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

TL;DR

This paper introduces an adaptive, explicit fourth-order Runge-Kutta-Fehlberg method combined with a compact finite difference scheme to efficiently and accurately solve American put options pricing problems, including free boundary determination.

Contribution

It develops a novel numerical approach that integrates adaptive time-stepping with high-order spatial discretization for American options pricing, improving accuracy and computational speed.

Findings

The method achieves higher accuracy in option pricing and boundary computation.

It outperforms existing methods in computational speed.

The approach maintains at least third-order spatial accuracy at boundaries.

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.