Research is proposed in five areas. These are unified by a stochastic calculus methodology. (1) Callable convertible bonds. This is a two-person game, played by a bond-holder, who can convert the bond for stock, and the issuing firm, which can call the bond and thereby force conversion or surrender. (2) Contingent claim replication under portfolio constraints. A financial institution takes a position in assets which produce a risky cash flow, and wishes to offset this risk by taking opposing positions. However, there may be constraints on its actions, or the offsetting positions mandated by the model employed may be too unstable to actually implement. The effect of such constraints can be studied via a dual problem. (3) Passport options. A passport option entitles its owner to actively manage a fund, receive any profit which accrues, and be forgiven any loss which occurs. The institution selling this option must observe the trading activity of the option owner and solve a stochastic control problem to determine how to react to the owner's trading so as to be able to cover losses the owner incurs. (4) Credit derivatives. Financial institutions sell a variety of contracts designed to pay off in the event of default of corporate bonds. Default cannot be perfectly predicted, and this element of 'surprise' requires that models reach beyond the standard Brownian-motion framework. (5) Energy derivatives. Deregulation of natural gas and electricity has given impetus to derivative securities in energy markets, and models for such derivatives are under development.
The United States has a tenuous lead in the financial services industry in large part because of the whole-scale integration of mathematical modelling and computer technology into this industry. The maintenance of this lead requires continued fundamental research and redoubled technological educational efforts. This proposal is part of a long-term project to develop an educational and research enterprise which is doubly connected to the finance industry. The first connection is through the provision of human resources. The proposed work would support a successful interdisciplinary Ph.D. program in Mathematical Finance. Carnegie Mellon also has a professional Master's degree program in Computational Finance, and an undergraduate option in finance is under development. These programs have a substantial synergy. The second connection with industry is through research projects which use advanced mathematics on fundamental problems drawn from the industrial context. Among these are energy derivatives, which have become important with the deregulation of natural gas an electricity, and credit derivatives, which are a form of insurance against default of corporate bonds
