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Multivariate Stochastic Volatility Models and Large Deviation Principles

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  • Archil Gulisashvili

Abstract

We establish a comprehensive sample path large deviation principle (LDP) for log-processes associated with multivariate time-inhomogeneous stochastic volatility models. Examples of models for which the new LDP holds include Gaussian models, non-Gaussian fractional models, mixed models, models with reflection, and models in which the volatility process is a solution to a Volterra type stochastic integral equation. The LDP for log-processes is used to obtain large deviation style asymptotic formulas for the distribution function of the first exit time of a log-process from an open set and for the price of a multidimensional binary barrier option. We also prove a sample path LDP for solutions to Volterra type stochastic integral equations with predictable coefficients depending on auxiliary stochastic processes.

Suggested Citation

  • Archil Gulisashvili, 2022. "Multivariate Stochastic Volatility Models and Large Deviation Principles," Papers 2203.09015, arXiv.org, revised Nov 2022.
    • Handle: RePEc:arx:papers:2203.09015
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    References listed on IDEAS

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    1. Archil Gulisashvili, 2017. "Large deviation principle for Volterra type fractional stochastic volatility models," Papers 1710.10711, arXiv.org, revised Aug 2018.
    2. Antoine Jacquier & Mikko S. Pakkanen & Henry Stone, 2017. "Pathwise large deviations for the Rough Bergomi model," Papers 1706.05291, arXiv.org, revised Dec 2018.
    3. Christian Bayer & Fabian Andsem Harang & Paolo Pigato, 2020. "Log-modulated rough stochastic volatility models," Papers 2008.03204, arXiv.org, revised May 2021.
    4. Wang, Zhidong, 2008. "Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients," Statistics & Probability Letters, Elsevier, vol. 78(9), pages 1062-1071, July.
    5. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," The Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    6. Baldi, P. & Caramellino, L., 2011. "General Freidlin-Wentzell Large Deviations and positive diffusions," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1218-1229, August.
    7. Antoine Jacquier & Alexandre Pannier, 2020. "Large and moderate deviations for stochastic Volterra systems," Papers 2004.10571, arXiv.org, revised Apr 2022.
    8. Archil Gulisashvili, 2020. "Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super roughness," Papers 2002.05143, arXiv.org, revised Dec 2020.
    9. Omar El Euch & Mathieu Rosenbaum, 2019. "The characteristic function of rough Heston models," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 3-38, January.
    10. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    11. Robertson, Scott, 2010. "Sample path Large Deviations and optimal importance sampling for stochastic volatility models," Stochastic Processes and their Applications, Elsevier, vol. 120(1), pages 66-83, January.
    12. Gulisashvili, Archil, 2020. "Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3648-3686.
    13. Archil Gulisashvili, 2020. "Large deviation principles for stochastic volatility models with reflection and three faces of the Stein and Stein model," Papers 2006.15431, arXiv.org.
    14. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    15. Gulisashvili, Archil, 2021. "Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super roughness," Stochastic Processes and their Applications, Elsevier, vol. 139(C), pages 37-79.
    16. Gerhold, Stefan & Gerstenecker, Christoph & Gulisashvili, Archil, 2021. "Large deviations for fractional volatility models with non-Gaussian volatility driver," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 580-600.
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    Cited by:

    1. Peter K. Friz & Thomas Wagenhofer, 2023. "Reconstructing volatility: Pricing of index options under rough volatility," Mathematical Finance, Wiley Blackwell, vol. 33(1), pages 19-40, January.
    2. Giacomo Giorgio & Barbara Pacchiarotti & Paolo Pigato, 2023. "Short-Time Asymptotics for Non-Self-Similar Stochastic Volatility Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 30(3), pages 123-152, May.
    3. Peter K. Friz & Thomas Wagenhofer, 2022. "Reconstructing Volatility: Pricing of Index Options under Rough Volatility," Papers 2212.07817, arXiv.org.

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