IDEAS home Printed from https://ideas.repec.org/a/wly/jnlamp/v2023y2023i1n1812518.html

Lattice Boltzmann Method for the Generalized Black‐Scholes Equation

Author

Listed:
  • Fangfang Wu
  • Duoduo Xu
  • Ming Tian
  • Yingying Wang

Abstract

In this paper, an efficient lattice Boltzmann model for the generalized Black‐Scholes equation governing option pricing is proposed. The Black‐Scholes equation is firstly equivalently transformed into an initial value problem for a partial differential equation with a source term using the variable substitution and the derivative rules, respectively. Then, applying the multiscale Chapman‐Enskog expansion, the amending function is expanded to second order to recover the convective and source terms of the macroscopic equation. The D1Q3 lattice Boltzmann model with spatial second‐order accuracy is constructed, and the accuracy of the established D1Q5 model is greater than second order. The numerical simulation results demonstrate the effectiveness and numerical accuracy of the proposed models and indicate that our proposed models are suitable for solving the Black‐Scholes equation. The proposed lattice Boltzmann model can also be applied to a class of partial differential equations with variable coefficients and source terms.

Suggested Citation

  • Fangfang Wu & Duoduo Xu & Ming Tian & Yingying Wang, 2023. "Lattice Boltzmann Method for the Generalized Black‐Scholes Equation," Advances in Mathematical Physics, John Wiley & Sons, vol. 2023(1).
    • Handle: RePEc:wly:jnlamp:v:2023:y:2023:i:1:n:1812518
    DOI: 10.1155/2023/1812518
    as

    Download full text from publisher

    File URL: https://doi.org/10.1155/2023/1812518
    Download Restriction: no
    File URL: https://libkey.io/10.1155/2023/1812518?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
    ---><---

    References listed on IDEAS

    as
    1. Hull, John & White, Alan, 1988. "The Use of the Control Variate Technique in Option Pricing," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 23(3), pages 237-251, September.
    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. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
    Full references (including those not matched with items on IDEAS)

    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. Ammann, Manuel & Kind, Axel & Wilde, Christian, 2003. "Are convertible bonds underpriced? An analysis of the French market," Journal of Banking & Finance, Elsevier, vol. 27(4), pages 635-653, April.
    2. Engstrom, Malin & Norden, Lars, 2000. "The early exercise premium in American put option prices," Journal of Multinational Financial Management, Elsevier, vol. 10(3-4), pages 461-479, December.
    3. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    4. San-Lin Chung & Mark Shackleton, 2005. "On the use and improvement of Hull and White's control variate technique," Applied Financial Economics, Taylor & Francis Journals, vol. 15(16), pages 1171-1179.
    5. Mark Joshi & Mike Staunton, 2012. "On the analytical/numerical pricing of American put options against binomial tree prices," Quantitative Finance, Taylor & Francis Journals, vol. 12(1), pages 17-20, December.
    6. Kocagil, Ahmet E. & Eduardo, Benjamin E., 1996. "Impacts of new environmental standards on mining industry: The case of Peru," Resource and Energy Economics, Elsevier, vol. 18(3), pages 291-310, October.
    7. Chuang-Chang Chang & Jun-Biao Lin, 2010. "The valuation of multivariate contingent claims under transformed trinomial approaches," Review of Quantitative Finance and Accounting, Springer, vol. 34(1), pages 23-36, January.
    8. Hull, John & White, Alan, 1995. "The impact of default risk on the prices of options and other derivative securities," Journal of Banking & Finance, Elsevier, vol. 19(2), pages 299-322, May.
    9. Mark Broadie & Jérôme Detemple, 1996. "Recent Advances in Numerical Methods for Pricing Derivative Securities," CIRANO Working Papers 96s-17, CIRANO.
    10. Chung-Li Tseng & Daniel Wei-Chung Miao & San-Lin Chung & Pai-Ta Shih, 2021. "How Much Do Negative Probabilities Matter in Option Pricing?: A Case of a Lattice-Based Approach for Stochastic Volatility Models," JRFM, MDPI, vol. 14(6), pages 1-32, May.
    11. Martzoukos, Spiros H. & Trigeorgis, Lenos, 2002. "Real (investment) options with multiple sources of rare events," European Journal of Operational Research, Elsevier, vol. 136(3), pages 696-706, February.
    12. Leisen, Dietmar P. J., 1998. "Pricing the American put option: A detailed convergence analysis for binomial models," Journal of Economic Dynamics and Control, Elsevier, vol. 22(8-9), pages 1419-1444, August.
    13. S. Wang & X. Q. Yang & K. L. Teo, 2006. "Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation," Journal of Optimization Theory and Applications, Springer, vol. 129(2), pages 227-254, May.
    14. Andrea Gamba & Lenos Trigeorgis, 2007. "An Improved Binomial Lattice Method for Multi-Dimensional Options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(5), pages 453-475.
    15. Ting Chen & Mark Joshi, 2012. "Truncation and acceleration of the Tian tree for the pricing of American put options," Quantitative Finance, Taylor & Francis Journals, vol. 12(11), pages 1695-1708, November.
    16. Wei-Cheng Chen & Wei-Ho Chung, 2019. "Option Pricing via Multi-path Autoregressive Monte Carlo Approach," Papers 1906.06483, arXiv.org.
    17. Chiu, Chun-Yuan & Dai, Tian-Shyr & Lyuu, Yuh-Dauh & Liu, Liang-Chih & Chen, Yu-Ting, 2022. "Option pricing with the control variate technique beyond Monte Carlo simulation," The North American Journal of Economics and Finance, Elsevier, vol. 62(C).
    18. Zhongdi Cen & Anbo Le & Aimin Xu, 2012. "A Second-Order Difference Scheme for the Penalized Black–Scholes Equation Governing American Put Option Pricing," Computational Economics, Springer;Society for Computational Economics, vol. 40(1), pages 49-62, June.
    19. Chandra Sekhara Rao, S. & Manisha,, 2018. "Numerical solution of generalized Black–Scholes model," Applied Mathematics and Computation, Elsevier, vol. 321(C), pages 401-421.
    20. Buckley, Adrian & Eijgenhuijsen, Hans, 1997. "A conceptual framework for evaluating foreign investments," Serie Research Memoranda 0008, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.

    More about this item

    Statistics

    Access and download statistics

    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:wly:jnlamp:v:2023:y:2023:i:1:n:1812518. 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: Wiley Content Delivery (email available below). General contact details of provider: https://onlinelibrary.wiley.com/journal/3197 .

    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.