Implied Correlations: Smiles or Smirks? Şenay Ağca George Washington University Deepak Agrawal Diversified Credit Investments Saiyid Islam Standard & Poor’s This version: Aug 15, 2007 . Abstract With standardized collateralized debt obligation (CDO) tranches on credit default swap (CDS) indices trading very actively, the concept of implied correlation and correlation trading have gained popularity in recent times. In this paper, we examine the commonly observed ‘implied correlation smile’ pattern that is oftentimes interpreted to signal relative value opportunities across tranches. We investigate whether implied correlation smile can arise as a result of model mis-specifications in the industry standard one factor Gaussian copula default time model rather than relative mis-pricing across tranches. Our evidence shows that empirical features like fat tails in return distributions, heterogeneous pair-wise correlations, heterogeneous spreads, and correlation between default probabilities and recovery rates can give rise to smile patterns in implied correlations even when all the tranches are fairly priced. The standard Gaussian copula model assumes away these features and is thus mis-specified. Our results suggest that implied correlations computed from this model are not very useful to determine relative mis-pricing across tranches. On the other hand, we also show that one factor Gaussian copula model and implied correlations obtained from this model perform well for pricing a given tranche across time, even in periods of market turmoil related to credit risk events. JEL Classification: G 12, G 13 Key words: Collateralized Debt Obligations, Gaussian Copula, Implied Correlation Smile Şenay Ağca is at George Washington University, Department of Finance, School of Business, 2201 G Street NW, Funger Hall Room 505, Washington, DC, 20052, Ph: (202) 994-9209, Fax: (202) 994-5017, Email: [email protected]; Deepak Agrawal is at Diversified Credit Investments, 201 Spear Street, Suite 250, San Francisco, CA 94105, Ph: (415) 321-7428, Email: [email protected]; Saiyid Islam is at Standard & Poor’s, 55 Water Street, New York, NY 10007, Ph: (212) 438-3453, Email: [email protected]. This study was initiated while Deepak Agrawal and Saiyid Islam were at KMV. We acknowledge KMV’s support in providing some of the data for this study. We would also like to thank Don Chance, Anurag Gupta, Bill Morokoff, Yim Lee and conference participants at the Hedge Funds World Conference, New York, 2005, Advanced Correlation Modeling Conference, New York, 2005, and seminar participants at the FDIC for helpful discussions. Any remaining errors are our responsibility. The views expressed in this paper are the authors' own and do not necessarily represent the views of Standard & Poor’s or Diversified Credit Investments.2 1. Introduction Recent developments in credit markets have led to a renewed focus on correlations among credits. A number of securities whose values depend on correlations among a set of credits have started trading actively. The most popular ones include collateralized debt obligation (CDO) tranches and basket default swaps. Correlation affects the values of these securities because of its pronounced impact on the shape of the default loss distribution of credit portfolios that underlie these securities. One of the most important drivers of active trading of such correlation dependent securities has been the rapid standardization in credit derivatives market. CDS indices like the ‘Dow Jones CDX North America Investment Grade (DJ.CDX.NA.IG)’ and ‘iTraxx Europe’ are standardized, equally-weighted, tradable portfolios of credit default swaps that have high liquidity. A further development has been the standardization of CDO tranches on standardized indices, which trade very actively1. With active trading of these tranches, market participants have the opportunity to trade correlations, just like option traders talk about trading volatility when they trade options. Borrowing further from the analogy of ‘implied volatility’ in the options market, it has become commonplace to talk about 'implied correlation' to refer to correlation extracted from the observed tranche prices using a simple one-factor Gaussian copula model. Furthermore, market participants rely on these implied correlations to propose relative value strategies among tranches.2 To infer implied correlations from observed CDO tranche prices, one needs a standard model for valuing CDOs similar to the Black and Scholes (1973) model (Black-Scholes model hereafter) that is used to obtain implied volatilities in the options market. It has become an industry practice to use a simple, one-factor Gaussian copula model first introduced by Li (2000)3 to obtain implied correlations from CDO prices. This model further assumes that all pair-wise correlations among the credits underlying a CDO tranche are homogeneous. 1 Amato and Gyntelberg (2005) provide an overview of how standardization has been one of the main drivers of active trading in CDS indices and index tranches markets. 2 For example, an article in December 2004 issue of Credit magazine suggests that one can infer relative mis-pricing across tranches by looking at the implied correlation chart. It says “investors are not being properly compensated for the risk of buying equity and junior mezzanine tranches…. The implied correlation smile shows the current strength of demand for equity and mezzanine tranches.” The article goes on to “…advising clients to take advantage of the richness of the 3-6% tranche in particular, either to express a bearish view on the market or as a cheap hedge against other tranches…”. 3 See Laurent and Gregory (2005) and Burtschell, Gregory and Laurent (2005) for a comparative analysis of alternative CDO pricing models. See Hull and White (2004) for details of CDO pricing.3Therefore, one can back-solve the standard Gaussian copula model to compute one implied correlation number from the observed price of each tranche. When implied correlations are computed from a set of CDO tranches on a given reference portfolio, the resulting correlations should theoretically be identical because they all refer to correlations among credits in the same reference portfolio. In practice, however, implied correlations computed from the observed prices of standardized tranches on CDS indices show a pronounced and