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Unbiased estimators for the Heston model with stochastic interest rates

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  • Chao Zheng
  • Jiangtao Pan

Abstract

We combine the unbiased estimators in Rhee and Glynn (Operations Research: 63(5), 1026-1043, 2015) and the Heston model with stochastic interest rates. Specifically, we first develop a semi-exact log-Euler scheme for the Heston model with stochastic interest rates. Then, under mild assumptions, we show that the convergence rate in the $L^2$ norm is $O(h)$, where $h$ is the step size. The result applies to a large class of models, such as the Heston-Hull-While model, the Heston-CIR model and the Heston-Black-Karasinski model. Numerical experiments support our theoretical convergence rate.

Suggested Citation

  • Chao Zheng & Jiangtao Pan, 2023. "Unbiased estimators for the Heston model with stochastic interest rates," Papers 2301.12072, arXiv.org, revised Aug 2023.
    • Handle: RePEc:arx:papers:2301.12072
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    References listed on IDEAS

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    1. 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..
    2. Bakshi, Gurdip & Cao, Charles & Chen, Zhiwu, 2000. "Pricing and hedging long-term options," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 277-318.
    3. Hull, John & White, Alan, 1990. "Pricing Interest-Rate-Derivative Securities," The Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 573-592.
    4. Annalena Mickel & Andreas Neuenkirch, 2021. "The Weak Convergence Rate of Two Semi-Exact Discretization Schemes for the Heston Model," Risks, MDPI, vol. 9(1), pages 1-38, January.
    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.
    6. Alexander van Haastrecht & Antoon Pelsser, 2011. "Generic pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility," Quantitative Finance, Taylor & Francis Journals, vol. 11(5), pages 665-691.
    7. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
    8. Chang-Han Rhee & Peter W. Glynn, 2015. "Unbiased Estimation with Square Root Convergence for SDE Models," Operations Research, INFORMS, vol. 63(5), pages 1026-1043, October.
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