The European Central Bank released a Working Paper proposing a semi-parametric framework to model persistent time variation in extreme tail risk, focusing on extreme Value-at-Risk (VaR) and Expected Shortfall (ES). The approach combines Extreme Value Theory with a conditional Generalized Pareto Distribution and uses integrated score-driven dynamics so tail fatness can evolve with near unit-root persistence. The model’s main simplification is to rescale peaks-over-threshold observations by their thresholds, allowing the entire tail to be captured by a single time-varying tail shape parameter rather than separate time-varying scale and shape parameters. The paper derives parameter regions ensuring stationarity, ergodicity and filter invertibility, and establishes consistency and asymptotic normality for maximum likelihood estimation despite integrated dynamics. In an application to hourly Bitcoin and Ether returns (2018–2025), the framework identifies sharp increases in tail risk during 2022 events linked to the collapses of Terra/Luna, FTX and Celsius, with the estimated 99.5% ES approximately tripling during those periods, and out-of-sample comparisons indicate performance broadly comparable to more complex dynamic EVT specifications and stronger than a standard GARCH model in the far tail.
European Central Bank 2026-01-06
European Central Bank publishes research on a single-parameter integrated tail model for extreme Value-at-Risk and Expected Shortfall dynamics
The European Central Bank's Working Paper introduces a semi-parametric framework for modeling extreme tail risk, integrating Extreme Value Theory with a conditional Generalized Pareto Distribution to capture time-varying tail risk in financial returns. Applied to Bitcoin and Ether returns, the model highlights significant tail risk increases during 2022's Terra/Luna, FTX, and Celsius collapses, with performance comparable to complex EVT models and better than standard GARCH in extreme scenarios.