# CDO tranche Pricing : Default probability

I'm trying to compute the price of a CDO at any time, using the one factor gaussian copula and the Large Homogenous Portfolio Approximation. You can find the CDO pricing formula in M. Neugebauer (2007): A comprehensive Analysis of Advanced Pricing Models for Collateralised Debt Obligations here.

As you can see in the formula $(2.13)$ page 9, the only difficulty is the computation of the expected tranche loss $ELT(t)$, but you can find a closed form (thanks to the one factor gaussian copula model and the large homogenous approximation), it is written in page 16 (equation $(2.37)$). We can see that $ELT(t)$ depends on the default probability of the reference name $P(\tau \le t)$, with $\tau$ being the default time of the reference name. If we assume that $\tau$ has an exponential distribution with hazard rate function $\lambda(t)$, then $P(\tau \le t) = 1 - e^{-\lambda (t)*t}$.

My question is : how do you calibrate $\lambda (t)$ ? I know that for CDSs, we have that $\lambda \simeq \frac{s_0}{1-RR}$ with $s_0$ the contractual spread, and $RR$ the recovery rate (and at time $t$, you have $\lambda_t = s_t / 1-RR$ with $s_t$ the market spread at time $t$). How does it work in our case? Thank you.

You have quotes, on the market, usually for the 6m, 1y, 2y, 3y, 4y, 5y, 7y, 10y, 15y, 20y, 30y tenors, of quoted spreads / upfront that allow for backing out $$\lambda$$, provided you have a model on $$\lambda$$, like the Standard (ISDA/JPM) model. As far as I remember there is an open gamma paper on this question.