Pricing Basket Options via Eigenvalue-Conditional Black-Scholes Mixing

Pricing European basket options requires the distribution of a weighted sum of correlated lognormals — the Fenton Distribution (Nagy, 2026a) — which has no closed form. We propose eigenvalue-conditional Black-Scholes mixing: decompose the correlation matrix, condition on dominant factor realizations via Gauss-Hermite quadrature, and price each conditional basket using Black-Scholes. We prove that conditioning reduces the conditional coefficient of variation by \(\sqrt{1 - \tau_K}\) (Proposition 1), with total pricing error bounded by \(\lvert V - \hat{V}\rvert \leq \varepsilon_{\text{FW}}(K) + \varepsilon_{\text{GH}}\) (Theorem 1). The method achieves 10/10 across a graduated test suite spanning 2--500 assets, with 1.76% mean error. A 500-asset basket prices in 61 ms; a 1,558-point correlation-volatility stress surface computes in 8.7 seconds (vs. \(\sim\)26 minutes for Monte Carlo). The eigenvalue-conditional architecture unifies risk measurement and derivative pricing within a single decomposition framework, validated at 60/60 across six independent test suites.