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Quantitative Risk Management / Extreme Value Theory

Overview

VaR and ES are only as good as the tail model behind them. Every approach discussed so far — the normal distribution, the parametric GARCH framework, even filtered historical simulation — eventually hits the same limit: the tail probabilities at extreme quantiles (99.5% for Solvency II, 99.9% for Basel IV Internal Models) are too far into the tail to be estimated reliably from the empirical data alone. A thousand days of history gives perhaps one observation beyond the 99.9th percentile. The solution is not to collect more data — it is to exploit the mathematical structure that governs tail behaviour universally.
That structure is captured by extreme value theory (EVT): the body of probability results describing how maxima, upper-order statistics, and high-threshold exceedances behave as sample sizes grow. The key insight, proved by Fisher, Tippett, Gnedenko, Pickands, Balkema, and de Haan, is that extremes obey universal limiting laws — the generalised extreme value (GEV) distribution for block maxima and the generalised Pareto distribution (GPD) for threshold exceedances — regardless of the underlying distribution from which data are drawn (provided it is in the appropriate domain of attraction, which essentially all finance distributions are).
In practice, the most powerful EVT tool for risk management is the peaks-over-threshold (POT) method: fit a GPD to all loss observations exceeding a high threshold, then extrapolate VaR and ES analytically. Combined with GARCH filtering (the conditional EVT or McNeil–Frey procedure), this produces the most reliable tail risk estimates available for daily financial returns.

By the end of this topic you should be able to:

  • state the Fisher–Tippett–Gnedenko theorem and identify which common distributions fall into each of the three GEV extreme value types (Fréchet, Gumbel, Weibull);
  • define the GPD and explain the Pickands–Balkema–de Haan theorem;
  • implement the POT method: choose a threshold using the mean excess plot, fit a GPD by MLE, and compute EVT-based VaR and ES;
  • describe the Hill estimator for the tail index and explain when it applies;
  • explain conditional EVT (GARCH + GPD) and how it combines dynamic volatility modelling with parametric tail estimation;
  • describe the POT model for threshold exceedances as a marked Poisson process.

In this lesson

  1. Contents
  2. The big picture

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