there exists a copula function such that (7) where F is the joint cumulative distribution function and , i = 1, ..., ni are the univariate marginal distributions. is the inverse of . If the marginal distributions are continuous, it follows that C is unique. For properties and proofs of equation (11), see Sklar (1959) and Nelsen (2006). Numerous types of copula functions exist. They can be broadly categorized in one-parameter copulas as the Gaussian copula, and the Archimedean copula, which comprise Gumbel, Clayton and Frank copulas. Often cited two-parameter copulas are student-t, Fréchet, and Marshall-Olkin. For an overview of these copulas, see Nelsen (2006). for example in a collateralized debt obligation, CDO. This was first done by Li in 2006. He defined the uniform margins as cumulative default probabilities Q for entity i at a fixed time t, : (8). Hence, from equations (7) and (8) we derive the Gaussian default time copula CGD, (9). In equation (9) the terms map the cumulative default probabilities Q of asset i for time t, , percentile to percentile to standard normal. The mapped standard normal marginal distributions are then joined to a single n-variate distribution by applying the correlation structure of the multivariate normal distribution with correlation matrix R. The probability of n correlated defaults at time t is given by . Copulae and the 2007–08 financial crisis Numerous non-academic articles have been written demonizing the copula approach and blaming it for the 2007/2008 global financial crisis, see for example Salmon 2009, Jones 2009, and Lohr 2009. . (a) Tail dependence In a crisis, financial correlations typically increase, see studies by Das, Duffie, Kapadia, and Saita (2007) and Duffie, Eckner, Horel and Saita (2009) and references therein. Hence it would be desirable to apply a correlation model with high co-movements in the lower tail of the joint distribution. It can be mathematically shown that the Gaussian copula has relative low tail dependence, as seen in the following scatter plots. Figure 1: Scatter plots of different copula models As seen in Figure 1b, the student-t copula exhibits higher tail dependence and might be better suited to model financial correlations. Also, as seen in Figure 1(c), the Gumbel copula exhibits high tail dependence especially for negative co-movements. Assuming that correlations increase when asset prices decrease, the Gumbel copula might also be a good correlation approach for financial modeling. (b) Calibration A further criticism