On the Location of a Free Boundary for American Options

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.