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The American-style option pricing problem is can be phrased as an evolutionary variational inequality, a parabolic obstacle problem, or a parabolic free boundary value problem. Problems in finance are often especially challenging because the PDE is degenerate and the "obstacle function" is only Lipschitz continuous rather than smooth.

Elliptic and parabolic partial differential equations (PDEs) arising in option pricing problems involving "degenerate" stochastic diffusion processes are well-known to be "degenerate" elliptic or parabolic. A diffusion process can often be viewed as the solution to a stochastic ordinary differential equation, which is like an ordinary differential equation, except that it includes a term involving Brownian motion. A degenerate diffusion process is one where the coefficient matrix for the Brownian motion term may become zero, while a degenerate elliptic or parabolic PDE is one where the coefficient matrix of the second-order spatial derivatives may become zero. (These two concepts of degeneracy are closely related.)

We shall first illustrate the nature of American-style option pricing problems with a simple example which can be solved explicitly by elementary methods (calculus). The example will serve to motivate the kinds of questions we hope to answer in greater generality in more realistic situations. In particular, we shall describe some of our work on the existence, uniqueness, and regularity questions for stationary and evolutionary variational equalities and inequalities (obstacle problems) involving degenerate elliptic and parabolic differential operators and applications to the American-style option pricing problem. This is joint work with Panagiota Daskalopoulos (Professor of Mathematics, Columbia University) and Camelia Pop (Ph.D. Student in Mathematics, Rutgers University). The problem of developing numerical solutions to these obstacle problems (which we shall not have time to discuss) is joint work with Eduardo Osorio (Ph.D. Student in Mathematics, Rutgers University).

These topics will be discussed in more depth in my Fall 2011 course:

Math 16:642:611 - Topics in Applied Mathematics: Variational Inequalities, Obstacle, and Free Boundary Problems in Mathematical Finance Prerequisites: First-year graduate sequence on Real Analysis (for example, Math 640:501-502). Co-requisites: Partial Differential Equations (for example, Math 640:517 or a similar course based on the text by Evans) is recommended, but not required and the course will be self-contained in order to accommodate beginning second-year students exploring potential research topics

Speaker: Paul Feehan, Rutgers University

Slides: ( pdf)