Solving Complex PDE Systems for Pricing American Options with Regime-Switching by Efficient Exponential Time Differencing Schemes

In this paper we study the numerical solutions of a class of complex PDE systems with free boundary conditions. This problem arises naturally in pricing American options with regimeswitching, 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 differencing Crank-Nicolson (ETD-CN) method is employed 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. Nu-