Model Fragility Beyond Base Correlation: An Operator-Theoretic Approach to CDO Risk Measurement

Base correlation, the industry-standard measure of copula model risk for collateralized debt obligations, is structurally blind to an entire dimension of parameter sensitivity. We establish this blindness as a mathematical theorem and develop an operator-theoretic framework that captures the missing information.The CDO pricing operator is defined as an integral operator on L2([0, 1]) whose kernel is the conditional tranche loss function. We prove its Fréchet differentiability and define the conduction intensity τ as the operator norm of the derivative. A duality theorem shows that τ equals the first-order growth rate of local parametric model uncertainty in the sense of Cont [3], with explicit second-order corrections. Three functional independence theorems then establish that τ and base correlation carry provably distinct information:the Jacobian of the map (β, p)7→ (ρbase, τ ) has rank two, with the root cause traced to an amplification factor 1/φ(Φ−1(p)) entering τ but invisible to ρbase.A phase transition theorem shows that senior tranche τ jumps from near-zero to O(1) at a critical parameter curve. A granularity analysis reveals that the large homogeneous pool approximation overestimates τ in normal markets but underestimates it in crisis, with the scale factor varying by nearly 50% across parameters. Numerical experiments on 1,600 parameter configurations confirm that at identical base correlation levels, τ varies by over 150-fold.