Abstract:
We study the free boundary problem of the American type of options. We consider
a continuous dividend paying put option and provide much simpler way of
approximating the option payo and value. The essence of this study is to apply
geometric techniques to approximate option values in the exercise boundary.
This, being done with the nature of the exercise boundary in mind, more accurate
results are guaranteed. We de ne a transformation (map) from a unit square to
the free boundary. We then examine the transformation and its properties. We
take a linear case for a transformation as well as a nonlinear case which would be
more tting for option values. We consider stochasticity (an Ito process) as we
de ne this transformation and this yields better approximations for option values
and payo s. We also numerically compute optimal option prices using the same
transformation. We nally demonstrate that our transformation performs better
than most semi-analytic results.
