Overview — Multivariate Models and Copulas

This topic develops the vocabulary and models for multivariate financial distributions, structured around two questions:

1. Which multivariate distributions are realistic for financial returns? — The multivariate normal, which underpins the variance-covariance method of VaR and classical mean-variance portfolio theory, fails empirically: returns show joint heavy tails, volatility clustering, and stronger co-movement in downturns than the normal predicts. Normal variance mixtures (including the multivariate $t$) correct the marginal tail problem while preserving much of the normal's tractability. Elliptical distributions provide the right class of tractable multivariate models.
2. How do we model dependence separately from marginals? — Copulas answer this: Sklar's theorem factorises any joint distribution into its marginals and a copula capturing dependence. The Gaussian copula, despite having zero tail dependence, became the default model for CDO tranches in the 2000s — with catastrophic consequences. The $t$-copula has positive symmetric tail dependence; Archimedean copulas (Clayton, Gumbel, Frank) offer asymmetric tail dependence patterns more relevant to stressed market scenarios.

• define and simulate from the multivariate normal, multivariate $t$, and normal variance mixture distributions;
• test multivariate normality and understand why it fails for daily financial returns;
• explain elliptical distributions and their properties (spectral representation, tail index, linear subspace invariance);
• state Sklar's theorem and construct a copula from a joint distribution;
• compute and interpret rank correlation (Kendall's $\tau$, Spearman's $\rho$) and tail dependence coefficients;
• distinguish Gaussian copula (zero tail dependence) from $t$-copula (positive symmetric tail dependence);
• identify Clayton, Gumbel, and Frank Archimedean copulas by their tail dependence structure;
• fit a copula to multivariate return data by MLE.

1. Contents
2. The big picture