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Extreme-strike asymptotics for general Gaussian stochastic volatility models

Author

Listed:
  • Archil Gulisashvili

    (Ohio University)

  • Frederi Viens

    (Michigan State University)

  • Xin Zhang

    (Purdue University)

Abstract

We consider a stochastic volatility asset price model in which the volatility is the absolute value of a continuous Gaussian process with arbitrary prescribed mean and covariance. By exhibiting a Karhunen–Loève expansion for the integrated variance, and using sharp estimates of the density of a general second-chaos variable, we derive asymptotics for the asset price density for large or small values of the variable, and study the wing behavior of the implied volatility in these models. Our main result provides explicit expressions for the first three terms in the expansion of the implied volatility, based on three basic spectral-type statistics of the Gaussian process: the top eigenvalue of its covariance operator, the multiplicity of this eigenvalue, and the $$L^{2}$$ L 2 norm of the projection of the mean function on the top eigenspace. Numerical illustrations using the Stein–Stein and fractional Stein–Stein models are presented, including strategies for parameter calibration.

Suggested Citation

  • Archil Gulisashvili & Frederi Viens & Xin Zhang, 2019. "Extreme-strike asymptotics for general Gaussian stochastic volatility models," Annals of Finance, Springer, vol. 15(1), pages 59-101, March.
    • Handle: RePEc:kap:annfin:v:15:y:2019:i:1:d:10.1007_s10436-018-0338-z
    DOI: 10.1007/s10436-018-0338-z
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    References listed on IDEAS

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    1. Rosenbaum, Mathieu, 2008. "Estimation of the volatility persistence in a discretely observed diffusion model," Stochastic Processes and their Applications, Elsevier, vol. 118(8), pages 1434-1462, August.
    2. Fabienne Comte & Eric Renault, 1998. "Long memory in continuous‐time stochastic volatility models," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 291-323, October.
    3. Alexandra Chronopoulou & Frederi G. Viens, 2012. "Stochastic volatility and option pricing with long-memory in discrete and continuous time," Quantitative Finance, Taylor & Francis Journals, vol. 12(4), pages 635-649, December.
    4. Jim Gatheral & Antoine Jacquier, 2014. "Arbitrage-free SVI volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 14(1), pages 59-71, January.
    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. S. Benaim & P. Friz, 2009. "Regular Variation And Smile Asymptotics," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 1-12, January.
    7. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
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    Citations

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    Cited by:

    1. Archil Gulisashvili, 2020. "Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super roughness," Papers 2002.05143, arXiv.org, revised Dec 2020.
    2. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Working Papers hal-02946146, HAL.
    3. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Post-Print hal-02946146, HAL.
    4. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Finance and Stochastics, Springer, vol. 26(4), pages 733-769, October.
    5. 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.
    6. Eduardo Abi Jaber, 2019. "The Laplace transform of the integrated Volterra Wishart process," Papers 1911.07719, arXiv.org, revised Jul 2024.
    7. Eduardo Abi Jaber, 2022. "The Laplace transform of the integrated Volterra Wishart process," Mathematical Finance, Wiley Blackwell, vol. 32(1), pages 309-348, January.
    8. Eduardo Abi Jaber, 2020. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Papers 2009.10972, arXiv.org, revised May 2022.
    9. Eduardo Abi Jaber, 2022. "The Laplace transform of the integrated Volterra Wishart process," Post-Print hal-02367200, HAL.
    10. Eduardo Abi Jaber, 2020. "The Laplace transform of the integrated Volterra Wishart process," Working Papers hal-02367200, HAL.
    11. Eduardo Abi Jaber, 2022. "The Laplace transform of the integrated Volterra Wishart process," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-02367200, HAL.
    12. Huy N. Chau & Duy Nguyen & Thai Nguyen, 2024. "On short-time behavior of implied volatility in a market model with indexes," Papers 2402.16509, arXiv.org, revised Apr 2024.
    13. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-02946146, HAL.
    14. 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.

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    More about this item

    Keywords

    Stochastic volatility; Implied volatility; Large strike; Karhunen–Loève expansion; Chi-squared variates;
    All these keywords.

    JEL classification:

    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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