Asymptotics of forward implied volatility - Jacquier,A; Roome,P (SIAM Journal on Financial Mathematics, 2012)
We prove here a general closed-form expansion formula for forward-startoptions and the forward implied volatility smile in a large class of models,including the Heston stochastic volatility and time-changed exponential L\'evymodels. This expansion applies to both small and large maturities and is basedsolely on the properties of the forward characteristic function of theunderlying process. The method is based on sharp large deviations techniques,and allows us to recover (in particular) many results for the spot impliedvolatility smile. In passing we (i) show that the forward-start date has to berescaled in order to obtain non-trivial small-maturity asymptotics, (ii) provethat the forward-start date may influence the large-maturity behaviour of theforward smile, and (iii) provide some examples of models with finite quadraticvariation where the small-maturity forward smile does not explode.