  In this paper, an integral equation representation for the early exercise
boundary of an American option contract is considered. Thus far, a number of
different techniques have been proposed in the literature to obtain a variety
of integral equation forms for the early exercise boundary, all starting from
the Black-Scholes partial differential equation. We first present a coherent
categorization of exiting integral equation methodologies in the American
option pricing literature. In the reminder and based on the fact that the early
exercise boundary satisfies a fully nonlinear weakly singular non-standard
Volterra integral equation, we propose a product integration approach based on
linear barycentric rational interpolation to solve the problem. The price of
the option will then be computed using the obtained approximation of the early
exercise boundary and a barycentric rational quadrature. The convergence of the
approximation scheme will also be analyzed. Finally, some numerical experiments
based on the introduced method are presented and compared to some exiting
approaches.
