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A Stable and Convergent Finite Difference Method for Fractional Black–Scholes Model of American Put Option Pricing

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  • R. Kalantari

    (University of Tabriz)

  • S. Shahmorad

    (University of Tabriz)

Abstract

We introduce the mathematical modeling of American put option under the fractional Black–Scholes model, which leads to a free boundary problem. Then the free boundary (optimal exercise boundary) that is unknown, is found by the quasi-stationary method that cause American put option problem to be solvable. In continuation we use a finite difference method for derivatives with respect to stock price, Grünwal Letnikov approximation for derivatives with respect to time and reach a fractional finite difference problem. We show that the set up fractional finite difference problem is stable and convergent. We also show that the numerical results satisfy the physical conditions of American put option pricing under the FBS model.

Suggested Citation

  • 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.
    • Handle: RePEc:kap:compec:v:53:y:2019:i:1:d:10.1007_s10614-017-9734-0
    DOI: 10.1007/s10614-017-9734-0
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    References listed on IDEAS

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    1. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    2. Broadie, Mark & Detemple, Jerome, 1996. "American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods," The Review of Financial Studies, Society for Financial Studies, vol. 9(4), pages 1211-1250.
    3. Kaushik Amin & Ajay Khanna, 1994. "Convergence Of American Option Values From Discrete‐ To Continuous‐Time Financial Models1," Mathematical Finance, Wiley Blackwell, vol. 4(4), pages 289-304, October.
    4. Jumarie, Guy, 2008. "Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 271-287, February.
    5. 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.
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    Cited by:

    1. Xubiao He & Pu Gong, 2020. "A Radial Basis Function-Generated Finite Difference Method to Evaluate Real Estate Index Options," Computational Economics, Springer;Society for Computational Economics, vol. 55(3), pages 999-1019, March.
    2. 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.
    3. Hanbyeol Jang & Sangkwon Kim & Junhee Han & Seongjin Lee & Jungyup Ban & Hyunsoo Han & Chaeyoung Lee & Darae Jeong & Junseok Kim, 2020. "Fast Monte Carlo Simulation for Pricing Equity-Linked Securities," Computational Economics, Springer;Society for Computational Economics, vol. 56(4), pages 865-882, December.

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