An introduction to pricing correlation products using a pair-wise correlation matrix

The pricing of synthetic collateralized debt obligations and other correlation products is an evolving area of credit derivatives markets that has yet to be fully resolved. One of the biggest weaknesses of the current one-factor copula pricing model is its unrealistic and problematic capture of corporate default-time correlation. Problems such as multiple implied correlations and correlation skews introduced by the current one-factor pricing model have been partially addressed by "model work-arounds" (eg, base correlation), but the underlying issue of using a single default-time correlation factor among a pool of numerous corporate credits still needs to be permanently and fully resolved. This paper evaluates a pricing model using a full pair-wise correlation matrix based on historical asset correlations. That is, instead of using one correlation factor an entire correlation matrix is used. This allows for both unique pair-wise correlations between any two reference entities and even negative correlations. The model's prices are compared with that of the market in the hope of identifying mispriced (ie, cheap or expensive) synthetic collateralized debt obligation tranches. The model algorithm and mechanics are presented first. In particular the construction and heterogeneity of the full correlation matrix are examined in detail. Next, the model's prices are compared with real tranche prices obtained from the synthetic collateralized debt obligation market. Finally, the correlation skew implied by the full matrix model is compared with that of the market.