IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v130y2020i5p2675-2692.html
   My bibliography  Save this article

Optimal strong convergence rate of a backward Euler type scheme for the Cox–Ingersoll–Ross model driven by fractional Brownian motion

Author

Listed:
  • Hong, Jialin
  • Huang, Chuying
  • Kamrani, Minoo
  • Wang, Xu

Abstract

In this paper, we investigate the optimal strong convergence rate of numerical approximations for the Cox–Ingersoll–Ross model driven by fractional Brownian motion with Hurst parameter H∈(1∕2,1). To deal with the difficulties caused by the unbounded diffusion coefficient, we study an auxiliary equation based on Lamperti transformation. By means of Malliavin calculus, we prove that the backward Euler scheme applied to this auxiliary equation ensures the positivity of the numerical solution, and is of strong order one. Furthermore, a numerical approximation for the original model is obtained and converges with the same order.

Suggested Citation

  • Hong, Jialin & Huang, Chuying & Kamrani, Minoo & Wang, Xu, 2020. "Optimal strong convergence rate of a backward Euler type scheme for the Cox–Ingersoll–Ross model driven by fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 2675-2692.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:5:p:2675-2692
    DOI: 10.1016/j.spa.2019.07.014
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414918305283
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2019.07.014?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. Hu, Yaozhong & Nualart, David & Song, Xiaoming, 2008. "A singular stochastic differential equation driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2075-2085, October.
    2. Alfonsi, Aurélien, 2013. "Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process," Statistics & Probability Letters, Elsevier, vol. 83(2), pages 602-607.
    3. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    4. Peter Kloeden & Andreas Neuenkirch & Raffaella Pavani, 2011. "Multilevel Monte Carlo for stochastic differential equations with additive fractional noise," Annals of Operations Research, Springer, vol. 189(1), pages 255-276, September.
    5. 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.
    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. Huang, Chuying & Wang, Xu, 2023. "Strong convergence rate of the Euler scheme for SDEs driven by additive rough fractional noises," Statistics & Probability Letters, Elsevier, vol. 194(C).
    2. Gao, Xiangyu & Wang, Jianqiao & Wang, Yanxia & Yang, Hongfu, 2022. "The truncated Euler–Maruyama method for CIR model driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 189(C).

    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. Gao, Xiangyu & Wang, Jianqiao & Wang, Yanxia & Yang, Hongfu, 2022. "The truncated Euler–Maruyama method for CIR model driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 189(C).
    2. Marc Mukendi Mpanda & Safari Mukeru & Mmboniseni Mulaudzi, 2020. "Generalisation of Fractional-Cox-Ingersoll-Ross Process," Papers 2008.07798, arXiv.org, revised Jul 2022.
    3. Andrei Cozma & Christoph Reisinger, 2017. "Strong convergence rates for Euler approximations to a class of stochastic path-dependent volatility models," Papers 1706.07375, arXiv.org, revised Oct 2018.
    4. Andrei Cozma & Christoph Reisinger, 2017. "Strong order 1/2 convergence of full truncation Euler approximations to the Cox-Ingersoll-Ross process," Papers 1704.07321, arXiv.org, revised Oct 2018.
    5. Andrei Cozma & Christoph Reisinger, 2015. "Exponential integrability properties of Euler discretization schemes for the Cox-Ingersoll-Ross process," Papers 1601.00919, arXiv.org.
    6. Alessandro Bondi & Sergio Pulido & Simone Scotti, 2022. "The rough Hawkes Heston stochastic volatility model," Working Papers hal-03827332, HAL.
    7. Thomas Kokholm & Martin Stisen, 2015. "Joint pricing of VIX and SPX options with stochastic volatility and jump models," Journal of Risk Finance, Emerald Group Publishing Limited, vol. 16(1), pages 27-48, January.
    8. Darren Shannon & Grigorios Fountas, 2021. "Extending the Heston Model to Forecast Motor Vehicle Collision Rates," Papers 2104.11461, arXiv.org, revised May 2021.
    9. Chang, Chuang-Chang & Hsieh, Pei-Fang & Tang, Chih-Wei & Wang, Yaw-Huei, 2013. "The intraday behavior of information misreaction across various categories of investors in the Taiwan options market," Journal of Financial Markets, Elsevier, vol. 16(2), pages 362-385.
    10. Choi, Jaehyuk & Kwok, Yue Kuen, 2024. "Simulation schemes for the Heston model with Poisson conditioning," European Journal of Operational Research, Elsevier, vol. 314(1), pages 363-376.
    11. Tore Selland Kleppe & Jun Yu & H.J. Skaug, 2010. "Simulated maximum likelihood estimation of continuous time stochastic volatility models," Advances in Econometrics, in: Maximum Simulated Likelihood Methods and Applications, pages 137-161, Emerald Group Publishing Limited.
    12. Dotsis, George & Psychoyios, Dimitris & Skiadopoulos, George, 2007. "An empirical comparison of continuous-time models of implied volatility indices," Journal of Banking & Finance, Elsevier, vol. 31(12), pages 3584-3603, December.
    13. Yang, Nian & Chen, Nan & Wan, Xiangwei, 2019. "A new delta expansion for multivariate diffusions via the Itô-Taylor expansion," Journal of Econometrics, Elsevier, vol. 209(2), pages 256-288.
    14. Almeida, Thiago Ramos, 2024. "Estimating time-varying factors’ variance in the string-term structure model with stochastic volatility," Research in International Business and Finance, Elsevier, vol. 70(PA).
    15. Giandomenico, Rossano, 2015. "Financial Methods: A Quantitative Approach," MPRA Paper 71919, University Library of Munich, Germany.
    16. Chenxu Li, 2014. "Closed-Form Expansion, Conditional Expectation, and Option Valuation," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 487-516, May.
    17. Levendorskii, Sergei, 2004. "Consistency conditions for affine term structure models," Stochastic Processes and their Applications, Elsevier, vol. 109(2), pages 225-261, February.
    18. O. Samimi & Z. Mardani & S. Sharafpour & F. Mehrdoust, 2017. "LSM Algorithm for Pricing American Option Under Heston–Hull–White’s Stochastic Volatility Model," Computational Economics, Springer;Society for Computational Economics, vol. 50(2), pages 173-187, August.
    19. Almut Veraart & Luitgard Veraart, 2012. "Stochastic volatility and stochastic leverage," Annals of Finance, Springer, vol. 8(2), pages 205-233, May.
    20. Gaetano Bua & Daniele Marazzina, 2021. "On the application of Wishart process to the pricing of equity derivatives: the multi-asset case," Computational Management Science, Springer, vol. 18(2), pages 149-176, June.

    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:spapps:v:130:y:2020:i:5:p:2675-2692. 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.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.