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When solving the American options with or without dividends, numericalmethods often obtain lower convergence rates if further treatment is notimplemented even using high-order schemes. In this article, we present a fastand explicit fourth-order compact scheme for solving the free boundary options.In particular, the early exercise features with the asset option and optionsensitivity are computed based on a coupled of nonlinear PDEs with fixedboundaries for which a high order analytical approximation is obtained.Furthermore, we implement a new treatment at the left boundary by introducing athird-order Robin boundary condition. Rather than computing the optimalexercise boundary from the analytical approximation, we simply obtain it fromthe asset option based on the linear relationship at the left boundary. Assuch, a high order convergence rate can be achieved. We validate by examplesthat the improvement at the left boundary yields a fourth-order convergencerate without further implementation of mesh refinement, Rannachertime-stepping, and/or smoothing of the initial condition. Furthermore, weextensively compare, the performance of our present method with several 5(4)Runge-Kutta pairs and observe that Dormand and Prince and Bogacki and Shampine5(4) pairs are faster and provide more accurate numerical solutions. Based onnumerical results and comparison with other existing methods, we can validatethat the present method is very fast and provides more accurate solutions withvery coarse grids.

On the Efficiency of 5(4) RK-Embedded Pairs with High Order Compact Scheme and Robin Boundary Condition for Options Valuation