What is the difference between Base Correlation and Implied Correlation for a CDO tranche?
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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 $\rho_i(0,1)$.
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$\begingroup$ Can you please explain bit more the base correlation, as I understand it the super-senior tranche would have base correlation that "correctly prices" all tranches, so no need for any lower-tranche base correlations? What means "correctly price"? $\endgroup$ – emcor Sep 11 '14 at 7:31
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$\begingroup$ By "correctly price" I meant "matches the market price of". I'll update. $\endgroup$ – Brian B Sep 12 '14 at 15:29
