In this article, we study the numerical solutions of a class of complex partial differential equation (PDE) systems with free boundary conditions. This problem arises naturally in pricing American options with regime-switching, which adds significant complexity in the PDE systems due to regime coupling. Developing efficient numerical schemes will have important applications in computational finance. We propose a new method to solve the PDE systems by using a penalty method approach and an exponential time differencing scheme. First, the penalty method approach is applied to convert the free boundary value PDE system to a system of PDEs over a fixed rectangular region for the time and spatial variables. Then, a new exponential time differncing Crank-Nicolson (ETD-CN) method is used to solve the resulting PDE system. This ETD-CN scheme is shown to be second order convergent. We establish an upper bound condition for the time step size and prove that this ETD-CN scheme satisfies a discrete version of the positivity constraint for American option values. The ETD-CN scheme is compared numerically with a linearly implicit penalty method scheme and with a tree method. Numerical results are reported to illustrate the convergence of the new scheme.

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