Dependent Defaults and Credit Migrations∗

The paper deals with the modeling of mutually dependent default times of several credit names through the intensity-based approach. We extend to the case of multiple ratings some models results due to Kusuoka (1999) and Jarrow and Yu (2001). The issue of the arbitrage valuation of related basket credit derivatives is also briefly examined. We argue that our approach permits in some cases to reduce significantly the dimensionality of the valuation problem at hand. 1 Dependent Intensities of Default Times In the case of mutually dependent defaults, it is natural to assume that the default probability of a certain entity increases as soon as a related firm defaults on its obligations. Within the socalled reduced-form approach to the modeling of credit risk, this kind of dependence is reflected in the jump of the default intensity of a given firm at the default time of another entity. This specific method of modeling mutually dependent default times was examined by, among others, Schmidt (1998), Kusuoka (1999), Jeanblanc and Rutkowski (2000, 2001), Kijima and Muromachi (2000), Schönbucher and Schubert (2000), and Jarrow and Yu (2001). We present few results, which generalize the valuation formulae for corporate bonds obtained in the recent paper by Jarrow and Yu (2001). We focus on the modeling of mutually dependent default times and credit migrations of several credit names through the intensity-based approach. Although most presented results remain valid for a finite number of reference entities, for the sake of expositional clarity, we shall first concentrate on the special case of two reference entities. Subsequently, we shall present results concerning the general set-up; that is, the case of several entities and multiple ratings. We shall argue that for some contracts the calculations can be reduced to the familiar results concerning the case of conditionally independent random times through a judicious choice of an equivalent probability measure. Finally, it should be stressed that in our approach the migration process can also be seen as a standard Markov chain (or a conditional Markov chain). However, for large pools of obligors the dimension of the state space of the corresponding joint migration process ∗Research supported by State Committee for Scientific Research (Komitet Badań Naukowych) Grant PBZ-KBN016/P03/1999. The research of Tomasz R. Bielecki was also partially supported by NSF Grants DMS-9971307 and DMS-0202851.