Archimedean Copulas: Clayton, Gumbel, Frank, and Fitting

The Gaussian and $t$ copulas are implicit — extracted from known multivariate distributions. Archimedean copulas are explicit: they are built directly from a single function, the generator $\psi$, through the formula $C(u_1, u_2) = \psi(\psi^{-1}(u_1) + \psi^{-1}(u_2))$. This construction gives tractable closed-form expressions, analytically computable tail dependence, and simple simulation algorithms. It also gives flexibility that the Gaussian copula lacks: the Clayton copula concentrates dependence in the lower tail (simultaneous losses), the Gumbel copula in the upper tail (simultaneous gains), and the Frank copula symmetrically with no tail dependence at all.

For credit risk modelling, Archimedean copulas solve a specific problem: the Gaussian copula's zero tail dependence ($\lambda = 0$) means joint defaults are severely underpriced in the tail. Clayton's $\lambda_L = 2^{-1/\theta} > 0$ provides an explicit lower-tail dependence that can be calibrated to historical default clustering. The simulation algorithm — generate a common mixing variable $V$, then draw exponentials conditionally — provides the high-dimensional extension.

Learning objectives. By the end of this note you should be able to: