Mathematical Economy and Finances
Research in Mathematical Finances is being developed, in the wake of the installation of Master programs at U São Paulo, IMPA and other institutions. Some preliminary results especially by J. Zubelli, exist. A few research projects aiming at exploring the collaboration with G. Papanicolaou are mentioned:
An option is a contract that gives the holder the right to trade in the future at a previously agreed price. The use of such derivative instruments, in modern financial markets has become so important that its volume has grown to be comparable to the so-called primary markets. One of the key points being that options are widely used as a way of hedging against market and commodity fluctuations.
A central problem in derivative markets is that of pricing. It involves fairly sophisticated mathematical techniques such as stochastic analysis, partial differential equations and optimization. The 1997 Nobel Prize in Economics was awarded to Robert C. Merton and Myron S. Scholes for their work, along with Fischer Black, in developing the Fischer-Black options pricing model. Black, who died in 1995, would undoubtedly have shared in the prize had he still been alive.
The Black-Scholes formula is a parabolic-type partial differential equation that relates derivative prices to current stock prices in terms of different market parameters. Among those parameters, one of the most important, and difficult to obtain is the volatility. The problem of stably determining the volatility, which is known as the calibration of the model, is an extremely difficult one and belongs to the realm of Inverse Problems. A subject that has attracted the attention of many leading mathematicians and taken life of its own.
The standard Black-Scholes model assumes a constant volatility. It is now well known that, despite its simplicity, the Black-Scholes model is no longer sufficient to capture modern market phenomena. This became even more clear after the 1987 crash. One natural extension of the B-S model is to modify the specification of volatility to make it a stochastic process. This leads to a fascinating area of research which combines partial differential equations, statistics, and numerical analysis. A break-through in this area was made recently by Papanicolaou, Fouque, and Sircar, who describe in their book "Derivatives in financial markets with stochastic volatility" a method for modeling, analysis, and estimation that exploits the fast mean reversion of the volatility.
On a short term, one area we would like to investigate is the validation of the methodology developed by Papanicolaou's group through analysis of real data from different markets such as the IBOVESPA, FOOTSE, DAX, etc. This leads to interesting data analysis problems. Some preliminary results obtained by J. P. Zubelli jointly with C. Ibsen and H. Moreira indicate the validity of the fast mean reversion hypothesis and lead to new questions of interest in economics. On a long term we would like to analyze the different inverse problems that are natural consequences of the methodology proposed by Papanicolaou and collaborators.
There is also a well established and very successful research and graduate program in Mathematical Economics around A. Araujo and his collaborators.