Transform Methods for the Solution of the American Put Option

 

Abstract

The most common of the Exotic Options is the American Put which allows early exercise. The property of early exercise makes the option path dependent. This dependency corresponds to the problem of identifying the free or moving boundary of the associated second order PDE which describes the option and its constraints. It has been shown that within the Black-Scholes framework it is possible to identify the free boundary as the solution to a particular saltus problem (Kolodner, 1956), and that either by Forier Transform Methods (McKean, 1965) or by the Volume Potential Method (Jacka, 1989, Jamshidian, 1990), this results in a recursive second order Volterra integral equation.

There are no known methods for solving such integral equations unless one has a model for the structure of the free boundary. Using the near expiry models suggested by Carr et al(1992), Barles(1995), Wilmott et al(1997) and Kruske and Keller(1998), we propose a single parameter family of curves for the moving boundary. It will be shown that this is capable of a quick numerical solution using either Newton-Raphson or interpolation methods and that the results are within penny accuracy of the binomial tree and finite difference methods, although the latter require on average 100 seconds for similar accuracy.