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Fast numerical simulation of a new time-space fractional option pricing model governing European call option

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  • Zhang, H.
  • Liu, F.
  • Chen, S.
  • Anh, V.
  • Chen, J.

Abstract

When the fluctuation of option price is regarded as a fractal transmission system and the stock price follows a Lévy distribution, a time-space fractional option pricing model (TSFOPM) is obtained. Then we discuss the numerical simulation of the TSFOPM. A discrete implicit numerical scheme with a second-order accuracy in space and a 2−γ order accuracy in time is constructed, where γ is a transmission exponent. The stability and convergence of the obtained numerical scheme are analyzed. Moreover, a fast bi-conjugate gradient stabilized method is proposed to solve the numerical scheme in order to reduce the storage space and computational cost. Then a numerical example with exact solution is presented to demonstrate the accuracy and effectiveness of the proposed numerical method. Finally, the TSFOPM and the above numerical technique are applied to price European call option. The characteristics of the fractional option pricing model are analyzed in comparison with the classical Black–Scholes (B-S) model.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:339:y:2018:i:c:p:186-198
    DOI: 10.1016/j.amc.2018.06.030
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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. Peter Carr & Hélyette Geman & Dilip B. Madan & Marc Yor, 2003. "Stochastic Volatility for Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(3), pages 345-382, July.
    3. Cartea, Álvaro & del-Castillo-Negrete, Diego, 2007. "Fractional diffusion models of option prices in markets with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 374(2), pages 749-763.
    4. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    5. repec:bla:jfinan:v:58:y:2003:i:2:p:753-778 is not listed on IDEAS
    6. 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.
    7. Svetlana I Boyarchenko & Sergei Z Levendorskii, 2002. "Non-Gaussian Merton-Black-Scholes Theory," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4955, December.
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    Cited by:

    1. 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).

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