Financial Mathematics is a broad area of mathematics driven by the mathematical needs of practitioners in the finance industry, such as asset pricing models, derivatives, portfolio selection, and risk modeling. My introduction to this field started with work on electricity futures pricing at a regional utility in Austin, TX. Shortly thereafter I moved to Barra, Inc., in Berkeley, California, to work on fixed income portfolio risk models.
Since then I've been interested in portfolio risk, credit risk, credit spread models, optimal stopping time problems, problems arising in numerical linear algebra and statistics related to model estimation, and foundations of the field inspired by the need to teach Master's students in Financial Math. I'm also collaborating with Economist Paul Beaumont on dynamic asset pricing models in macroeconomics. More details below.
Jointly with Paul Beaumont, and with the help of some seed-grant support, we are studying the characteristics of the standard Lucas model of asset pricing, with a view to understanding out-of-equilibrium behavior of markets. This work is motivated by the need to understand market dynamics like high price volatility and pricing bubbles, which are not explained by traditional models. Our approach is to analyze the demand function via agent-based simulation code, written in Java in collaboration with recent PhD Andrew Culham. The long range goal is to bring ideas from nonlinear dynamics more fully into a a dynamical economy driving by adaptive learning by intelligent agents who have realistic limits on their knowledge about the other agents.
Current PhD students: Michelle Guan (Mathematics) and Aaron Schmerbeck (Economics, co-directed with Paul Beaumont)
A country's central bank may at times intervene in currency markets by buying or selling currency in order to influence the exchange rate with a foreign benchmark currency. This leads to a dynamic programming problem in which a value function is to be maximized by some intervention strategy. This can be set in the context of an optimal stopping problem, about which there has been a great deal of interesting progress lately. With recent PhD Juan Moreno, we are looking at intervention problems in the context of this recent optimal stopping research, with particular focus on South America.
Current PhD student: He Huang
Most financial random variables are not Normal. In particular, there may be skewness or heavy tails in the individual marginal distributions, and tail dependence in the dependence structure described by the copula function connecting the margins. This is particularly true for default related derivatives, such as basket credit default swaps (CDS): extreme events are more common than predicted by the normal distribution, and "default contagion" means that firms are more likely to default after others already have.
There are a variety of other multivariate distributions available to the modeler, but calibration becomes more difficult as new parameters are added. However, work with a former student Wenbo Hu has shown that there are some good choices that can be calibrated quickly using the EM algorithm -- in particular the skewed t distribution, which is a subfamily of the generalized hyperbolic distributions. Current joint work favorably compares a copula approach with the skewed t distribution to other methods currently in use.
Current PhD student: Yang Liu
Juan Moreno, PhD Fall 2007 FSU; now at Citibank, Bogota.
Andrew Culham, PhD Summer 2007 FSU; now at Florida Power and Light. Dissertation: [pdf]
Jianke Zhang, PhD Spring 2007 FSU; now at Florida Power and Light
Wenbo Hu, PhD Fall 2005 FSU; now at Bell Trading, Chicago
Brian Tandy, PhD 1997 UT Austin; now at Digeo Corp., Palo Alto
Paul Fabel, PhD 1994 UT Austin; now Associate Professor, Dept. of Mathematics, Mississippi State University