  In practical work with American put options, it is important to be able to
know when to exercise the option, and when not to do so. In computer simulation
based on the standard theory of geometric Brownian motion for simulating stock
price movements, this problem is fairly easy to handle for options with a short
lifespan, by analyzing binomial trees. It is considerably more challenging to
make the decision for American put options with long lifespan. In order to
provide a satisfactory analysis, we look at the corresponding free boundary
problem, and show that the free boundary -- which is the curve that separates
the two decisions, to exercise or not to -- has an asymptotic expansion, where
the coefficient of the main term is expressed as an integral in terms of the
free boundary. This raises the perspective that one could use numerical
simulation to approximate the integral and thus get an effective way to make
correct decisions for long life options.
