Compact Finite Difference Scheme with Hermite Interpolation for Pricing American Put Options Based on Regime Switching Model

Abstract

American put options with the regime-switching model is a system of coupled free boundary problems. In this study, we present an accurate finite difference method coupled with the Hermite interpolation for solving this system. To this end, we first employ the logarithmic transformation to map the free boundary for each regime to a fixed interval and then eliminate the first-order derivatives in the transformed model by taking derivatives to obtain a system of partial differential equations which we call the asset-delta-gamma-speed equations. We then discretize the system using the fourth-order compact scheme coupled with the Crank–Nicholson method. At the same time, the influence of other asset options and option sensitivities are estimated based on the third-order Hermite interpolation. As such, the overall scheme consists of four tridiagonal linear systems, which can be easily solved using the Thomas algorithm and the Gauss–Seidel iteration. The obtained scheme is then applied for the model with two, four, and sixteen regimes, respectively. Our results show that the scheme provides an accurate solution that is fast in computation as compared with other existing numerical methods.