University of Michigan

**Abstract**: We will discuss the free boundary problem
associated with the American option pricing problem for jump
diffusions. Because of the jump in the underlying process, this problem is a
free boundary problem for parabolic integro-differential equations. Using the
tools in proving regularity results for parabolic differential equations and
especially the Stefan problem, we will show that the optimal exercise boundary/
free boundary of the American option is continuously differentiable (except at
the maturity). Moreover, we will upgrade its regularity and show it is
infinitely differentiable under an appropriate regularity assumption on the
jump distribution. Our regularity result improves the result of Yang, Jiang and
Bian in 2006 by removing a technical assumption on the parameters. This is a
joint work with Prof. Erhan Bayraktar.