  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.
