Sklar’s theorem
For any joint CDF $F(x_1,\ldots,x_n)$ with marginals $F_1,\ldots,F_n$, there exists a copula $C: [0,1]^n \to [0,1]$ such that:
\[F(x_1,\ldots,x_n) = C(F_1(x_1),\ldots,F_n(x_n))\]Conversely, any copula combined with any marginals gives a valid joint distribution. The copula is unique if all marginals are continuous.
Common copulas
Gaussian copula: $C(u_1,\ldots,u_n) = \Phi_R(\Phi^{-1}(u_1),\ldots,\Phi^{-1}(u_n))$ where $\Phi_R$ is the multivariate normal CDF with correlation matrix $R$.
Student-$t$ copula: heavier tails and stronger tail dependence than Gaussian.
Clayton copula: strong lower-tail dependence — assets co-crash but don’t co-boom.
Gumbel copula: strong upper-tail dependence.
Tail dependence
The upper tail dependence coefficient:
\[\lambda_U = \lim_{u\to1^-} P(U_2 > u \mid U_1 > u)\]Gaussian copula: $\lambda_U = 0$ (tail independence) regardless of $\rho$ — a dangerous feature for modelling joint crashes. The $t$-copula has $\lambda_U > 0$.
The 2008 crisis exposed over-reliance on the Gaussian copula for CDO tranching, which drastically understated joint default probabilities.