Local volatility models are extensively used and well-recognized for
hedging and pricing in financial markets. They are frequently used,
for instance, in the evaluation of exotic options so as to avoid
arbitrage opportunities with respect to other instruments. From a
mathematical viewpoint, the model consists in assuming that the
stochastic volatility of the underlying price is given by a
(deterministic) function of time and the (stochastic) price of such
underlying. We are thus concerned with the inverse problem of
calibrating such volatility surface from observed market data.

The ill-posed character of local volatility surface calibration from
market prices requires the use of regularization techniques either
implicitly or explicitly. Such regularization techniques have been
widely studied for a while and are still a  topic of intense research.
The subject falls naturally in the beautiful intersection of
Statistics, Optimization, and Numerical Analysis.

In the final part of the talk we shall describe ongoing work on the
use local volatility models in the context of commodity markets, in
particular applied to energy and oil ones. This work is part of
ongoing collaboration with V. Albani (Vienna), U. Ascher (Toronto),
and Xu Yang (IMPA).