Pricing and Semimartingale Representations of Vulnerable Contingent Claims in Regime-Switching Markets

TL;DR

This paper introduces a novel Poisson series representation for pricing vulnerable contingent claims in regime-switching markets, enabling efficient computation and analysis of complex path-dependent and defaultable instruments.

Contribution

It develops a new Poisson series approach for pricing and representing vulnerable claims in regime-switching markets, including path-dependent and defaultable options.

Findings

Accurate pricing of defaultable bonds and barrier options.

Efficient computational method for path-dependent claims.

Rigorous derivation of semimartingale and Feynman-Kac representations.

Abstract

Using a suitable change of probability measure, we obtain a novel 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 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 pre-default price function of European vulnerable claims, which enables us to rigorously deduce Feynman-Kac representations for the pre-default pricing function and new semimartingale representations for the price process of the vulnerable claim under both risk-neutral and objective probability measures.