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An adaptive positive preserving numerical scheme based on splitting method for the solution of the CIR model

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  • Kamrani, Minoo
  • Hausenblas, Erika

Abstract

This paper aims to investigate an adaptive numerical method based on a splitting scheme for the Cox–Ingersoll–Ross (CIR) model. The main challenge associated with numerically simulating the CIR process lies in the fact that most existing numerical methods fail to uphold the positive nature of the solution. Within this article, we present an innovative adaptive splitting scheme. Due to the existence of a square root in the CIR model, the step size is adaptively selected to ensure that, at each step, the value under the square-root does not fall under a given positive level and it is bounded. Moreover, an alternate numerical method is employed if the chosen step size becomes excessively small or the solution derived from the splitting scheme turns negative. This alternative approach, characterized by convergence and positivity preservation, is called the “backstop method”.

Suggested Citation

  • Kamrani, Minoo & Hausenblas, Erika, 2025. "An adaptive positive preserving numerical scheme based on splitting method for the solution of the CIR model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 229(C), pages 673-689.
    • Handle: RePEc:eee:matcom:v:229:y:2025:i:c:p:673-689
    DOI: 10.1016/j.matcom.2024.10.021
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    References listed on IDEAS

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    1. 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.
    2. 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..
    3. Griselda Deelstra & Freddy Delbaen, 1998. "Convergence of discretised stochastic interest rate: processes with stochastic drift term," ULB Institutional Repository 2013/7584, ULB -- Universite Libre de Bruxelles.
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    5. 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.
    6. C'onall Kelly & Gabriel Lord & Heru Maulana, 2020. "The role of adaptivity in a numerical method for the Cox-Ingersoll-Ross model," Papers 2002.10206, arXiv.org, revised Jan 2022.
    7. 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.
    8. Jean-Francois Chassagneux & Antoine Jacquier & Ivo Mihaylov, 2014. "An explicit Euler scheme with strong rate of convergence for financial SDEs with non-Lipschitz coefficients," Papers 1405.3561, arXiv.org, revised Apr 2016.
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