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 systemof 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.
We consider a system of coupled free boundary problems for pricing American put options with regime-switching. To solve this system, we first employ the logarithmic transformation to map the free boundary for each regime to multi-fixed intervals and then eliminate the first-order derivative in the transformed model by taking derivatives to obtain a system of partial differential equations which we call the asset-delta-gamma-speed equations. As such, the fourth-order compact finite difference scheme can be used for solving this system. The influence of other asset, delta, gamma , and speed options in the present regime is estimated based on Hermite interpolations. Finally, the numerical method is tested with several examples. Our results show that the scheme provides an accurate solution that is fast in computation as compared with other existing numerical methods.
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