The Black-Scholes equation, a partial differential equation commonly used in financial engineering to price various financial derivatives, does not admit an analytical solution for American put options. These options allow for early exercise, which leads to a free boundary problem, a nonlinear phenomenon that introduces inherent complexity, making the analysis particularly challenging. Some previous studies have employed approximation methods such as discretizing integral equations related to the early exercise boundary or approximating early exercise using barrier options. However, even the best results have absolute error of around \(10^{-4}\) , which are not sufficiently accurate. In this study, the integral term representing the early exercise premium is computed with high-precision by applying the double-exponential (DE) transformation with the composite trapezoidal rule (DE numerical integration formula). Furthermore, since the evaluation of the integral requires the function values at arbitrary discretization points, high-precision interpolation is carried out by combining the DE transformation with the Sinc approximation. For the function values required at arbitrary discretization points in the integral computation, high-precision interpolation is performed by combining the DE transformation with the Sinc approximation (DE-Sinc method). By applying these method to the numerical computation of the optimal early exercise boundary, we achieved an absolute error of approximately \(10^{-13}\) in double-precision floating-point arithmetic, demonstrating exceptionally high-precision numerical solutions.
