Erlangization/Canadization of Phase-Type Jump Diffusions, with Applications to Barrier Options

For a jump diffusion with upward phase-type jumps with p phases, the maximum before an independent Erlang time eηq with q stages and rate parameter η is again phase-type with q(p+1) phases. An iterative scheme for computing the phase generator is presented and applied to representing the price of a barrier option with time horizon eηq as a single ordinary integral. Canadization then means to approximate a fixed horizon T with an eηq satisying Eeηq=T for a sufficiently large q. Similar results holds for Greeks like the delta and the gamma. A numerical example is given for a down-and-in call option and the Canadization is combined with Richardson extrapolation. Finally, a recursion is developed that only requires the iteration to be performed in p+1 dimensions.