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
\textit{out-of-the-money} options is found to be linear in time, the short time
asymptotics of \textit{at-the-money} options is shown to depend on the fine
structure of the semimartingale.

Starting from the requirement that risk measures of financial portfolios
should be based on their losses, not their gains, we define the notion of
loss-based risk measure and study the properties of this class of risk
measures. We characterize loss-based risk measures by a representation theorem
and give examples of such risk measures.

We propose two nonparametric tests for investigating the pathwise properties
of a signal modeled as the sum of a L\'{e}vy process and a Brownian
semimartingale. Using a nonparametric threshold estimator for the continuous
component of the quadratic variation, we design a test for the presence of a
continuous martingale component in the process and a test for establishing
whether the jumps have finite or infinite variation, based on observations on a
discrete-time grid.

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.