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On arbitrarily slow convergence rates for strong numerical approximations of Cox–Ingersoll–Ross processes and squared Bessel processes

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Abstract

Cox–Ingersoll–Ross (CIR) processes are extensively used in state-of-the-art models for the pricing of financial derivatives. The prices of financial derivatives are very often approximately computed by means of explicit or implicit Euler- or Milstein-type discretization methods based on equidistant evaluations of the driving noise processes. In this article, we study the strong convergence speeds of all such discretization methods. More specifically, the main result of this article reveals that each such discretization method achieves at most a strong convergence order of \(\delta /2\), where \(0<\delta <2\) is the dimension of the squared Bessel process associated to the considered CIR process.

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Acknowledgements

Special thanks are due to André Herzwurm for a series of fruitful discussions on this work. This project has been supported through the SNSF-Research project 200021_156603 “Numerical approximations of nonlinear stochastic ordinary and partial differential equations”. We gratefully acknowledge the Institute for Mathematical Research (FIM) at ETH Zurich which provided office space and partially organized the short visit of the first author to ETH Zurich in 2016 when part of this work was done.

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Authors and Affiliations

  1. Fachbereich Mathematik, Universität Kaiserslautern, Postfach 3049, 67653, Kaiserslautern, Germany

    Mario Hefter

  2. Seminar for Applied Mathematics, Eidgenössische Technische Hochschule Zürich, Rämistrasse 101, 8092, Zurich, Switzerland

    Arnulf Jentzen

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  1. Mario Hefter
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  2. Arnulf Jentzen
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Correspondence to Mario Hefter.

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Hefter, M., Jentzen, A. On arbitrarily slow convergence rates for strong numerical approximations of Cox–Ingersoll–Ross processes and squared Bessel processes. Finance Stoch 23, 139–172 (2019). https://doi.org/10.1007/s00780-018-0375-5

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  • DOI: https://doi.org/10.1007/s00780-018-0375-5

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