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Base and Implied Correlation for CDO Tranche Pricing

Article Quant Q&A · Author: emcor
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Summary

The document distinguishes tranche implied correlation from base correlation in collateralized debt obligation pricing. Implied correlation is the correlation input in a chosen copula model that reproduces the market price of one tranche. Depending on the tranche and pricing relationship, a mezzanine tranche can have more than one correlation value that matches its price.

Base correlation instead fits the price of the tranche together with all lower attachment tranches, treated as a combined portfolio extending from the first-loss point to the tranche’s detachment point. Since the cumulative pricing function is monotonic in correlation, this construction yields a unique base correlation. The answer notes that this makes base correlation useful for describing a correlation skew in a way analogous to implied volatility across option strikes. The super-senior tranche’s base correlation corresponds to the correlation that matches the price of the whole underlying portfolio. These measures depend on the chosen copula and pricing framework; the document does not discuss model calibration or the risks of interpreting correlation as directly observable.

Key ideas

  • Implied correlation is the model correlation that reproduces the price of a specific tranche.
  • A mezzanine tranche can sometimes have multiple implied correlations that fit its market price.
  • Base correlation fits the combined price of a tranche and all tranches with lower attachment points.
  • The monotonic cumulative pricing function makes base correlation unique in this construction.
  • The super-senior tranche’s base correlation matches the price of the entire underlying portfolio.

Tags

Fixed incomeDerivatives pricingStatistics
Full text
# What is Base- vs. Implied Correlation of a CDO tranche?


# What is Base- vs. Implied Correlation of a CDO tranche?












What is the difference between Base Correlation and Implied Correlation for a CDO tranche?

## Answer by Brian B (score 4, accepted)

https://quant.stackexchange.com/a/14699

An implied correlation $\rho_i(k_1,k_2)$ is a correlation that matches the $(k_1,k_2)$ tranche price $P_{k_1}^{k_2}$ (usually computed under a gaussian or student t copula)

$$ C(k_1,k_2,\rho_i(k_1,k_2)) = P_{k_1}^{k_2} $$

For mezzanine tranches, there can sometimes be two different implied correlations matching the tranche price.

A base correlation $b_i(k_2)$ is a correlation that matches the price of the tranche, plus all higher-risk tranches "beneath" it, so we can write it as

$$ b_i(k_2) = \rho_i(0,k_2) $$

where we obtain $P_{0}^{k_2}$ as $$ P_{0}^{k_2} = \sum_{k_i\leq{k_2}}P_{k_{i-1}}^{k_i} $$

The pricing function $C(0,k_2,\rho)$ is monotonic in $\rho$, hence the base correlation is unique. This allows practitioners to think about correlations a bit more like they previously thought about implied volatility (and volatility skews) for options.

The super-senior tranche has (trivially) a base correlation that matches the price of the entire underlying instrument, since it is $\rho_i(0,1)$.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.