  Financial markets have developed a lot of strategies to control risks induced
by market fluctuations. Mathematics has emerged as the leading discipline to
address fundamental questions in finance as asset pricing model and hedging
strategies. History began with the paradigm of zero-risk introduced by Black &
Scholes stating that any random amount to be paid in the future may be
replicated by a dynamical portfolio. In practice, the lack of information leads
to ill-posed problems when model calibrating. The real world is more complex
and new pricing and hedging methodologies have been necessary. This challenging
question has generated a deep and intensive academic research in the 20 last
years, based on super-replication (perfect or with respect to confidence level)
and optimization. In the interplay between theory and practice, Monte Carlo
methods have been revisited, new risk measures have been back-tested. These
typical examples give some insights on how may be used mathematics in financial
risk management.
