We consider the problem of modelling the term structure of bonds subject to default risk, under minimal assumptions on the default time. In particular, we do not assume the existence of a default intensity and we therefore allow for the possibility of default at predictable times. It turns out that this requires the introduction of an additional term to the forward-rate approach by Heath, Jarrow and Morton (1992). This term is driven by a random measure encoding information about those times where default can happen with positive probability. In this framework, we derive necessary and sufficient conditions for a reference probability measure to be a local martingale measure for the large financial market of credit risky bonds, also considering general recovery schemes. To this end, we establish a new Fubini theorem with respect to a random measure by means of enlargement of filtrations techniques.
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