When solving the American options with or without dividends, numerical methods often obtain lower convergence rates if further treatment is not implemented even using high-order schemes. In this article, we present a fast and explicit fourth-order compact scheme for solving the free boundary options. In particular, the early exercise features with the asset option and option sensitivity are computed based on a coupled of nonlinear PDEs with fixed boundaries for which a high order analytical approximation is obtained. Furthermore, we implement a new treatment at the left boundary by introducing a third-order Robin boundary condition. Rather than computing the optimal exercise boundary from the analytical approximation, we simply obtain it from the asset option based on the linear relationship at the left boundary. As such, a high order convergence rate can be achieved. We validate by examples that the improvement at the left boundary yields a fourth-order convergence rate without further implementation of mesh refinement, Rannacher time-stepping, and/or smoothing of the initial condition. Furthermore, we extensively compare, the performance of our present method with several 5(4) Runge-Kutta pairs and observe that Dormand and Prince and Bogacki and Shampine 5(4) pairs are faster and provide more accurate numerical solutions. Based on numerical results and comparison with other existing methods, we can validate that the present method is very fast and provides more accurate solutions with very coarse grids.
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