Large Deviations and Stochastic Volatility with Jumps: Asymptotic Implied Volatility for Affine Models

30 Pages Posted: 30 Aug 2011

See all articles by Antoine (Jack) Jacquier

Antoine (Jack) Jacquier

Imperial College London; The Alan Turing Institute

Martin Keller-Ressel

Dresden University of Technology - Department of Mathematics

Aleksandar Mijatovic

Imperial College London

Date Written: August 30, 2011

Abstract

We show that the implied volatility has a uniform (in log moneyness x) limit as the maturity tends to infinity, given by an explicit closed-form formula, for x in some compact neighborhood of zero in the class of affine stochastic volatility models. This expression is function of the convex dual of the limiting cumulant generating function h of the scaled log-spot process. We express h in terms of the functional characteristics of the underlying model. The proof of the limiting formula rests on the large deviation behavior of the scaled log-spot process as time tends to infinity. We apply our results to obtain the limiting smile for several classes of stochastic volatility models with jumps used in applications (e.g. Heston with state-independent jumps, Bates with state-dependent jumps and Barndorff-Nielsen-Shephard model).

Keywords: large deviation principle, stochastic volatility with jumps, affine processes, implied volatility in the large maturity limit

JEL Classification: 60G44, 60F10, 91G20

Suggested Citation

Jacquier, Antoine and Keller-Ressel, Martin and Mijatovic, Aleksandar, Large Deviations and Stochastic Volatility with Jumps: Asymptotic Implied Volatility for Affine Models (August 30, 2011). Available at SSRN: https://ssrn.com/abstract=1919494 or http://dx.doi.org/10.2139/ssrn.1919494

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

Martin Keller-Ressel

Dresden University of Technology - Department of Mathematics ( email )

Zellescher Weg 12-14
Willers-Bau C 112
Dresden, 01062
Germany

Aleksandar Mijatovic

Imperial College London ( email )

Department of Mathematics
180 Queen's Gate
London, SW7 2AZ
United Kingdom

HOME PAGE: http://www3.imperial.ac.uk/people/a.mijatovic

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