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 co
mpact 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.
We present a multigrid iterative algorithm for solving a system of coupled free boundary problems for pricing American put options with regime-switching. The algorithm is based on our recently developed compact finite difference scheme coupled with H
ermite interpolation for solving the coupled partial differential equations consisting of the asset option and the delta, gamma, and speed sensitivities. In the algorithm, we first use the Gauss-Seidel method as a smoother and then implement a multigrid strategy based on modified cycle (M-cycle) for solving our discretized equations. Hermite interpolation with Newton interpolatory divided difference (as the basis) is used in estimating the coupled asset, delta, gamma, and speed options in the set of equations. A numerical experiment is performed with the two- and four- regime examples and compared with other existing methods to validate the optimal strategy. Results show that this algorithm provides a fast and efficient tool for pricing American put options with regime-switching.
