  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.
