This grant provides funding for the development of stochastic models in financial engineering. Financial engineering uses mathematical, statistical, computational methods, 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 proposed work consists of four projects: (1) Using compound Poisson processes to model asset prices (e.g. stocks, bonds, etc.) and credit risk. The goal is to improve the empirical performance of the classical Black-Scholes model, while still retaining analytical tractability. (2) Using birth-death and diffusion 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, to understand the recent burst of the ``tech bubble'', and to understand the long-term return of high tech stocks. (3) Using renewal theory to price discrete path-dependent options, such as discrete barrier and lookback options. The goal is to get accurate analytical approximations and fast algorithms for pricing complex financial contracts. (4) Using financial engineering in revenue management. The goal is to apply financial concepts to design better products for revenue management.
If successful, the results of this research will lead to (1) a better understanding of jump risk in stock prices and in credit markets; (2) a detailed study of what factors may influence the valuation of high tech stocks, and of connections between economic growth and valuation of high tech stocks; (3) faster algorithms to compute path-dependent options, which are widely used in financial and insurance industries; (4) designs of new financial contracts to enhance revenue and to minimize effects of uncertainty for service industries, such as airlines and hotels. This work will also contribute to methodologies available for stochastic processes.
