Shapes of Implied Volatility with Positive Mass at Zero

23 Pages Posted: 4 Oct 2013 Last revised: 29 Aug 2016

See all articles by Stefano De Marco

Stefano De Marco

Ecole Polytechnique, Paris - Centre de Mathematiques Appliquees

Caroline Hillairet

Ecole Polytechnique, Paris - Centre de Mathematiques Appliquees

Antoine (Jack) Jacquier

Imperial College London; The Alan Turing Institute

Date Written: August 29, 2016

Abstract

We study the shapes of the implied volatility when the underlying distribution has an atom at zero and analyse the impact of a mass at zero on at-the-money implied volatility and the overall level of the smile. We further show that the behaviour at small strikes is uniquely determined by the mass of the atom up to high asymptotic order, under mild assumptions on the remaining distribution on the positive real line. We investigate the structural difference with the no-mass-at-zero case, showing how one can -- theoretically -- distinguish between mass at the origin and a heavy-left-tailed distribution. We numerically test our model-free results in stochastic models with absorption at the boundary, such as the CEV process, and in jump-to-default models. Note that while Lee's moment formula tells that implied variance is at most asymptotically linear in log-strike, other celebrated results for exact smile asymptotics do not apply in this setting -- essentially due to the breakdown of Put-Call duality.

Keywords: Atomic distribution, heavy-tailed distribution, Implied Volatility, smile asymptotics, absorption at zero, CEV model

Suggested Citation

De Marco, Stefano and Hillairet, Caroline and Jacquier, Antoine, Shapes of Implied Volatility with Positive Mass at Zero (August 29, 2016). Available at SSRN: https://ssrn.com/abstract=2335445 or http://dx.doi.org/10.2139/ssrn.2335445

Stefano De Marco

Ecole Polytechnique, Paris - Centre de Mathematiques Appliquees ( email )

Palaiseau Cedex, 91128
France

Caroline Hillairet

Ecole Polytechnique, Paris - Centre de Mathematiques Appliquees ( email )

Palaiseau Cedex, 91128
France

Antoine Jacquier (Contact Author)

Imperial College London ( email )

South Kensington Campus
London SW7 2AZ, SW7 2AZ
United Kingdom

HOME PAGE: http://wwwf.imperial.ac.uk/~ajacquie/

The Alan Turing Institute ( email )

British Library, 96 Euston Road
London, NW12DB
United Kingdom

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