Spectral Market Risk: Deterministic VaR, ES, and Stress Testing Without Monte Carlo

We present a benchmark of the Spectral Fenton method --- a deterministic alternative to Monte Carlo simulation for computing Value-at-Risk, Expected Shortfall, and full stress surfaces --- on realistic multi-asset portfolios representative of bank trading desks. The method condenses the complete portfolio loss distribution into 130 Fourier coefficients (1.04 KB), from which VaR at any confidence level costs 0.46 ms and ES is available in closed form. On a 4-asset heterogeneous portfolio (equities, bonds, crypto), the spectral method matches Monte Carlo (\(10^6\) paths) to 0.94\% VaR accuracy and 0.78\% ES accuracy while producing zero sampling noise --- compared to MC's 0.07\% inter-seed variability. For FRTB Expected Shortfall at \(\alpha = 2.5\%\), the spectral estimate is deterministic and maximizes the statistical power of the Acerbi--Székely ridge backtest, eliminating the 15--30\% power degradation caused by MC estimation noise at typical production path counts.

At bank desk scale (5,000 portfolios \(\times\) 10 stress scenarios), the spectral approach reduces the daily risk batch from 10 hours to 1 hour on a single CPU --- without GPU infrastructure. A 20-level VaR fan chart for the entire desk computes in 46 seconds versus 18 hours for Monte Carlo. The 130-coefficient "risk certificate" enables independent regulatory verification without portfolio disclosure.

We characterize the method's applicability envelope: sub-1\% accuracy for portfolios with \(\sigma_i \leq 0.3\) and moderate correlation heterogeneity; 1--5\% accuracy for portfolios with \(\sigma_i \leq 0.8\) and adaptive factor selection; and a documented extreme-volatility frontier (\(\sigma > 1.0\)) where MC validation is recommended. The spectral approach is positioned not as a universal MC replacement, but as a deterministic accelerator for the high-volume, time-critical segment of the market risk stack: daily VaR/ES for linear portfolios under geometric Brownian motion.