Heath-Jarrow-Morton Type Models

Abstract

In the context of the modeling of the defaultable term structure, the HJM methodology was first examined by Jarrow and Turnbull (1995) and Duffie and Singleton (1999). Their studies were undertaken by Schönbucher (1996, 1998a), who has studied in a systematic way various forms of the no-arbitrage condition between the default-free and defaultable term structures. More recently, some of these results were re-discovered by Maksymiuk and Gątarek (1999) and Pugachevsky (1999), who focused on the arbitrage-free dynamics under the spot martingale measure of the instantaneous forward credit spreads. Subsequently, the HJM methodology was extended by Bielecki and Rutkowski (1999, 2000a, 2000b) and Schönbucher (2000a) to cover the cases of term structure models with multiple ratings for corporate bonds. Eberlein and Õzkan (2001) generalize this approach by considering models driven by Lévy motions (for related results, also see Eberlein and Raible (1999) and Eberlein (2001)). In contrast with models presented in the previous chapter, the credit migration process is not exogenously specified, but it is endogenous in a model. It follows a conditionally Markov process with respect to a reference filtration under the spot (or forward) martingale measure.