Financial Mathematics is a widely interdisciplinary area of mathematics driven by the mathematical needs of practitioners in the finance industry. After working on fixed income portfolio risk at MSCIBarra in Berkeley, California, I came to Florida State University to join their Financial Mathematics group.
Here I've been interested in portfolio risk, credit risk, optimal stopping time problems, problems arising in numerical linear algebra and statistics related to model estimation, statistical models of the limit order book, and foundations of the field inspired by the needs of graduate 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, 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. 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. Recent PhD graduates include Andrew Culham and Michelle Guan.
Current PhD students: Dawna Jones, co-directed with Paul Beaumont
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. Correlated jump processes can be used to price basket CDS, and to model risk for portfolios of assets with jumps.
Current student: Pierre Garreau
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 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.
Generalized Hyperbolic distributions can be used to fit multivariate asset returns data for portfolio risk management. We are currently working on ways to use them in portfolio risk models at different time scales after calibration with high-frequency returns.
Current PhD student: Yang Liu
High-frequency trading is becoming dominant in financial markets, where intra-day matters such as order book dynamics become important. We are looking at models of order-book dynamics via generalized birth-death processes as a framework for high-frequency trading strategies.
Current PhD student: Huang (Henry) He
Stochastic volatility models are a popular way to recover the smile or smirk in option implied volatilities. In these models the volatility of a stock or interest rate is itself a stochastic process driven by a second source of noise. We are examining fully coupled models where the observable appears explicitly in the volatility dynamics. Such models are similar to standard CIR models to lowest order, but to higher order they differ from models for which the volatility is independent of the price.
Current PhD student: Tianyu Liang, co-directed with Xiaoming Wang
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.
Michelle Guan, PhD Summer 2011 FSU; now at Indiana University Northwest
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. Dissertation: [amazon.com]
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