An American put option is a derivative financial instrument that
gives its holder the right but not the obligation to sell an
underlying security at a pre-determined price. American options
may be exercised at any time prior to expiry at the discretion of
the holder, and the decision as to whether or not to exercise
leads to a free boundary problem. In this paper, we examine the
behavior of the free boundary close to expiry. Working directly
with the underlying PDE, by using asymptotic expansions, we are
able to deduce this behavior of the boundary in this limit.