Geometric Stress Coordinates for Conditional Tail Forecasting: Soft Distributional Routing across Equity Markets

Abstract

The results show that an online Wasserstein stress coordinate contributes to one-step tail forecasting after current volatility scale is filtered. Daily returns for the S&P 500, FTSE 100, Nikkei 225, and Hang Seng from 2000 through June 2026 are evaluated strictly out of sample from January 2007. Frozen pre-2007 distributionalregime thresholds softly route observations between calm and stress empirical Value-at-Risk (VaR) and Expected Shortfall (ES) experts. Under the original 22-day realized-volatility filter, soft DRP routing improves pinball loss in all eight market-tail cells, with FTSE at 1% and HSI at 5% surviving cross-market Holm correction. The joint FZ0 loss also improves in seven of eight cells, with the FTSE 1% gain surviving correction; the same corrected FTSE 1% joint-loss result is retained under the RiskMetrics EWMA filter. The prespecified extensions show that the magnitude of the gain depends on scale filtering, market, and training memory: frozen GARCH is weaker and a 1,000-day rolling window reverses the SPX pattern. Soft routing adds comparative tail-loss information without systematically resolving absolute conditional-coverage or DQ failures. The evidence gives geometric stress coordinates a positive but conditional role in one-step VaR-ES forecasting.

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