CDO tranche Pricing : Default probability

Question

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.

Adam.

Answer 1

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.