Given the individual distribution functions F1,…,FmF_1, \dots, F_m and a copula CC, the joint distribution function is F(x1,…,xm)=C(F1(x1),…,Fm(xm))F(x_1,\dots,x_m) = C(F_1(x_1),\dots,F_m(x_m))1. The dependence can be replaced without changing the individual distributions.

In credit, the distribution of each name's time to default is taken from its CDS, and the copula links the names2. In a hypothetical example, even if five names have the same distribution, a higher copula correlation makes it more likely that they default in the same period.

The form of the copula (the Gaussian copula, for example) changes how likely simultaneous defaults are. The choice of form and of correlation is an assumption of the valuation.

Notes

  1. David X. Li, "The Valuation of Basket Credit Derivatives: A Copula Function Approach" (Fields Institute presentation, December 8, 2000), slides 17–18. ↩

  2. Damiano Brigo, Andrea Pallavicini and Roberto Torresetti, "Credit Models and the Crisis" (arXiv:0912.5427v3, February 2010), section 3 (page 20). ↩