We consider a system of coupled free boundary problems for pricing American put options with regime-switching. To solve this system, we first fix the optimal exercise boundary for each regime resulting in multi-variable fixed domains. We further eliminate the first-order derivatives associated with the regime-switching model by taking derivatives to obtain a system of coupled partial differential equations which we called the asset-delta-gamma-speed option equations. The fourth-order compact finite difference scheme is then employed in each regime for solving the system of the equations. In particular, the performance of cubic and quintic Hermite and cubic spline interpolation is explored in estimating the coupled asset, delta, gamma and speed options in the set of equations. The numerical method is finally tested with several examples. Our results show that the scheme provides an accurate solution with the convergent rate of 3.7 which is very fast in computation as compared with other existing numerical methods.
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