Class ExtremeValueTheory

java.lang.Object

com.quantfinlib.risk.ExtremeValueTheory

public final class ExtremeValueTheory
extends Object

Extreme value theory via peaks-over-threshold — the statistically principled way to ask about quantiles beyond the sample. Historical VaR at 99.9% from 500 observations is reading the worst half-observation; EVT instead fits the Generalized Pareto Distribution to the exceedances over a high threshold (the Pickands-Balkema-de Haan theorem says the tail of any well-behaved distribution converges to a GPD), then extrapolates along the fitted tail:

  VaR_p = u + (β/ξ)·[((n/Nᵤ)(1−p))^{−ξ} − 1]

The shape ξ is the number to stare at: ξ ≈ 0 is an exponential tail (Gaussian-ish), ξ > 0 is a power-law tail (fat; equity returns typically fit ξ ≈ 0.2-0.4), and ξ ≥ 1 means the tail mean does not even exist (expectedShortfall refuses rather than returning a finite lie). fitPot(double[], double) uses probability-weighted moments (Hosking and Wallis 1987): closed form, good for ξ < 0.5 but biased and inefficient beyond. fitPotMle(double[], double) maximises the GPD likelihood through Grimshaw's (1993, Technometrics 35, 185-191) one-dimensional profile in θ = ξ/β and reports the asymptotic standard errors of Smith (1987, Ann. Statist. 15, 1174-1207), valid for ξ > −0.5. meanExcess(double[], double[]) gives the mean-excess function used to choose the threshold (linear in u above a valid threshold). The threshold choice is the caller's judgment call — the classic diagnostic is fitting at several thresholds and checking ξ stability; a quantile between 0.90 and 0.95 of the losses is the usual start. Research lane, deterministic.

Nested Class Summary

Nested Classes

Modifier and Type	Class	Description
static final record	ExtremeValueTheory.GpdFit	A fitted POT tail model.
static final record	ExtremeValueTheory.GpdMle	A maximum-likelihood GPD fit with asymptotic standard errors Var ξ̂ = (1+ξ)²/m, Var β̂ = 2β²(1+ξ)/m (Smith 1987).

Method Summary

Modifier and Type	Method	Description
static ExtremeValueTheory.GpdFit	fitPot(double[] losses, double thresholdQuantile)	Fits a GPD to the losses exceeding the thresholdQuantile of the sample (e.g. 0.90), via probability-weighted moments.
static ExtremeValueTheory.GpdMle	fitPotMle(double[] losses, double thresholdQuantile)	Maximum-likelihood GPD fit to the exceedances over the thresholdQuantile of the sample.
static double	gpdLogLikelihood(double[] y, double shape, double scale)	GPD log-likelihood of exceedances y at (ξ, β); −∞ outside the support.
static double[]	meanExcess(double[] losses, double[] thresholds)	Empirical mean-excess function e(u) = mean(L − u | L > u) at each threshold (NaN where no loss exceeds u).

Methods inherited from class Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Details

fitPot

public static ExtremeValueTheory.GpdFit fitPot(double[] losses,
 double thresholdQuantile)

Fits a GPD to the losses exceeding the thresholdQuantile of the sample (e.g. 0.90), via probability-weighted moments. Losses are positive numbers (feed -returns or a loss series directly).

fitPotMle

public static ExtremeValueTheory.GpdMle fitPotMle(double[] losses,
 double thresholdQuantile)

Maximum-likelihood GPD fit to the exceedances over the thresholdQuantile of the sample. The log-likelihood of exceedances y is −m·ln β − (1 + 1/ξ)·Σ ln(1 + ξ·y/β); with θ = ξ/β the maximising ξ for fixed θ is (1/m)·Σ ln(1 + θ·y) and the profile −m·ln(ξ(θ)/θ) − m·(1 + ξ(θ)) is maximised over θ > −1/max(y) by a grid scan followed by golden-section search.

gpdLogLikelihood

public static double gpdLogLikelihood(double[] y,
 double shape,
 double scale)

GPD log-likelihood of exceedances y at (ξ, β); −∞ outside the support.

meanExcess

public static double[] meanExcess(double[] losses,
 double[] thresholds)

Empirical mean-excess function e(u) = mean(L − u | L > u) at each threshold (NaN where no loss exceeds u). For a GPD tail it is linear in u with slope ξ/(1−ξ); the threshold should be chosen where the plot becomes linear.