Free boundary problems in mathematical finance

We provide a unified approach to studying a wide variety of free boundary problems that arise in modern mathematical finance. For the most part, the main ideas will be presented in the simplest case of the early exercise boundary for the American put option on a geometric Brownian motion. In addition to discussing the existence and uniqueness of the solution to the problem, and the convexity of the free boundary, we will describe several fast and accurate numerical and analytical approximations for the location of these early exercise boundaries. The same approach can be used to treat similar problems with more general underliers such as jump diffusion processes. We will also show how the techniques can be carried over to treat other classes of free boundary problems such as the inverse first crossing problem of the default barrier of a credit process as well as the pricing of mortgage prepayment options. Various parts of this work are joint efforts with Xinfu Chen (Pittsburgh) and David Saunders (Pittsburgh and Waterloo) as well as our recent Ph.D. students Lan Cheng, Ge Han and Dejun Xie.