In this project a new set of mathematical tools for studying, in a unified manner, a wide class of free boundary problems that are of current interest in modern Finance will be developed. In particular, variational and viscosity solution methods will be adapted to treat the fundamental questions of existence and uniqueness that remain open for many of these problems. The free boundaries will be obtained as the solutions to a new set of integral and integro-differential equations whose structures are motivated by the original financial problem. A full spectrum of methods from mathematical analysis (functional, asymptotic and numerical) will be developed to study the solutions. Of special interest will be the development of accurate analytical approximations, and fast robust numerical schemes, for the location of the free boundaries. In many cases, basic properties of the free boundaries such as regularity, monotonicity, convexity, etc., are open. These issues will be addressed not only because they are of fundamental mathematical interest but also because they play a key role in making rigorous, and in simplifying, the proofs of the results mentioned above. Because of the wide variety of financial applications of interest in the equity, bond, mortgage, credit, and energy markets, special attention will be paid to ensure that the development of these methods will apply to all of the various stochastic processes that might arise as underliers: Uhlenbeck-Ornstein, geometric Brownian motion, mean reversion, jump-diffusion, stochastic volatility, variance gamma, etc.
The prototype, and most widely studied, free boundary in Finance is the early exercise boundary for an American put option; i.e., for any time before expiry of the option, the free boundary is the stock price below which it is optimal to exercise the option early. In spite of the enormous amount of attention already paid to this problem, the convexity of this early exercise boundary, and its use to obtain a global analytic estimate for its location, is still lacking when the stock pays a dividend. Also under study is the free boundary problem that arises in determining the optimal interest rate for prepaying a mortgage or for converting a variable rate mortgage to a fixed rate one. Existing strategies are based on using only today's yield curve whereas this free boundary approach incorporates one's anticipation of future yields. Not only will this provide a more accurate decision tool for individual mortgage holders but also lead to more accurate pricing in the mortgage backed securities (MBS) sector. Energy prices are modeled using jump-diffusion and stochastic volatility processes and the methods will be extended to this framework in order to price financial derivatives like options on forward contracts and swing options. Free boundaries arise in the credit sector in a completely different manner. A firm defaults when its total value decreases below a prescribed barrier. Matching a firm's time-dependent default barrier to the probability of default assigned to the firm by a rating agency leads to another type of free boundary problem for its default barrier. This provides a context for the rigorous comparison of default correlation among several firms in this value-of-firms model with the other, currently popular, copula approach and hence will lead to a better understanding and pricing of credit debt obligations (CDOs). By changing the underlying risky process to a pure jump or an extreme value model, a similar analysis will lead to a better understanding of the catastrophic risks associated with high impact / low probability weather or terrorist related events. A strategically important goal is to use this information to develop insurance products in this catastrophic insurance sector.
