Pricing and semimartingale representations of vulnerable contingent claims in regime-switching markets

Using a suitable change of probability measure, we obtain a Poisson series representation for the arbitrage-free price process of vulnerable contingent claims in a regime-switching market driven by an underlying continuous-time Markov process. As a result of this representation, along with a short-time asymptotic expansion of the claim's price process, we develop an efficient novel method for pricing claims whose payoffs may depend on the full path of the underlying Markov chain. The proposed approach is applied to price not only simple European claims such as defaultable bonds, but also a new type of path-dependent claims that we term self-decomposable, as well as the important class of vulnerable call and put options on a stock. We provide a detailed error analysis and illustrate the accuracy and computational complexity of our method on several market traded instruments, such as defaultable bond prices, barrier options, and vulnerable call options. Using again our Poisson series representation, we show differentiability in time of the predefault price function of European vulnerable claims, which enables us to rigorously deduce Feynman-Kač representations for the predefault pricing function and new semimartingale representations for the price process of the vulnerable claim under both risk-neutral and objective probability measures.