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Implied Correlation The implied correlations for the CDO index tranches are the correlations backed out by the market quoted prices using CDO valuation models available in the credit library, namely, the Poisson model and the Normal Copula model. Because there is no explicit solution for the implied correlation and the Monte Carlo simulation is employed to solve this inverse problem, it will be time consuming to run a root finding procedure by iteration. Hence a linear interpolation approximation is used instead. The model serves the purpose of calculating the implied correlations for standard index collateral debt obligation (CDO) tranches. In the market the credit default swaps (CDS) indexes CDX and Trac-X portfolios have been introduced and the standard tranches linked to these reference sets have also started to be actively quoted. Given the market price of each tranche, which is either quoted as a b/e spread or a 5% fee payment plus an upfront payment, we could back out an implied correlation value using CDO valuation models. Because there is no explicit solution for correlation and we employ Monte Carlo simulation (MC) for CDO valuation in model, it is very time consuming to run root finding by iteration. It is also impossible to get an accurate solution due to the presence of MC noise. Hence in the submitted model each tranche is first valued at various constant correlations ranging from 0 to 1 with interval 0.05. The implied correlation for each tranche is then found through linear interpolation by matching the calculated value with the market price. The main approximation of this methodology is the linear interpolation. As shown in the next section, the tranche value is not a linear function of correlation, especially for mezzanine tranches. Therefore, a non-linearity error will occur. However, this error could be reduced with increased granularity. Hence the main target is to assess if the correlation change interval (0.05 in the submitted model) is adequate. It has been known that there exist duel solutions of the implied correlation for the mezzanine tranches. In the submitted model, the lower of the two correlations is being used in all cases, which is consistent with market practice. However, this will cause a problem when the implied correlation is used to find the correlation information of a non-standard tranche. A more reasonable approach to avoid dual solutions, namely base correlation methodology, has been proposed and quickly become a market standard.
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