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The short‐time behavior of VIX‐implied volatilities in a multifactor stochastic volatility framework

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  • Andrea Barletta
  • Elisa Nicolato
  • Stefano Pagliarani

Abstract

We consider a modeling setup where the volatility index (VIX) dynamics are explicitly computable as a smooth transformation of a purely diffusive, multidimensional Markov process. The framework is general enough to embed many popular stochastic volatility models. We develop closed‐form expansions and sharp error bounds for VIX futures, options, and implied volatilities. In particular, we derive exact asymptotic results for VIX‐implied volatilities, and their sensitivities, in the joint limit of short time‐to‐maturity and small log‐moneyness. The expansions obtained are explicit based on elementary functions and they neatly uncover how the VIX skew depends on the specific choice of the volatility and the vol‐of‐vol processes. Our results are based on perturbation techniques applied to the infinitesimal generator of the underlying process. This methodology has previously been adopted to derive approximations of equity (SPX) options. However, the generalizations needed to cover the case of VIX options are by no means straightforward as the dynamics of the underlying VIX futures are not explicitly known. To illustrate the accuracy of our technique, we provide numerical implementations for a selection of model specifications.

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  • Andrea Barletta & Elisa Nicolato & Stefano Pagliarani, 2019. "The short‐time behavior of VIX‐implied volatilities in a multifactor stochastic volatility framework," Mathematical Finance, Wiley Blackwell, vol. 29(3), pages 928-966, July.
    • Handle: RePEc:bla:mathfi:v:29:y:2019:i:3:p:928-966
    DOI: 10.1111/mafi.12196
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    Cited by:

    1. Mora-Valencia, Andrés & Rodríguez-Raga, Santiago & Vanegas, Esteban, 2021. "Skew index: Descriptive analysis, predictive power, and short-term forecast," The North American Journal of Economics and Finance, Elsevier, vol. 56(C).
    2. Matthew Lorig & Natchanon Suaysom, 2022. "Explicit Caplet Implied Volatilities for Quadratic Term-Structure Models," Papers 2212.04425, arXiv.org.
    3. Takuji Arai, 2019. "Pricing And Hedging Of Vix Options For Barndorff-Nielsen And Shephard Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(08), pages 1-26, December.
    4. 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.

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