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 \(e^q_\eta\) $e_{\eta }^q$ with q stages and rate parameter \(\eta\) $\eta$ is again phase-type with \(q(p+1)\) $q(p+1)$ phases. An iterative scheme for computing the phase generator is presented and applied to representing the price of a barrier option with time horizon \(e^q_\eta\) $e_{\eta }^q$ as a single ordinary integral. Canadization then means to approximate a fixed horizon T with an \(e^q_\eta\) $e_{\eta }^q$ satisying \({\mathbb {E}}e^q_\eta =T\) $\mathbb{E}{e}_{\eta }^q=T$ 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 \(p+1\) $p+1$ dimensions.