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Robustness of Hilbert space-valued stochastic volatility models

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  • Fred Espen Benth
  • Heidar Eyjolfsson

Abstract

In this paper we show that Hilbert space-valued stochastic models are robust with respect to perturbation, due to measurement or approximation errors, in the underlying volatility process. Within the class of stochastic volatility modulated Ornstein-Uhlenbeck processes, we quantify the error induced by the volatility in terms of perturbations in the parameters of the volatility process. We moreover study the robustness of the volatility process itself with respect to finite dimensional approximations of the driving compound Poisson process and semigroup generator respectively, when considering operator-valued Barndorff-Nielsen and Shephard stochastic volatility models. We also give results on square root approximations. In all cases we provide explicit bounds for the induced error in terms of the approximation of the underlying parameter. We discuss some applications to robustness of prices of options on forwards and volatility.

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  • Fred Espen Benth & Heidar Eyjolfsson, 2022. "Robustness of Hilbert space-valued stochastic volatility models," Papers 2211.16071, arXiv.org.
    • Handle: RePEc:arx:papers:2211.16071
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    References listed on IDEAS

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    1. Cox, Sonja & Karbach, Sven & Khedher, Asma, 2022. "Affine pure-jump processes on positive Hilbert–Schmidt operators," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 191-229.
    2. Christa Cuchiero & Sara Svaluto-Ferro, 2021. "Infinite-dimensional polynomial processes," Finance and Stochastics, Springer, vol. 25(2), pages 383-426, April.
    3. Benth, Fred Espen & Schroers, Dennis & Veraart, Almut E.D., 2022. "A weak law of large numbers for realised covariation in a Hilbert space setting," Stochastic Processes and their Applications, Elsevier, vol. 145(C), pages 241-268.
    4. Fred Espen Benth & Paul Kruhner, 2014. "Representation of infinite dimensional forward price models in commodity markets," Papers 1403.4111, arXiv.org.
    5. Fred Espen Benth & Jūratė Šaltytė Benth & Steen Koekebakker, 2008. "Stochastic Modeling of Electricity and Related Markets," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 6811, January.
    6. Benth, Fred Espen & Rüdiger, Barbara & Süss, Andre, 2018. "Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 461-486.
    7. Luca Di Persio & Isacco Perin, 2015. "An Ambit Stochastic Approach to Pricing Electricity Forward Contracts: The Case of the German Energy Market," Journal of Probability and Statistics, Hindawi, vol. 2015, pages 1-17, October.
    8. Sonja Cox & Sven Karbach & Asma Khedher, 2022. "An infinite‐dimensional affine stochastic volatility model," Mathematical Finance, Wiley Blackwell, vol. 32(3), pages 878-906, July.
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