The Short‐Time Behavior of Vix‐Implied Volatilities in a Multifactor Stochastic Volatility Framework

39 Pages Posted: 28 May 2020

See all articles by Andrea Barletta

Andrea Barletta

Nordea Bank

Elisa Nicolato

Aarhus University - Department of Business and Economics

Stefano Pagliarani

DEAMS, Università di Trieste

Date Written: July 2019

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.

Keywords: asymptotic expansions, implied volatility asymptotics, multifactor stochastic volatility, VIX options

JEL Classification: C60, G12, G13

Suggested Citation

Barletta, Andrea and Nicolato, Elisa and Pagliarani, Stefano, The Short‐Time Behavior of Vix‐Implied Volatilities in a Multifactor Stochastic Volatility Framework (July 2019). Mathematical Finance, Vol. 29, Issue 3, pp. 928-966, 2019, Available at SSRN: https://ssrn.com/abstract=3613161 or http://dx.doi.org/10.1111/mafi.12196

Andrea Barletta (Contact Author)

Nordea Bank ( email )

Christianbro
Strandgade 3
Copenhagen, 1401
Denmark

HOME PAGE: http://https://sites.google.com/view/andreabarletta/home

Elisa Nicolato

Aarhus University - Department of Business and Economics ( email )

Nordre Ringgade 1
Aarhus C, DK-8000
Denmark

Stefano Pagliarani

DEAMS, Università di Trieste ( email )

Via Valerio n. 4/1
Trieste
Italy

HOME PAGE: http://www.cmap.polytechnique.fr/~pagliarani/

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