copula-modeling

Draw correlated normals via Cholesky decomposition of the correlation matrix, transform to uniforms with the copula's CDF, then threshold against each market's marginal probability. The skill provides working NumPy and SciPy implementations for Gaussian, Student-t, and Clayton copulas.

Gaussian copulas have zero tail dependence, assigning zero probability to extreme co-movements. A Student-t copula with nu=4 and rho=0.6 gives roughly 0.18 tail dependence, typically producing 2-5x higher probabilities of joint extreme outcomes.

Use a Clayton copula with theta between 1 and 5, which has lower tail dependence only (lambda_L = 2^{-1/theta}). For joint positive resolutions use Gumbel, and for symmetric fat tails use Student-t with nu between 3 and 6.

Use vine copulas, which decompose a d-dimensional dependency into d(d-1)/2 bivariate conditional copulas arranged in trees. Build maximum spanning trees ordered by Kendall's tau, select pair-copula families via AIC, and use pyvinecopulib or the R VineCopula package.

Gaussian copulas model tail dependence as zero, so joint extreme events were priced as essentially impossible. When correlated defaults clustered in 2008, portfolios built on Gaussian assumptions collapsed precisely in the tail scenarios the model excluded.