We derive a forward partial integro-differential equation for prices of call options in a model where the dynamics of the underlying asset under the pricing measure is described by a -possibly discontinuous-semimartingale. This result generalizes Dupire's forward equation to a large class of non-Markovian models with jumps.

We study the short-time asymptotics of conditional expectations of smooth and non-smooth functions of a (discontinuous) Ito semimartingale; we compute the leading term in the asymptotics in terms of the local characteristics of the semimartingale. We derive in particular the asymptotic behavior of call options with short maturity in a semimartingale model: whereas the behavior of out-of-the-money options is found to be linear in time, the short time asymptotics of at-the-money options is shown to depend on the fine structure of the semimartingale.