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Numerical Approximation to a Variable-Order Time-Fractional Black–Scholes Model with Applications in Option Pricing

Author

Listed:
  • Meihui Zhang

    (Shandong University of Finance and Economics)

  • Xiangcheng Zheng

    (Peking University)

Abstract

We propose and analyze a fully-discrete finite element method to a variable-order time-fractional Black–Scholes model, which provides adequate descriptions for the option pricing, and the variable fractional order may accommodate the effects of the uncertainties or fluctuations in the financial market on the memory of the fractional operator. Due to the impact of the variable order, the temporal discretization coefficients of the fractional operator lose the monotonicity that is critical in error estimates of the time-fractional problems. Furthermore, the Riemann–Liouville fractional derivative is usually adopted in time-fractional Black–Scholes equations, while rigorous numerical analysis for Riemann–Liouville variable-order fractional problems are rarely founded in the literature. Thus the main contributions of this work lie in developing novel techniques to resolve the aforementioned issues and proving the stability and optimal-order convergence estimate of the fully-discrete finite element scheme. Numerical experiments are carried out to substantiate the numerical analysis and to demonstrate the potential applications in option pricing.

Suggested Citation

  • Meihui Zhang & Xiangcheng Zheng, 2023. "Numerical Approximation to a Variable-Order Time-Fractional Black–Scholes Model with Applications in Option Pricing," Computational Economics, Springer;Society for Computational Economics, vol. 62(3), pages 1155-1175, October.
    • Handle: RePEc:kap:compec:v:62:y:2023:i:3:d:10.1007_s10614-022-10295-x
    DOI: 10.1007/s10614-022-10295-x
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    References listed on IDEAS

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    1. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-777, April.
    2. 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.
    3. H. Ghafouri & M. Ranjbar & A. Khani, 2020. "The Use of Partial Fractional Form of A-Stable Padé Schemes for the Solution of Fractional Diffusion Equation with Application in Option Pricing," Computational Economics, Springer;Society for Computational Economics, vol. 56(4), pages 695-709, December.
    4. Rahman Farnoosh & Hamidreza Rezazadeh & Amirhossein Sobhani & M. Hossein Beheshti, 2016. "A Numerical Method for Discrete Single Barrier Option Pricing with Time-Dependent Parameters," Computational Economics, Springer;Society for Computational Economics, vol. 48(1), pages 131-145, June.
    5. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-778, April.
    6. R. Kalantari & S. Shahmorad, 2019. "A Stable and Convergent Finite Difference Method for Fractional Black–Scholes Model of American Put Option Pricing," Computational Economics, Springer;Society for Computational Economics, vol. 53(1), pages 191-205, January.
    7. Wilmott,Paul & Howison,Sam & Dewynne,Jeff, 1995. "The Mathematics of Financial Derivatives," Cambridge Books, Cambridge University Press, number 9780521497893.
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