High-Precision Greeks for Multi-Asset Spread Options via Eigenvalue-Conditioned Fourier Inversion

Spread options on three or more correlated assets are fundamental hedging instruments in energy, commodity, and agricultural markets. The 3:2:1 crack spread, the crush spread, and the multi-tenor calendar spread all require the distribution of a weighted sum of correlated lognormals with mixed signs \(-\) a problem that defeats standard analytical methods for \(n \geq 3\). Kirk's approximation (1995) handles two assets; Monte Carlo handles arbitrary \(n\) but produces noisy Greeks at high cost. We apply the Eigen-COS method (Nagy, 2026a) \(-\) eigenvalue decomposition of the correlation matrix combined with Fourier-cosine series expansion \(-\) to compute prices and the complete Greek surface for European spread options on \(n \geq 3\) correlated lognormal assets. Every first- and second-order sensitivity, including the correlation Greek \(\partial V/\partial \rho_{ij}\), reduces to an \(N\)-term Fourier series (typically \(N = 128\)) whose coefficients are obtained by differentiating the conditional characteristic function under Gauss-Hermite quadrature. The cost of all Greeks combined equals a single precomputation. We introduce the implied correlation surface \(-\) backed out from market spread option prices \(-\) as a new diagnostic for correlation risk. For a three-asset crack spread, the full Greek surface is computed in \(\sim\)1.5 seconds versus \(\sim\)27 seconds for Monte Carlo bump-and-reprice, an \(\sim\)18\(\times\) speedup with deterministic, noise-free output. The correlation sensitivity is recovered to \(10^{-4}\) relative accuracy at sub-second cost.