Asymptotics of forward implied volatility

TL;DR

This paper derives a comprehensive closed-form expansion for forward implied volatility and forward-start options across various models, revealing how forward-start date scaling affects asymptotic behaviors for different maturities.

Contribution

It introduces a general expansion formula for forward implied volatility in diverse models, utilizing large deviations techniques, and explores the impact of forward-start date scaling on asymptotics.

Findings

Forward implied volatility can be expanded using properties of the forward characteristic function.

The forward-start date must be rescaled for meaningful small-maturity asymptotics.

Forward-start date influences large-maturity forward smile behavior.

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 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.