Completeness of a Reduced-Form Credit Risk Model with Discontinuous Asset Prices
by Tomasz R. Bielecki of the Illinois Institute of Technology,
Monique Jeanblanc of the UniversitÚ d'╔vry Val d'Essonne, and
Marek Rutkowski of the University of New South Wales & Warsaw University of Technology
August 20, 2005
Introduction: The goal of this work is to examine the issue of attainability of a generic defaultable claim within the reduced-form approach to credit risk modeling, as well as a closely related issue of market completeness. In contrast to our previous work Bielecki et al. (2004a), we consider here the case where the prices of default-free assets and the pre-default values of defaultable assets follow general (that is, not necessarily continuous) semimartingales.
In Section 2, we concentrate on trading in primary assets with discontinuous prices. Our main goals is to analyze the issue of replication of a generic contingent claim by means a self-financing trading strategy that is additionally subject to an algebraic constraint, interpreted as the balance condition. We examine the relationships between the completeness of a market model with unconstrained strategies, and a corresponding market with strategies satisfying an exogenous constraint, referred to as the balance condition. Though in this section we do not deal with specific issues related to the modeling of credit risk, it is nevertheless essential for a full appreciation of results of Section 3. For the proofs of all results presented in Section 2, the interested reader is referred to Bielecki et al. (2004c), where completeness of a generic market model is studied. Let us only mention that the case of credit risk model has peculiarities, which do not show up explicitly when dealing with a general set-up.
In Section 3, we also include defaultable assets in our portfolio. In this case, our primary goal is to examine the issue of replication of a generic defaultable contingent claim. The crucial property of a replicating strategy is that has to satisfy a suitable version of the balance condition (see equation (17) in Section 3). Its role is to ensure that a portfolio of default-free and defaultable assets is continuously rebalanced in such a way that, in case of default occurring at some random time, the post-default wealth will match the recovery payoff of a hedged defaultable claim.
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