An Adaptive and Explicit Fourth Order Runge-Kutta-Fehlberg Method Coupled with Compact Finite Differencing for Pricing American Put Options

Abstract: We propose an adaptive and explicit fourth-order Runge-Kutta-Fehlberg method coupled with a fourth-order compact scheme to solve the American put options problem. First, the free boundary problem is converted into a system of partial differential equations with a fixed domain by using logarithm transformation and taking additional derivatives. With the addition of an intermediate function with a fixed free boundary, a quadratic formula is derived to compute the velocity of the optimal exercise boundary analytically. Furthermore, we implement an extrapolation method to ensure that at least, a third-order accuracy in space is maintained at the boundary point when computing the optimal exercise boundary from its derivative. As such, it enables us to employ fourth-order spatial and temporal discretization with Dirichlet boundary conditions for obtaining the numerical solution of the asset option, option Greeks, and the optimal exercise boundary. The advantage of the Runge-Kutta-Fehlberg method is based on error control and the adjustment of the time step to maintain the error at a certain threshold. By comparing with some existing methods in the numerical experiment, it shows that the present method has a better performance in terms of computational speed and provides a more accurate solution.