By Debra J. Housel
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IG 0–3% 3–7% 7–10% 10–15% 15–30% 30–100% For the DJ-iTraxx and CDX pools, the equity tranche (A = 0, B = 3%) is quoted by means of the fair U0A,B , while assuming S0A,B = 500 bp. The reason for the equity tranche to be quoted as up-front is to reduce the counterparty credit risk that the protection seller is facing. All other tranches are generally quoted by means of the fair running spread S0A,B , assuming no up-front fee (U0A,B = 0). Following the recent market turmoil the 3–6% and the 3–7% have also been quoting in terms of an up-front amount and a running S0A,B = 500 bp given the exceptional risk also priced by the market for mezzanine tranches.
S: Systemic factor affecting default times of all names in onefactor copula models. In the Gaussian Copula model it is a Gaussian random variable, while in the Double-t Copula model it is a Student t-variable. • Y : Idiosyncratic factor affecting just one name in one-factor copula models. In the Gaussian Copula model it is Gaussian random vari able, while in the Double-t Copula model it is a Student t-variable. • X : Random variable entering the deﬁnition of copula models. For example, in the Gaussian copula, for each name i we get: X i := �−1 1 − e−�i (τi ) If we consider a one-factor version of the model, then the following equation relates the systemic and idiosyncratic factors to the copula with ρi as correlation parameter for the ith name of the pool: Xi = √ ρi S + 1 − ρi Yi Notation and List of Symbols xxix Dynamic loss models • M : Inverse of the minimum jump for the pool’s loss process.
Such a procedure cannot easily be extended to a fully dynamical model in general. We cannot do justice to the huge copula literature in credit derivatives here; we only mention that there have been at tempts to go beyond the Gaussian Copula introduced in the CDO world by Li (2000) and leading to the implied (base and compound) correlation framework, some important limits of which have been pointed out in Torresetti et al. (2006b). Li and Hong Liang (2005) also proposed a mixture approach in connection with CDO squared.