This project focuses on two aspects of applied probability in financial engineering: (1) applying useful techniques from operations research to option pricing; and (2) applying results from option pricing to engineering. The tools to used are those from applied probability, including renewal theory, marked point processes, duality and stochastic calculus. More precisely, the following four problems will be studied within the above two categories, with a focus on the first category: (1) Using renewal theory to price discrete exotic options, such as discrete barrier and lookback options, which are widely used in foreign currency markets. (2) Using marked point processes to study interest rate models, and to price interest rate derivatives, such as caps and caplets. (3) Using duality to study pricing of options in incomplete markets, in market with portfolio constraints, and in markets with different interest rates for borrowing and leading. (4) Although the main focus will be to apply operations research techniques to finance, there is an additional plan to study pricing of electricity options, and its applications to pricing of electricity generators and transmissions, as an application of financial methods to operations research. Potential improvements in teaching and lecture notes that may result from the project are also suggested. If successful, the results of this research will lead to a better understanding of the risks and benefits of financial derivatives; and, the role of using financial derivatives, such as electricity options, interest rate derivatives, and currency options, to help U.S. utility and manufacture industries to reduce their financial risks. For example, commodity, foreign currency, and interest rate risks, particularly to hedge the financial risks in deregulated utility markets.
