The Short-Time Behaviour of VIX Implied Volatilities in a Multifactor Stochastic Volatility Framework

Mathematical Finance (to appear, accepted June 7, 2018)

31 Pages Posted: 30 Mar 2017 Last revised: 27 Jul 2018

See all articles by Andrea Barletta

Andrea Barletta

Nordea Bank

Elisa Nicolato

University of Aarhus - Department of Theoretical Statistics

Stefano Pagliarani

DEAMS, Università di Trieste

Date Written: April 20, 2017

Abstract

We consider a modelling setup where the VIX index 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 obtained expansions 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 been previously 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: VIX Options, Multifactor Stochastic Volatility, Asymptotic Expansions

JEL Classification: C60, G12, G13

Suggested Citation

Barletta, Andrea and Nicolato, Elisa and Pagliarani, Stefano, The Short-Time Behaviour of VIX Implied Volatilities in a Multifactor Stochastic Volatility Framework (April 20, 2017). Mathematical Finance (to appear, accepted June 7, 2018). Available at SSRN: https://ssrn.com/abstract=2942262 or http://dx.doi.org/10.2139/ssrn.2942262

Andrea Barletta

Nordea Bank ( email )

Christianbro
Strandgade 3
Copenhagen, 1401
Denmark

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

Elisa Nicolato

University of Aarhus - Department of Theoretical Statistics ( email )

Nordre Ringgade 1
DK-8000 Aarhus C
Denmark
+45 8942 1111 (Phone)
+45 8942 1109 (Fax)

Stefano Pagliarani (Contact Author)

DEAMS, Università di Trieste ( email )

Via Valerio n. 4/1
Trieste
Italy

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

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