Small-Time Asymptotics for Implied Volatility under the Heston Model

International Journal of Theoretical and Applied Finance, Vol. 12, No. 6, pp. 861-876, 2009

Posted: 30 Nov 2009 Last revised: 23 Mar 2012

See all articles by Martin Forde

Martin Forde

Dublin City University - Department of Mathematical Sciences

Antoine (Jack) Jacquier

Imperial College London; The Alan Turing Institute

Abstract

We rigorize the work of Lewis (2007) and Durrleman (2005) on the small-time asymptotic behavior of the implied volatility under the Heston stochastic volatility model (Theorem 2.1). We apply the Gärtner-Ellis theorem from large deviations theory to the exponential affine closed-form expression for the moment generating function of the log forward price, to show that it satisfies a small-time large deviation principle. The rate function is computed as Fenchel-Legendre transform, and we show that this can actually be computed as a standard Legendre transform, which is a simple numerical root-finding exercise. We establish the corresponding result for implied volatility in Theorem 3.1, using well known bounds on the standard Normal distribution function. In Theorem 3.2 we compute the level, the slope and the curvature of the implied volatility in the small-maturity limit At-the-money, and the answer is consistent with that obtained by formal PDE methods by Lewis (2000) and probabilistic methods by Durrleman (2004).

Keywords: Implied volatility asymptotics, Heston, large deviation, small-time behavior

Suggested Citation

Forde, Martin and Jacquier, Antoine, Small-Time Asymptotics for Implied Volatility under the Heston Model. International Journal of Theoretical and Applied Finance, Vol. 12, No. 6, pp. 861-876, 2009 , Available at SSRN: https://ssrn.com/abstract=1515569

Martin Forde (Contact Author)

Dublin City University - Department of Mathematical Sciences ( email )

Dublin
Ireland

Antoine Jacquier

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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