Asymptotics of Forward Implied Volatility

SIAM Journal on Financial Mathematics, 6(1): 307-351, 2015.

37 Pages Posted: 5 Dec 2012 Last revised: 4 Jun 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: December 4, 2012

Abstract

We prove here a general closed-form expansion formula for forward-start options and the forward implied volatility smile in a large class of models, including the Heston stochastic volatility and time-changed exponential Levy models.This expansion applies to both small and large maturities and is based solely on the properties of the forward characteristic function of the underlying process. The method is based on sharp large deviations techniques, and allows us to recover (in particular) many results for the spot implied volatility smile.In passing we (i) show that the forward-start date has to be rescaled in order to obtain non-trivial small-maturity asymptotics,(ii) prove that the forward-start date may influence the large-maturity behaviour of the forward smile,and (iii) provide some examples of models with finite quadratic variation where the small-maturity forward smile does not explode.

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

JEL Classification: C60, G10, G12, G13

Suggested Citation

Jacquier, Antoine and Roome, Patrick, Asymptotics of Forward Implied Volatility (December 4, 2012). SIAM Journal on Financial Mathematics, 6(1): 307-351, 2015., Available at SSRN: https://ssrn.com/abstract=2185075 or http://dx.doi.org/10.2139/ssrn.2185075

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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