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Calibrating Default Intensities from CDO and CDS Market Quotes

Article Quant Q&A · Author: AdamElKaroui
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Summary

This discussion concerns calibrating the default probability input to a collateralized debt obligation model using a one-factor Gaussian copula and the large homogeneous portfolio approximation. In that setup, expected tranche loss depends on the reference name’s default probability, which the question represents through a hazard rate. The response says market quotes across maturities, including spreads and upfront payments, can be used to infer the intensity curve when paired with a specified pricing model.

It distinguishes this curve-based calibration from applying a simple spread-to-intensity approximation directly. The quotes are inputs to a model such as the standard market model, and the inferred hazard rates depend on that model’s assumptions. The note provides no worked calibration, quote data, or treatment of recovery uncertainty, and the suggested reference is only mentioned rather than explained.

Key ideas

  • Expected tranche loss in the stated portfolio approximation depends on the reference names’ default probabilities.
  • Market spreads and upfront quotes at multiple maturities can constrain the default intensity curve.
  • Backing out hazard rates requires a pricing model that maps intensities to quoted prices.
  • The simple CDS spread approximation is not presented as a complete CDO calibration method.

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Fixed incomeDerivatives pricingStatistics
Full text
# CDO tranche Pricing : Default probability


# 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.

Adam.

## Answer by Olórin (score 0, accepted)

https://quant.stackexchange.com/a/42644

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.

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