We propose an adaptive and explicit 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.

我们提出了一种自适应和显式的 Runge-Kutta-Fehlberg 方法，结合四阶紧凑方案来解决美式看跌期权问题。首先，通过对数变换和附加导数，将自由边界问题转化为具有固定域的偏微分方程组。通过增加一个具有固定自由边界的中间函数，推导出一个二次公式来解析计算最优运动边界的速度。此外，我们实施了一种外推方法，以确保在从其导数计算最佳运动边界时，至少在边界点处保持空间的三阶精度。因此，它使我们能够使用带狄利克雷边界条件的四阶空间和时间离散化来获得资产期权、期权希腊人和最优行使边界的数值解。Runge-Kutta-Fehlberg 方法的优点是基于误差控制和时间步长的调整，以将误差保持在一定的阈值。在数值实验中与现有的一些方法进行对比，表明该方法在计算速度方面具有更好的性能，并提供了更准确的解。