美式看跌期权定价的自适应显式四阶Runge-Kutta-Fehlberg紧差分方法
An Adaptive and Explicit Fourth Order Runge-Kutta-Fehlberg Method Coupled with Compact Finite Differencing for Pricing American Put Options
摘要：
本文提出了一种自适应显式四阶Runge-Kutta-Fehlberg方法和一种四阶紧致格式来求解美式看跌期权问题。首先，利用对数变换和附加导数，将自由边界问题转化为固定区域的偏微分方程组。通过增加一个中间函数，并在固定的自由边界上，推导出一个二次公式来解析地计算最佳运动边界的速度。此外，我们实现了外推法，以确保至少，在空间中的三阶精度保持在边界点时，计算最佳的行使边界从其衍生物。因此，它使我们能够采用四阶空间和时间离散Dirichlet边界条件获得的资产期权，期权希腊人，和最佳的行使边界的数值解。Runge-Kutta-Fehlberg方法的优点是基于误差控制和时间步长的调整，以将误差保持在某个阈值。数值实验结果表明，该方法在计算速度和精度上都优于现有方法。
英文摘要：
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.
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