  We give conditions under which the normalized marginal distribution of a
semimartingale converges to a Gaussian limit law as time tends to zero. In
particular, our result is applicable to solutions of stochastic differential
equations with locally bounded and continuous coefficients. The limit theorems
are subsequently extended to functional central limit theorems on the process
level. We present two applications of the results in the field of mathematical
finance: to the pricing of at-the-money digital options with short maturities
and short time implied volatility skews.
