Gaussian copula

In the one-factor form, the variable of name $i$ is $X_i = \sqrt{\rho}\, M + \sqrt{1-\rho}\, Z_i$. $M$ is a factor common to all names and $Z_i$ is a name-specific factor, both independent standard normal variables¹. The lower $X_i$ is, the more likely name $i$ is to default, and given $M$ the names are independent.

Consider a hypothetical example with 100 names and $\rho = 0.3$. If the common factor $M$ is −2, each name's 5-year default probability rises from 9.5% to 39.9%, and if $M$ is 2 it falls to 0.2%.

In John Hull's February 2005 presentation, the market was then valuing index tranches with a Gaussian copula that had the same correlation between every pair of names². David X. Li's presentation calls it the Normal copula³. The article covers the valuation steps and the limits.

Notes

1. Damiano Brigo, Andrea Pallavicini and Roberto Torresetti, "Credit Models and the Crisis, or: How I learned to stop worrying and love the CDOs" (arXiv:0912.5427v3, February 2010), section 3.1 (page 22). ↩

2. John Hull, "Credit Derivatives Markets" (presentation, February 7, 2005), slide 12. ↩

3. David X. Li's presentation (December 8, 2000), slide 19. ↩