The importance of Statistics and Stochastic Processes in Quantitative Finance and Risk Analysis is by now well established. Yet, perhaps not so well-known, we have the fact that it is crucial for the effectiveness of Stochastic Processes in Finance the estimation of the relevant parameters in such processes. The estimation of such parameters has to be done in very large parametric spaces, often infinite dimensional ones. As a consequence, the theory of Inverse Problems comes into play in a fundamental way.

In this talk we shall explore one (large) class of examples where Inverse Problems play a role, namely in the estimation of Local Volatility in the option pricing.

The Black-Scholes model for option pricing led to a tremendous development of trading of financial instruments in stock exchanges throughout the world. Such a model provided a fair way of evaluating option prices making use of simplified assumptions. Mathematically, it consists of a parabolic diffusion equation that after a suitable change of variables becomes a heat equation. Its diffusion coefficient is the volatility and describes the agitation of the market.

However, soon it was realised that the Black-Scholes model was inadequate and required realistic extensions. One of the most well-accepted of such extensions is to consider variable diffusion coefficients thus leading to the so-called local volatility models.

Local volatility models are extensively used and well-recognised 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. The PDE (inverse) problem consists in recovering the time and space varying diffusion coefficient in a parabolic equation from limited data.  It is known that this corresponds to an ill-posed problem.

The ill-posed character of local volatility surface calibration from market prices requires the use of regularisation techniques either implicitly or explicitly. Such regularisation techniques have been widely studied for a while and are still a topic of intense research. We have employed convex regularisation tools and recent inverse problem advances to deal with the local volatility calibration problem.

We investigate theoretical as well as practical methods for the calibration of local Volatility models by convex regularisation. Such methods can also be applied to commodities, thus being very relevant also in the accurate pricing of commodity derivatives. We illustrate our results both with real and with simulated data. This is joint work with V. Albani (IMPA), U. Ascher (UBC), Xu Yang (IMPA).