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Extreme-Strike Asymptotics for General Gaussian Stochastic Volatility Models

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  • Archil Gulisashvili
  • Frederi Viens
  • Xin Zhang

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-Lo\`{e}ve 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 five terms in the expansion of the implied volatility. The expressions for the leading three terms are simple, and 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 $L^{2}$ norm of the projection of the mean function on the top eigenspace. The fourth term requires knowledge of all eigen-elements. We present detailed numerics based on realistic liquidity assumptions in which classical and long-memory volatility models are calibrated based on our expansion.

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  • Archil Gulisashvili & Frederi Viens & Xin Zhang, 2015. "Extreme-Strike Asymptotics for General Gaussian Stochastic Volatility Models," Papers 1502.05442, arXiv.org, revised Feb 2017.
    • Handle: RePEc:arx:papers:1502.05442
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Archil Gulisashvili & Josep Vives, 2014. "Asymptotic analysis of stock price densities and implied volatilities in mixed stochastic models," Papers 1403.5302, arXiv.org.
    4. Jim Gatheral & Antoine Jacquier, 2014. "Arbitrage-free SVI volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 14(1), pages 59-71, January.
    5. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
    6. Fabienne Comte & Eric Renault, 1998. "Long memory in continuous‐time stochastic volatility models," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 291-323, October.
    7. S. Benaim & P. Friz, 2009. "Regular Variation And Smile Asymptotics," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 1-12, January.
    8. J. D. Deuschel & P. K. Friz & A. Jacquier & S. Violante, 2011. "Marginal density expansions for diffusions and stochastic volatility, part I: Theoretical Foundations," Papers 1111.2462, arXiv.org, revised May 2013.
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    Cited by:

    1. Ankush Agarwal & Stefano de Marco & Emmanuel Gobet & Gang Liu, 2017. "Rare event simulation related to financial risks: efficient estimation and sensitivity analysis," Working Papers hal-01219616, HAL.
    2. Archil Gulisashvili, 2020. "Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super roughness," Papers 2002.05143, arXiv.org, revised Dec 2020.
    3. 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.
    4. Archil Gulisashvili, 2018. "Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions," Papers 1808.00421, arXiv.org, revised Jun 2019.
    5. Gordienko M. S., 2015. "Structuring elements of the intellectual capital," Annals of marketing-mba, Department of Marketing, Marketing MBA (RSconsult), vol. 4, December.
    6. Archil Gulisashvili & Frederi Viens & Xin Zhang, 2015. "Small-time asymptotics for Gaussian self-similar stochastic volatility models," Papers 1505.05256, arXiv.org, revised Mar 2016.
    7. 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.

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