Daru Finance
Lab

M/04 — Extreme Value Theory

Peaks-over-threshold and pairwise tail-coupling

Generalised Pareto fits to the upper tail of cross-asset returns, plus a model-free coefficient χ for asymptotic tail dependence.

The mathematics

Pickands–Balkema–de Haan

Let X be a random variable with CDF F satisfying F(x) → 1 as x → x* (the right endpoint, possibly +∞). For a high threshold u, define the excess random variable Y = X − u given X > u. The Pickands–Balkema–de Haan theorem (1974–75) states that F is in the maximum domain of attraction of an extreme-value distribution if and only if there exist scaling functions σ(u) such that

where G is the Generalised Pareto Distribution (GPD)

with y* = ∞ for ξ ≥ 0 and y* = −σ/ξ for ξ < 0. The shape parameter ξ governs tail behaviour:

  • ξ > 0: heavy tail, regularly varying with index 1/ξ. Equity returns and crypto live here, ξ ≈ 0.2–0.4.
  • ξ = 0: exponential tail. Many noise-driven series.
  • ξ < 0: bounded tail. Rare in finance but seen in volatility-floored series.

Method of moments estimator

The mean and variance of a GPD with parameters (ξ, σ) are

Inverting these relations yields the Hosking–Wallis (1987) method-of-moments estimator

MLE is asymptotically more efficient (and is what the production library uses), but method-of-moments is closed-form, deterministic, and adequate for the demo below.

Tail dependence coefficient χ

For a bivariate (X, Y) with continuous marginals FX, FY, define the upper-tail dependence coefficient

This is symmetric in (X, Y) — easy to verify from the definition. Two facts make χ the right object:

  • Independence ⇒ χ = 0. If X ⫫ Y the conditional probability is just 1 − u → 0.
  • Comonotonicity ⇒ χ = 1. If Y is a monotone function of X, the conditional probability is identically 1.

Crucially, χ is invariant under monotone transformations of the marginals — it depends only on the copula. That makes it a marginal-free measure of joint extremity. For Gaussian copulas (any correlation ρ < 1) χ = 0; for the bivariate t copula with ν degrees of freedom and correlation ρ,

which is > 0 for any ρ > −1. This is the empirical content of the “everything correlates in a crisis” observation: Gaussian correlation says nothing about joint extremes, but a Student-t copula at moderate ρ already produces meaningful tail-coupling.

Empirical estimator

Replace FX, FY with empirical CDFs (uniform ranks divided by n+1) and compute, for threshold u close to 1,

Worked example

Generate 2,000 pairs (Xt, Yt) with Yt = λ Xt + √(1 − λ²) Zt, X and Z independent Student-tν. Take ν = 5, λ = 0.5. Theoretical χ for the implied bivariate-t copula is non-zero. The fitted GPD on |X| above the 95th percentile should return ξ around 1/ν = 0.2.

Demo — Peaks-over-threshold + tail-coupling χ

Synthetic Student-t pair (X,Y) with linear coupling λ. Fit GPD on |X| excesses above the 95th percentile; compute empirical χ(u) on (X,Y) ranks.

n (sample size)2000
λ (coupling)0.50
ν (df, tail)5
threshold pct (u)95
seed=13
threshold u
2.537
# excesses
100
ξ (shape)
0.044
σ (scale)
0.834
empirical χ(q) — tail-coupling
q = 0.90.320
q = 0.950.260
q = 0.990.150
0.000.330.660.991.320.01.22.43.74.96.1excess y = |X| − u

Histogram of 100 excesses above threshold u=2.537. Amber line: fitted GPD density with ξ=0.044, σ=0.834. ξ > 0 indicates a heavy (Pareto-type) tail. The χ(0.99) value at left measures how often X and Y simultaneously exceed their 99th percentile — a model-free tail-coupling estimator.

Note: GPD fit uses method-of-moments (Hosking & Wallis 1987). The reference implementation uses MLE. Both estimators are consistent; MoM is preferred here for in-browser determinism.

Figures

Log-log tail survival of cross-asset strategy P&L with GPD fit
Fig. 1 —Empirical tail survival P(|R| > r) on log-log axes for daily strategy P&L pooled across BTC, ETH, SOL, DOGE. Each thin curve is the empirical step function; the bold curve is the POT-GPD fit above the 95th percentile, with the MLE shape ξ̂ printed in the legend. Crypto strategy P&L lives unambiguously in the heavy-tailed regime (ξ̂ > 0).
Empirical chi-bar(0.95) tail-coupling matrix between assets
Fig. 2 —Empirical χ̂(0.95) between cross-asset daily aggregate strategy returns over a 4-year common window. Off-diagonal entries are visibly positive — joint extremes between BTC, ETH, SOL, DOGE and friends occur strictly more often than independence would predict, the empirical content of 'everything correlates in a crisis'.

Why this matters for systematic strategies

Two practical uses in the firm’s pipeline. First, GPD fits on per-strategy drawdown distributions give honest tail-VaR and expected-shortfall estimates that don’t collapse during regime shifts — the Gaussian-VaR alternative under-reports tail risk by an order of magnitude during the same crises. Second, χ on cross-asset returns at u = 0.99 is a leading indicator of contagion. When BTC–XAU χ(0.99) drifts above 0.4, the empirical pattern is consistent with broad “flight-to-cash” episodes in which historically uncorrelated assets co-move in their tails.

Reproducibility

DaruFinance / tail-evt

Rust — open source reference implementation

Minimal invocation

use tail_evt::{pot_fit, tail_chi};

// X: &[f64] of returns
let fit = pot_fit(&x, /*pct=*/ 0.95);
println!("xi = {:.3}, sigma = {:.3}", fit.xi, fit.sigma);

// Pairwise tail dependence on (X, Y)
let chi_99 = tail_chi(&x, &y, 0.99);
println!("chi(0.99) = {:.3}", chi_99);

References

  1. [1]Pickands, J. (1975). Statistical inference using extreme order statistics. Annals of Statistics 3, 119–131.
  2. [2]Balkema, A. A. & de Haan, L. (1974). Residual life time at great age. Annals of Probability 2, 792–804.
  3. [3]Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics.
  4. [4]McNeil, A. J., Frey, R. & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools (rev. ed.). Princeton University Press.
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