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The role of adaptivity in a numerical method for the Cox-Ingersoll-Ross model

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  • C'onall Kelly
  • Gabriel Lord
  • Heru Maulana

Abstract

We demonstrate the effectiveness of an adaptive explicit Euler method for the approximate solution of the Cox-Ingersoll-Ross model. This relies on a class of path-bounded timestepping strategies which work by reducing the stepsize as solutions approach a neighbourhood of zero. The method is hybrid in the sense that a convergent backstop method is invoked if the timestep becomes too small, or to prevent solutions from overshooting zero and becoming negative. Under parameter constraints that imply Feller's condition, we prove that such a scheme is strongly convergent, of order at least 1/2. Control of the strong error is important for multi-level Monte Carlo techniques. Under Feller's condition we also prove that the probability of ever needing the backstop method to prevent a negative value can be made arbitrarily small. Numerically, we compare this adaptive method to fixed step implicit and explicit schemes, and a novel semi-implicit adaptive variant. We observe that the adaptive approach leads to methods that are competitive in a domain that extends beyond Feller's condition, indicating suitability for the modelling of stochastic volatility in Heston-type asset models.

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  • 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.
    • Handle: RePEc:arx:papers:2002.10206
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    References listed on IDEAS

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    1. Roger Lord & Remmert Koekkoek & Dick Van Dijk, 2010. "A comparison of biased simulation schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 177-194.
    2. Mark Broadie & Özgür Kaya, 2006. "Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes," Operations Research, INFORMS, vol. 54(2), pages 217-231, April.
    3. G. Deelstra & F. Delbaen, 1998. "Convergence of discretized stochastic (interest rate) processes with stochastic drift term," Applied Stochastic Models and Data Analysis, John Wiley & Sons, vol. 14(1), pages 77-84, March.
    4. 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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