Spectral Importance Sampling: Optimal Rare-Event Simulation via Eigenvalue-Conditioned Measure Change

We develop a variance reduction framework for simulating rare events in correlated portfolios by exploiting the eigenvalue decomposition of the correlation matrix. The central observation is that the eigenvalue modes \(Z_k\) — projections of the asset vector onto the eigenvectors of the correlation matrix — are mutually independent. This mode independence allows the importance sampling measure change to factorize into independent per-mode exponential tilts, each admitting a closed-form saddle-point solution. The optimal tilt is computable in \(O(K)\) operations, independent of the portfolio dimension \(n\).

We establish three main results. First, we prove that the factored importance sampling estimator is unbiased and logarithmically efficient: at the optimal tilt, the IS second moment satisfies \(\mathbb{E}_{\mathbb{Q}_{\theta^}}[(d\mathbb{P}/d\mathbb{Q})^2 \cdot \mathbf{1}\{L > \ell\}] \leq C(\ell) \cdot \rho^{-2\ell}\), where \(\rho > 1\) is the analyticity radius of the portfolio characteristic function, \(\ell\) is the loss threshold, and \(C(\ell)\) is a polynomial prefactor (Theorem 1). Second, we show that the optimal per-mode tilt parameter \(\theta_k^\) is the saddle point of the per-mode cumulant generating function, and that the collection \(\{\theta_k^*\}\) solves a separable convex program in the large-\(\ell\) regime (Theorem 2). Third, we prove a duality between the Bernstein ellipse radius \(\rho\) governing spectral coefficient decay and the Cramér rate function governing tail probabilities, establishing that the optimal importance sampling domain coincides with the analyticity domain of the Latent representation (Theorem 3).

The method applies to any portfolio model where the correlation matrix has a spectral gap (\(\lambda_1 / \lambda_2 > 1\)) and the marginals have analytic characteristic functions (\(\rho > 1\)). Heavy-tailed distributions without a moment generating function (Student-\(t\), Pareto, stable) have \(\rho = 1\) and receive polynomial rather than exponential variance reduction. We demonstrate the method on (i) deep-tail VaR/ES estimation at the \(10^{-8}\) level, (ii) multi-asset barrier option pricing with correlated knock-out, and (iii) CDO tranche loss estimation under correlated defaults.

Keywords: importance sampling, rare events, eigenvalue decomposition, spectral methods, variance reduction, portfolio risk, large deviations

MSC 2020: 65C05, 60F10, 91G60, 62P05