cds - What is Base- vs. Implied Correlation of a CDO tranche?

What is the difference between Base Correlation and Implied Correlation for a CDO tranche?

Answer

An implied correlation $\rho_i(k_1,k_2)$ is a correlation that matches the $(k_1,k_2)$ tranche price $P_{k_1}^{k_2}$ (usually computed under a gaussian or student t copula)

$$C(k_1,k_2,\rho_i(k_1,k_2)) = P_{k_1}^{k_2}$$

For mezzanine tranches, there can sometimes be two different implied correlations matching the tranche price.

A base correlation $b_i(k_2)$ is a correlation that matches the price of the tranche, plus all higher-risk tranches "beneath" it, so we can write it as

$$b_i(k_2) = \rho_i(0,k_2)$$

where we obtain $P_{0}^{k_2}$ as

$$P_{0}^{k_2} = \sum_{k_i\leq{k_2}}P_{k_{i-1}}^{k_i}$$

The pricing function $C(0,k_2,\rho)$ is monotonic in $\rho$, hence the base correlation is unique. This allows practitioners to think about correlations a bit more like they previously thought about implied volatility (and volatility skews) for options.

The super-senior tranche has (trivially) a base correlation that matches the price of the entire underlying instrument, since it is $\rho_i(0,1)$.