On pricing barrier options with regime switching

We consider the valuation of both European-style and American-style barrier options in a Markovian, regime-switching, Black-Scholes-Merton economy, where the price process of an underlying risky asset is governed by a Markovian, regime-switching, geometric Brownian motion. Both the probabilistic and partial differential equation (PDE), approaches are used to price the barrier options. For the probabilistic approach to value a European-style barrier option, we employ the fundamental matrix solution and the Fourier transform space to derive a (semi)-analytical solution. The PDE approach is employed to value an American barrier option, where we obtain a system of free-boundary, coupled PDEs and an analytical quadratic approximation to the price by solving the free-boundary problem.