IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v207y2023icp388-416.html
   My bibliography  Save this article

Convergence of a fitted finite volume method for pricing two dimensional assets with stochastic volatilities

Author

Listed:
  • Nyoumbi, Christelle Dleuna
  • Tambue, Antoine

Abstract

In this article, we provide the rigorous mathematical convergence proof both in space and time of the two dimensional Black Scholes equation with stochastic volatility. The spatial approximation of this three dimensional problem is performed using the finite volume method coupled with a fitted technique to tackle the degeneracy in the Black Scholes operator, while the temporal discretization is performed using implicit Euler method. We provide a mathematical rigorous convergence proof in space and time of the full discretized scheme. Numerical results are presented to validate our theoretical results.

Suggested Citation

  • Nyoumbi, Christelle Dleuna & Tambue, Antoine, 2023. "Convergence of a fitted finite volume method for pricing two dimensional assets with stochastic volatilities," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 388-416.
    • Handle: RePEc:eee:matcom:v:207:y:2023:i:c:p:388-416
    DOI: 10.1016/j.matcom.2023.01.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475423000010
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2023.01.001?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. 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.
    2. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    3. 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.
    4. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    5. 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)

    Citations

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


    Cited by:

    1. Michał Musiał & Lech Lichołai & Dušan Katunský, 2023. "Modern Thermal Energy Storage Systems Dedicated to Autonomous Buildings," Energies, MDPI, vol. 16(11), pages 1-28, May.

    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. 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.
    2. Jimmy E. Hilliard & Adam L. Schwartz & Alan L. Tucker, 1996. "Bivariate Binomial Options Pricing With Generalized Interest Rate Processes," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 19(4), pages 585-602, December.
    3. Wei-Cheng Chen & Wei-Ho Chung, 2019. "Option Pricing via Multi-path Autoregressive Monte Carlo Approach," Papers 1906.06483, arXiv.org.
    4. Peter Christoffersen & Ruslan Goyenko & Kris Jacobs & Mehdi Karoui, 2018. "Illiquidity Premia in the Equity Options Market," The Review of Financial Studies, Society for Financial Studies, vol. 31(3), pages 811-851.
    5. Lin, Zih-Ying & Chang, Chuang-Chang & Wang, Yaw-Huei, 2018. "The impacts of asymmetric information and short sales on the illiquidity risk premium in the stock option market," Journal of Banking & Finance, Elsevier, vol. 94(C), pages 152-165.
    6. 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.
    7. Saeed Marzban & Erick Delage & Jonathan Yumeng Li, 2020. "Equal Risk Pricing and Hedging of Financial Derivatives with Convex Risk Measures," Papers 2002.02876, arXiv.org, revised Sep 2020.
    8. Lin, Zhongguo & Han, Liyan & Li, Wei, 2021. "Option replication with transaction cost under Knightian uncertainty," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 567(C).
    9. Antonio Cosma & Stefano Galluccio & Paola Pederzoli & O. Scaillet, 2012. "Valuing American Options Using Fast Recursive Projections," Swiss Finance Institute Research Paper Series 12-26, Swiss Finance Institute.
    10. Bing-Huei Lin & Mao-Wei Hung & Jr-Yan Wang & Ping-Da Wu, 2013. "A lattice model for option pricing under GARCH-jump processes," Review of Derivatives Research, Springer, vol. 16(3), pages 295-329, October.
    11. Cheng Few Lee & Yibing Chen & John Lee, 2020. "Alternative Methods to Derive Option Pricing Models: Review and Comparison," World Scientific Book Chapters, in: Cheng Few Lee & John C Lee (ed.), HANDBOOK OF FINANCIAL ECONOMETRICS, MATHEMATICS, STATISTICS, AND MACHINE LEARNING, chapter 102, pages 3573-3617, World Scientific Publishing Co. Pte. Ltd..
    12. Donald Aingworth & Sanjiv Das & Rajeev Motwani, 2006. "A simple approach for pricing equity options with Markov switching state variables," Quantitative Finance, Taylor & Francis Journals, vol. 6(2), pages 95-105.
    13. Guidolin, Massimo & Timmermann, Allan, 2003. "Option prices under Bayesian learning: implied volatility dynamics and predictive densities," Journal of Economic Dynamics and Control, Elsevier, vol. 27(5), pages 717-769, March.
    14. 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.
    15. Katarzyna Toporek, 2012. "Simple is better. Empirical comparison of American option valuation methods," Ekonomia journal, Faculty of Economic Sciences, University of Warsaw, vol. 29.
    16. Jonathan Ziveyi, 2011. "The Evaluation of Early Exercise Exotic Options," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 12, July-Dece.
    17. Li, Hongshan & Huang, Zhongyi, 2020. "An iterative splitting method for pricing European options under the Heston model☆," Applied Mathematics and Computation, Elsevier, vol. 387(C).
    18. Hongshan Li & Zhongyi Huang, 2020. "An iterative splitting method for pricing European options under the Heston model," Papers 2003.12934, arXiv.org.
    19. Zoran Stoiljkovic, 2023. "Applying Reinforcement Learning to Option Pricing and Hedging," Papers 2310.04336, arXiv.org.
    20. Chuang-Chang Chang & Jun-Biao Lin & Wei-Che Tsai & Yaw-Huei Wang, 2012. "Using Richardson extrapolation techniques to price American options with alternative stochastic processes," Review of Quantitative Finance and Accounting, Springer, vol. 39(3), pages 383-406, October.

    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:eee:matcom:v:207:y:2023:i:c:p:388-416. 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: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    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.