Since the pioneering work of Fischer Black, Robert Merton, and Myron Scholes in 1973, which heralded the introduction of sophisticated mathematical ideas to the problem of pricing options and, more broadly, to risk analysis in the financial industry, advanced mathematical techniques have played an increasingly important role in managing risk across a wide range of insurance and financial industries. In recognition of their work, Merton and Scholes were awarded the 1997 Nobel Prize in Economics, Black having died in 1995. These developments in finance have benefited from ideas in many different fields, including the traditional ones of mathematics and statistics, but also from fields such as electrical engineering and condensed matter physics. The celebrated Black-Scholes-Merton model is no longer viewed as adequate, however, and attempts to replace it with improved models form the basis of intense current research activity in mathematical finance. I plan to bring my research experience in both pure and applied areas of mathematics -- mathematical physics, partial differential equations, and control theory -- to investigate the following interrelated research topics: First, risk analysis for portfolio insurance and related structured investment products; second, the application of methods of stochastic control theory to optimal portfolio management and trading models; third, the development of new option pricing, trading and risk analysis models for risky-asset stochastic processes, including stochastic volatility and jump diffusion models, as replacements for the Black-Scholes-Merton model; fourth, the numerical implementation of these models.
I will work on this project while at Bloomberg LP, New York, on leave from Rutgers University, working with Patrick Hagan and Peter Carr, Head of Quantitative Research at Bloomberg. Upon my return to Rutgers in Autumn 2005, I plan to continue my work on the development of a new applied mathematics master's degree program in mathematical finance, as well as my research in that field. Our master's degree program will lead to better career opportunities for our students in the financial, insurance, and risk analysis professions. The research activities described in my proposal can benefit society by generating improved algorithms for financial modeling and risk analysis, leading to greater protection of pension funds and to lower borrowing costs for home purchasers and for industry.
