoretical results and illustrate robustness of the presented method. The results produced by current approach are found to be more accurate than some available results.

Suggested Citation

  • Kumar, Yashveer & Singh, Vineet Kumar, 2021. "Computational approach based on wavelets for financial mathematical model governed by distributed order fractional differential equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 531-569.
    • Handle: RePEc:eee:matcom:v:190:y:2021:i:c:p:531-569
    DOI: 10.1016/j.matcom.2021.05.026
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    References listed on IDEAS

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    1. Alikhanov, Anatoly A., 2015. "Numerical methods of solutions of boundary value problems for the multi-term variable-distributed order diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 12-22.
    2. Li, Yuanlu & Liu, Fawang & Turner, Ian W. & Li, Tao, 2018. "Time-fractional diffusion equation for signal smoothing," Applied Mathematics and Computation, Elsevier, vol. 326(C), pages 108-116.
    3. Singh, Somveer & Patel, Vijay Kumar & Singh, Vineet Kumar, 2018. "Application of wavelet collocation method for hyperbolic partial differential equations via matrices," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 407-424.
    4. Singh, Somveer & Patel, Vijay Kumar & Singh, Vineet Kumar, 2016. "Operational matrix approach for the solution of partial integro-differential equation," Applied Mathematics and Computation, Elsevier, vol. 283(C), pages 195-207.
    5. Halil Mete Soner & Guy Barles, 1998. "Option pricing with transaction costs and a nonlinear Black-Scholes equation," Finance and Stochastics, Springer, vol. 2(4), pages 369-397.
    6. Pourbabaee, Marzieh & Saadatmandi, Abbas, 2019. "A novel Legendre operational matrix for distributed order fractional differential equations," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 215-231.
    7. Singh, Somveer & Patel, Vijay Kumar & Singh, Vineet Kumar & Tohidi, Emran, 2017. "Numerical solution of nonlinear weakly singular partial integro-differential equation via operational matrices," Applied Mathematics and Computation, Elsevier, vol. 298(C), pages 310-321.
    8. Asma Ali Elbeleze & Adem Kılıçman & Bachok M. Taib, 2013. "Fractional Variational Iteration Method and Its Application to Fractional Partial Differential Equation," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-10, July.
    9. Chesney, Marc & Scott, Louis, 1989. "Pricing European Currency Options: A Comparison of the Modified Black-Scholes Model and a Random Variance Model," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 24(3), pages 267-284, September.
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    Cited by:

    1. Marasi, H.R. & Derakhshan, M.H. & Ghuraibawi, Amer A. & Kumar, Pushpendra, 2024. "A novel method based on fractional order Gegenbauer wavelet operational matrix for the solutions of the multi-term time-fractional telegraph equation of distributed order," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 217(C), pages 405-424.
    2. Kumar, Yashveer & Yadav, Poonam & Singh, Vineet Kumar, 2023. "Distributed order Gauss-Quadrature scheme for distributed order fractional sub-diffusion model," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    3. Zhang, Meihui & Jia, Jinhong & Zheng, Xiangcheng, 2023. "Numerical approximation and fast implementation to a generalized distributed-order time-fractional option pricing model," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    4. Faïçal Ndaïrou & Delfim F. M. Torres, 2021. "Pontryagin Maximum Principle for Distributed-Order Fractional Systems," Mathematics, MDPI, vol. 9(16), pages 1-12, August.
    5. Kang, Helei & Liu, Renyun & Yao, Yifei & Yu, Fanhua, 2023. "Improved Harris hawks optimization for non-convex function optimization and design optimization problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 619-639.

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