A Note on the Correlation Smile
Svenja Hager and Rainer Schöbel ∗
First Version: May 2006
This Version: December 2006
The correct modeling of default dependence is essential for the valuation of multi-
name credit derivatives. However for the pricing of synthetic CDOs a one-factor
Gaussian copula model with constant and equal pairwise correlations, default
intensities and recovery rates for all assets in the reference portfolio has become
the standard market model. If this model were a reflection of market opinion
there wouldn’t be the implied correlation smile that is observed in the market.
The purpose of this paper is to explain the structure of the smile by discussing
the influence of different correlation matrices on CDO spreads.
JEL Classification: G13
Keywords: default risk, CDOs, implied correlation smile, correlation matrix, hetero-
In 2003, default swaps on the Trac-X and Dow Jones iBoxx portfolios were introduced.
Tranches linked to these reference sets also started to be actively quoted. A more
liquid and transparent market for tranched credit risk evolved from this portfolio stan-
dardization. The recent merger of the Trac-X and Dow Jones iBoxx to the Dow Jones
iTraxx in Europe and the Dow Jones CDX in the US drove this process even further.
The new liquidity of the market led increasingly to the quotation of tranched products
in terms of an implied correlation parameter instead of the quotation in terms of the
spread or the price of a tranche. This practice is well known from the use of implied
volatilities in options markets. The implied compound correlation of a tranche is the
uniform asset correlation that makes the tranche spread computed by the standard
market model equal to its observed market spread. The standard market model is a
Gaussian copula model that uses only one single parameter to summarize all correla-
tions among the various borrowers’ default times. Moreover the model assumes that
∗Department of Corporate Finance, Faculty of Economics and Business