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Small-time asymptotics for Gaussian self-similar stochastic volatility models

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  • Archil Gulisashvili
  • Frederi Viens
  • Xin Zhang

Abstract

We consider the class of self-similar Gaussian stochastic volatility models, and compute the small-time (near-maturity) asymptotics for the corresponding asset price density, the call and put pricing functions, and the implied volatilities. Unlike the well-known model-free behavior for extreme-strike asymptotics, small-time behaviors of the above depend heavily on the model, and require a control of the asset price density which is uniform with respect to the asset price variable, in order to translate into results for call prices and implied volatilities. Away from the money, we express the asymptotics explicitly using the volatility process' self-similarity parameter $H$, its first Karhunen-Loeve eigenvalue at time 1, and the latter's multiplicity. Several model-free estimators for $H$ result. At the money, a separate study is required: the asymptotics for small time depend instead on the integrated variance's moments of orders 1/2 and 3/2, and the estimator for $H$ sees an affine adjustment, while remaining model-free.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:1505.05256
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    References listed on IDEAS

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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. 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.
    3. Archil Gulisashvili, 2018. "Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions," Papers 1808.00421, arXiv.org, revised Jun 2019.
    4. 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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