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 L\'evy
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.Comment: 37 pages, 13 figure
