The Small-Maturity Heston Forward Smile

SIAM Journal on Financial Mathematics, 4(1): 831-856, 2013.

23 Pages Posted: 19 Mar 2013 Last revised: 21 Feb 2015

See all articles by Antoine (Jack) Jacquier

Antoine (Jack) Jacquier

Imperial College London; The Alan Turing Institute

Patrick Roome

Imperial College London - Department of Mathematics

Date Written: April 15, 2013

Abstract

In this paper we investigate the asymptotics of forward-start options and the forward implied volatility smile in the Heston model as the maturity approaches zero. We prove that the forward smile for out-of-the-money options explodes and compute a closed-form high-order expansion detailing the rate of the explosion. Furthermore the result shows that the square-root behaviour of the variance process induces a singularity such that for certain parameter configurations one cannot obtain high-order out-of-the-money forward smile asymptotics. In the at-the-money case a separate model-independent analysis shows that the small-maturity limit is well defined for any Ito diffusion. The proofs rely on the theory of sharp large deviations (and refinements) and incidentally we provide an example of degenerate large deviations behaviour.

Keywords: forward smile, forward-start options, large deviations, small-maturity asymptotics, Heston model

JEL Classification: C60, G10, G12, G13

Suggested Citation

Jacquier, Antoine and Roome, Patrick, The Small-Maturity Heston Forward Smile (April 15, 2013). SIAM Journal on Financial Mathematics, 4(1): 831-856, 2013., Available at SSRN: https://ssrn.com/abstract=2235196 or http://dx.doi.org/10.2139/ssrn.2235196

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

Patrick Roome (Contact Author)

Imperial College London - Department of Mathematics ( email )

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

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