# The condition of the integral representation of American Options

## Pascal Heider

#### Abstract

ABSTRACT

The pricing of an American option can be formulated as a free boundary problem for the Black–Scholes equation. If the free boundary is known, the option price can be evaluated by an integral representation involving the free boundary. Usually, the free boundary is approximated by some kind of numerical method, eg, as a solution of a suitable integral equation (Kim (1990); McKean (1965)) or a series expansion (Zhu (2006)). In this paper it is shown how the approximation error of the boundary influences the error in the option evaluation. A condition number κ depending only on the option parameters is introduced to measure this deviation.