Stochastic Models in Financial Engineering

A new area of operations research is financial engineering, whose origin goes back to the landmark papers of Markowitz, Merton, Black and Scholes (which were awarded two Nobel prizes in Economics). It uses mathematical, statistical, computational methods and algorithms, along with economic principles and intuition, to study complex problems in the financial service industry and to help other industries to better manage their financial risks. The research, which focuses on the development of stochastic models in financial engineering, consists of three projects: (1) Using jump diffusion processes to model asset (e.g. stocks, bonds, etc.) prices. The goal is to improve the empirical performance of the classical Black-Scholes model based on the normal distribution, while still retaining analytical tractability. (2) Using renewal theory to price discrete path-dependent options, such as discrete barrier and look back options. The goal is to get accurate analytical approximations and fast algorithms to price complex financial contracts. (3) Using birth-death processes to model growth stocks (e.g. biotechnology and Internet stocks). The goal is to explain an empirical puzzle reported in the Wall Street Journal, and to try to understand the recent burst of the ``internet bubble''.

The United States holds a leadership position in financial services industry, an important sector of the service economy; future preeminence in this industry will rely on innovation in financial services and in the technology that supports these services. This research aims at developing mathematical tools that can be implemented by the financial service industry to better understand complex financial products, and to better manage financial risks for both financial and non-financial companies. The emerging field of financial engineering is attracting many bright students and professionals in engineering, mathematical and physical sciences. The field presents interesting new research challenges and many opportunities to see academic research impact industry practice. The research is integrated with the PI's educational activities at both the doctoral and master level, and seeks to meet a high academic standard while addressing issues of practical importance and providing graduate training that will prepare students for research careers in either a university or industry.