IDEAS home Printed from https://ideas.repec.org/a/kap/compec/v53y2019i1d10.1007_s10614-017-9734-0.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10614-017-9734-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10614-017-9734-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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. 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.
    3. 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.
    4. 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.
    5. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    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. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Duan, Jin-Chuan & Simonato, Jean-Guy, 2001. "American option pricing under GARCH by a Markov chain approximation," Journal of Economic Dynamics and Control, Elsevier, vol. 25(11), pages 1689-1718, November.
    2. Minqiang Li, 2010. "A quasi-analytical interpolation method for pricing American options under general multi-dimensional diffusion processes," Review of Derivatives Research, Springer, vol. 13(2), pages 177-217, July.
    3. Weihan Li & Jin E. Zhang & Xinfeng Ruan & Pakorn Aschakulporn, 2024. "An empirical study on the early exercise premium of American options: Evidence from OEX and XEO options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 44(7), pages 1117-1153, July.
    4. Wang, Jun & Liang, Jin-Rong & Lv, Long-Jin & Qiu, Wei-Yuan & Ren, Fu-Yao, 2012. "Continuous time Black–Scholes equation with transaction costs in subdiffusive fractional Brownian motion regime," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(3), pages 750-759.
    5. Henry Lam & Zhenming Liu, 2014. "From Black-Scholes to Online Learning: Dynamic Hedging under Adversarial Environments," Papers 1406.6084, arXiv.org.
    6. Björn Lutz, 2010. "Pricing of Derivatives on Mean-Reverting Assets," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-642-02909-7, October.
    7. In oon Kim & Bong-Gyu Jang & Kyeong Tae Kim, 2013. "A simple iterative method for the valuation of American options," Quantitative Finance, Taylor & Francis Journals, vol. 13(6), pages 885-895, May.
    8. Suresh M. Sundaresan, 2000. "Continuous‐Time Methods in Finance: A Review and an Assessment," Journal of Finance, American Finance Association, vol. 55(4), pages 1569-1622, August.
    9. Oleksandr Zhylyevskyy, 2010. "A fast Fourier transform technique for pricing American options under stochastic volatility," Review of Derivatives Research, Springer, vol. 13(1), pages 1-24, April.
    10. Qianru Shang & Brian Byrne, 2021. "American option pricing: Optimal Lattice models and multidimensional efficiency tests," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 41(4), pages 514-535, April.
    11. Jia‐Hau Guo & Lung‐Fu Chang, 2020. "Repeated Richardson extrapolation and static hedging of barrier options under the CEV model," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 40(6), pages 974-988, June.
    12. Lv, Longjin & Xiao, Jianbin & Fan, Liangzhong & Ren, Fuyao, 2016. "Correlated continuous time random walk and option pricing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 447(C), pages 100-107.
    13. Andrew Ziogas, 2005. "Pricing American Options Using Fourier Analysis," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2-2005, January-A.
    14. David S. Bates, 1995. "Testing Option Pricing Models," NBER Working Papers 5129, National Bureau of Economic Research, Inc.
    15. repec:dau:papers:123456789/5374 is not listed on IDEAS
    16. Barone-Adesi, Giovanni, 2005. "The saga of the American put," Journal of Banking & Finance, Elsevier, vol. 29(11), pages 2909-2918, November.
    17. Haq, Sirajul & Hussain, Manzoor, 2018. "Selection of shape parameter in radial basis functions for solution of time-fractional Black–Scholes models," Applied Mathematics and Computation, Elsevier, vol. 335(C), pages 248-263.
    18. Chung, San-Lin & Hung, Mao-Wei & Wang, Jr-Yan, 2010. "Tight bounds on American option prices," Journal of Banking & Finance, Elsevier, vol. 34(1), pages 77-89, January.
    19. Manuel Moreno & Javier Navas, 2003. "On the Robustness of Least-Squares Monte Carlo (LSM) for Pricing American Derivatives," Review of Derivatives Research, Springer, vol. 6(2), pages 107-128, May.
    20. Mark Broadie & Jérôme Detemple, 1996. "Recent Advances in Numerical Methods for Pricing Derivative Securities," CIRANO Working Papers 96s-17, CIRANO.
    21. Lina Song, 2018. "A Semianalytical Solution of the Fractional Derivative Model and Its Application in Financial Market," Complexity, Hindawi, vol. 2018, pages 1-10, April.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:kap:compec:v:53:y:2019:i:1:d:10.1007_s10614-017-9734-0. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.