An Eigenvalue-Conditioned Copula with Positive Tail Dependence: A Machine-Verified Alternative to the Gaussian Copula

The Gaussian copula's failure to capture tail dependence was a central factor in the 2008 credit crisis: CDO tranche losses far exceeded model predictions because the model assigned near-zero probability to simultaneous defaults. We formally prove that the Gaussian copula has zero upper tail dependence (\(\lambda_U = 0\)) for any correlation \(|\rho| < 1\), confirming mathematically what the crisis demonstrated empirically.

We then construct an alternative --- the eigenvalue-conditioned copula --- that satisfies five properties simultaneously: it is a valid copula (grounded, uniform margins, 2-increasing), has positive tail dependence (\(\lambda_U > 0\)), admits exponentially convergent COS pricing for CDO tranches in \(O(N)\) operations, is dimension-free (the representation size \(N\) does not depend on the number of names), and forms a parametric hierarchy indexed by \(K = 0, 1, \ldots, n\) that nests the Gaussian copula (\(K = 0\)) and converges to the exact joint distribution (\(K = n\)). All structural results are machine-verified. With \(K = 0\) conditioning eigenvalues, the model coincides with the standard Gaussian copula; for any \(K \geq 1\), the mixture formula activates, the copula gains positive tail dependence, and the COS collapse enables \(O(N)\) deterministic pricing. Each additional eigenvalue captures finer correlation structure.

Keywords: copula, tail dependence, CDO pricing, COS expansion, formal verification, eigenvalue conditioning

JEL: C63, G13, G32