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Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

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  • Benth, Fred Espen
  • Rüdiger, Barbara
  • Süss, Andre
Abstract
We propose a non-Gaussian operator-valued extension of the Barndorff-Nielsen and Shephard stochastic volatility dynamics, defined as the square-root of an operator-valued Ornstein–Uhlenbeck process with Lévy noise and bounded drift. We derive conditions for the positive definiteness of the Ornstein–Uhlenbeck process, where in particular we must restrict to operator-valued Lévy processes with “non-decreasing paths”. It turns out that the volatility model allows for an explicit calculation of its characteristic function, showing an affine structure. We introduce another Hilbert space-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this class of stochastic volatility dynamics. Under a strong commutativity condition between the covariance operator of the Wiener process and the stochastic volatility, we can derive an analytical expression for the characteristic functional of the Ornstein–Uhlenbeck process perturbed by stochastic volatility if the noises are independent. The case of operator-valued compound Poisson processes as driving noise in the volatility is discussed as a particular example of interest. We apply our results to futures prices in commodity markets, where we discuss our proposed stochastic volatility model in light of ambit fields.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:2:p:461-486
    DOI: 10.1016/j.spa.2017.05.005
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    References listed on IDEAS

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    1. Anders B. Trolle & Eduardo S. Schwartz, 2009. "Unspanned Stochastic Volatility and the Pricing of Commodity Derivatives," The Review of Financial Studies, Society for Financial Studies, vol. 22(11), pages 4423-4461, November.
    2. Samuel Hikspoors & Sebastian Jaimungal, 2008. "Asymptotic Pricing of Commodity Derivatives using Stochastic Volatility Spot Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 15(5-6), pages 449-477.
    3. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305, World Scientific Publishing Co. Pte. Ltd..
    4. Ole E. Barndorff–Nielsen & Fred Espen Benth & Almut E. D. Veraart, 2010. "Modelling electricity forward markets by ambit fields," CREATES Research Papers 2010-41, Department of Economics and Business Economics, Aarhus University.
    5. Ole E. Barndorff-Nielsen & Fred Espen Benth & Almut E. D. Veraart, 2013. "Modelling energy spot prices by volatility modulated L\'{e}vy-driven Volterra processes," Papers 1307.6332, arXiv.org.
    6. Xiong, Jie, 2008. "An Introduction to Stochastic Filtering Theory," OUP Catalogue, Oxford University Press, number 9780199219704.
    7. Fred Espen Benth & Paul Kruhner, 2014. "Representation of infinite dimensional forward price models in commodity markets," Papers 1403.4111, arXiv.org.
    8. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
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    Cited by:

    1. 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.
    2. Schmidt, Thorsten & Tappe, Stefan & Yu, Weijun, 2020. "Infinite dimensional affine processes," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7131-7169.
    3. Dennis Schroers, 2024. "Dynamically Consistent Analysis of Realized Covariations in Term Structure Models," Papers 2406.19412, arXiv.org.
    4. 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.
    5. Fred Espen Benth & Heidar Eyjolfsson, 2022. "Robustness of Hilbert space-valued stochastic volatility models," Papers 2211.16071, arXiv.org.

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