The Black-Scholes model for option pricing led to a tremendous development of trading of financial instruments in stock exchanges throughout the world. Such model provided a fair way of evaluating option prices making use of simplified assumptions. Mathematically, it consists of 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 realized 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-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. 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 regularization techniques either implicitly or explicitly. Such regularization techniques have been widely studied for a while and are still a topic of intense research. We have employed convex regularization 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 regularization. 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).

Professor Jorge Zubelli obtained his PhD from University of California at Berkeley. He is mainly working in the following area: mathematical methods in finance, mathematical methods and modelling of biophysical phenomena solitons, integrable systems, and nonlinear evolution equations.