Reading about CDOs and calibration to find the implied correlation, I came up with the following question.
Suppose we are pricing a CDO over a pool of $N=125$ names, using the usual Gaussian copula structure, with constant recovery rate, constant hazard rates and a unique value $\rho$ for the default times of the 125 names in the pool. This CDO has standard tranches $(l_k, u_k)$, with $k=1,\ldots n$. This is how the implied correlation algorithm is explained in several articles I read:
(1) The market quotes thefair spread $x^*_k$ for each CDO tranche, $k=1,\ldots n$.
(2) We can then plug these values $x^*_k$ in an algorithm and calculate an implied (base) correlation parameter $\rho_k$ corresponding to each tranche $(l_k, u_k)$.
Question: Now that I have found each tranche correlation $\rho_k$, how do I "use" it? What's its purpose? I mean, shall I use this parameter $\rho_k$ to price (for example) other CDOs on the same pool of 125 names, perhaps with different maturities/premium dates? Or what else?
My question came up because when we determine the implied volatility out of a set of vanilla call options, we shall use those implied volatilities to price more complex instruments (on the same underlying) having equal strike/expiry: then, implied volatility is "extracted" from one class of instruments (vanilla call options) to be "used" in other classes of instruments (e.g., exotic options, etc.).
But here? Do we also use implied correlation for pricing/evaluating any other imstruments? Which ones, exactly? How?
So, what is exactly the use of implied correlation of a CDO?? Thanks in advance for your help.