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. 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. During the last few years, together with a number of collaborators, we have investigated different theoretical as well as practical methods for the calibration of local volatility models in different markets. This talk will summarize some of our findings and provide some perspectives of future research on the topic.

Bio:

Jorge P. Zubelli is Professor of Mathematics at the National Institute for Pure and Applied Mathematics (IMPA - Brazil) and heads the Laboratory for Analysis and Mathematical Modeling in the Applied Sciences (LAMCA - IMPA). His main research area is Inverse Problems with focus on its applications to real world problems. He obtained his PhD in Applied Mathematics from the University of California at Berkeley (1989), his MSc from IMPA in 1984, and his Electrical Engineering degree from IME-RJ in 1983 with specialization on Communication Engineering. From 2002 till 2017 he coordinated the Mathematical Methods in Finance Professional Masters program at IMPA. He coordinated a number of academic projects and research networks such as a PROSUL Latin America network (2008-2011), a Math-AmSud France-Latin America network (2009-2010), and an ALFA European Union and Latin America network (2002-2007). He also coordinated a number of industrial projects in Energy and Finance with corporations such as Petrobras and the Brazilian stock exchange BMF-Bovespa. As of October 2018 he has over 70 published articles in academic journals, he supervised 10 PhD thesis and over 30 MSc students.