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High-order compact finite difference schemes for the time-fractional Black-Scholes model governing European options

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  • Abdi, N.
  • Aminikhah, H.
  • Sheikhani, A.H. Refahi

Abstract

Based on the price fluctuations of the underlying fractal transmission system, the α-order time-fractional Black-Scholes model is obtained. In this paper, we introduce two compact finite difference schemes for the solution of the time-fractional Black-Scholes equation governing European option pricing. In proposed schemes, in order to gain sixth-order and eighth-order accuracy in space, we first use exponential transformation to eliminate the convection term of the Black-Scholes equation. Then, the time-fractional derivative discretizes by a 3−αth order numerical formula (called the L1–2 formula here) which is constructed by a linear interpolating polynomial on the first subinterval and the quadratic interpolating polynomials on the other subintervals. We investigate stability and convergence of proposed schemes by Fourier method. Finally, some numerical examples perform to demonstrate the theoretical order of accuracy and illustrate the effectiveness of proposed methods. We also discuss the influence of different parameters on the option price in the time-fractional Black-Scholes model.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:chsofr:v:162:y:2022:i:c:s0960077922006336
    DOI: 10.1016/j.chaos.2022.112423
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    1. Bekiros, Stelios & Cardani, Roberta & Paccagnini, Alessia & Villa, Stefania, 2016. "Dealing with financial instability under a DSGE modeling approach with banking intermediation: A predictability analysis versus TVP-VARs," Journal of Financial Stability, Elsevier, vol. 26(C), pages 216-227.
    2. Ndolane Sene & Babacar Sène & Seydou Nourou Ndiaye & Awa Traoré, 2020. "Novel Approaches for Getting the Solution of the Fractional Black–Scholes Equation Described by Mittag-Leffler Fractional Derivative," Discrete Dynamics in Nature and Society, Hindawi, vol. 2020, pages 1-11, July.
    3. Zhang, H. & Liu, F. & Chen, S. & Anh, V. & Chen, J., 2018. "Fast numerical simulation of a new time-space fractional option pricing model governing European call option," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 186-198.
    4. Ahmad Golbabai & Omid Nikan, 2020. "A Computational Method Based on the Moving Least-Squares Approach for Pricing Double Barrier Options in a Time-Fractional Black–Scholes Model," Computational Economics, Springer;Society for Computational Economics, vol. 55(1), pages 119-141, January.
    5. Nuugulu, Samuel M & Gideon, Frednard & Patidar, Kailash C, 2021. "A robust numerical scheme for a time-fractional Black-Scholes partial differential equation describing stock exchange dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    6. 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.
    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. Yousefpour, Amin & Jahanshahi, Hadi & Munoz-Pacheco, Jesus M. & Bekiros, Stelios & Wei, Zhouchao, 2020. "A fractional-order hyper-chaotic economic system with transient chaos," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
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