  We derive a nonparametric higher-order asymptotic expansion for small-time
changes of conditional characteristic functions of It\^o semimartingale
increments. The asymptotics setup is of joint type: both the length of the time
interval of the increment of the underlying process and the time gap between
evaluating the conditional characteristic function are shrinking. The spot
semimartingale characteristics of the underlying process as well as their spot
semimartingale characteristics appear as leading terms in the derived
asymptotic expansions. The analysis applies to a general class of It\^o
semimartingales that includes in particular L\'evy-driven SDEs and time-changed
L\'evy processes. The asymptotic expansion results are of direct use for
constructing nonparametric estimates pertaining to the stochastic volatility
dynamics of an asset from high-frequency data of options written on the
underlying asset.
