On barrier option pricing by Erlangization in a regime-switching model with jumps

We consider the risk-neutral pricing of vanilla, digital and down-and-out call options when the underlying asset price evolves like the exponential of a Markov-modulated Brownian motion (MMBM) with two-sided phase-type jumps. The price of such options is intimately related to the first passage properties of the MMBM. To analyse these first passages, we randomize the time horizon using Erlang distributions with suitable parameters and apply matrix-analytic methods. This provides us with closed form approximations of the options prices, with a very high precision, as shown by several numerical illustrations. In particular, we consider an example in which the phase-type jump distribution is constructed in such a way that it mimics fat tails.