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Quantitative Risk Management / Portfolio Credit Risk and Credit Derivatives

CDOs, Credit Indices, and the Gaussian Copula Correlation Smile

In May 2005, a sudden reversal in correlation expectations wiped out hedge funds that had sold protection on mezzanine tranches of the CDX index and bought protection on the equity tranche — a trade believed to be correlation-neutral. The episode, known as the "correlation crisis," revealed a fundamental flaw: the Gaussian factor copula model used to price CDO tranches could not simultaneously calibrate to all tranches in the capital structure. The correlation smile — the V-shaped pattern of implied asset correlations across tranche attachment points — was the market's way of saying that a single correlation number is insufficient.

This note covers the tools you need to understand that failure:

  1. CDO tranching and tranche loss: how the portfolio loss LtL_t is sliced into layers with attachment points (l,u)(l, u), and why the tranche loss Lt[l,u]=(Lt−lm)+−(Lt−um)+L^{[l,u]}_t = (L_t - lm)^+ - (L_t - um)^+ is a call spread.
  2. Credit index swaps and single-tranche CDOs: the standardised iTraxx/CDX products, their fair spreads, and the key result that CDO tranche spreads reduce to option prices on the portfolio loss.
  3. Factor copula models: the Gaussian factor copula (Li 2000), simulation via random thresholds, and the two-stage calibration separating marginal hazard rates (from single-name CDS) from copula parameters (from index tranches).
  4. The correlation smile and its aftermath: implied tranche correlation, base correlation, and why the Gaussian copula failed for super-senior tranches — the mechanism behind the 2008 CDO collapse.
Learning objectives. By the end of this note you should be able to:
  • define a CDO tranche, compute its notional and loss from the portfolio cumulative loss;
  • write the fair index spread as a weighted average of single-name CDS spreads;
  • state the copula model Definition 12.1 and describe the simulation Algorithm 12.2;
  • explain the two-stage calibration of a factor copula model;
  • define implied tranche correlation and base correlation, and explain the correlation smile;
  • articulate why the Gaussian copula systematically underprices super-senior tail risk.

In this lesson

    1. CDO structure and tranche loss
    1. Credit index swaps and fair spreads
    1. Factor copula models for default times
    1. The correlation smile and the 2008 failure

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