Extreme Value Theory explained for betting

In one sentence

Extreme value theory estimates how big and how frequent rare, extreme events are by modelling only the tail of the data instead of the whole distribution.

How it works

Most risk calculations use the average and the spread, then assume a bell curve. That works in the middle and fails in the tails, where the big losses live. A trader's worst days are usually far worse than a normal curve says they should be.

Extreme value theory ignores the middle. Pick a threshold, say any day you lose more than £50, and model only how far losses go beyond it. A mathematical result says these excesses tend to follow one family, the generalised Pareto distribution, whatever the shape of the rest of the data.

The key output is a shape number, ξ (xi). If ξ is above zero, the tail is heavy: very large losses are more likely than a normal model admits. That feeds directly into bank size, stop limits and staking.

The maths

P(X > x \mid X > u) = \left(1 + \xi\,\frac{x-u}{\sigma}\right)^{-1/\xi}

P(X > x) \approx \frac{N_u}{N}\left(1 + \xi\,\frac{x-u}{\sigma}\right)^{-1/\xi}

X is the daily loss; u is the threshold you chose.
ξ is the tail shape: above 0 means heavy tails, 0 means exponential-like, below 0 means a hard upper limit.
σ is the scale of losses beyond the threshold.
Nu is the number of days beyond the threshold and N the total number of days.

In words: the chance of a very big loss is the chance of beating the threshold times the tail curve beyond it.

Worked betting example

An in-play football trader (Match Odds and Over/Under goals) has 500 days of results, illustrative figures. On 40 days (8%) the loss exceeded £50. Fitting a generalised Pareto to those 40 excesses gives ξ = 0.2 and σ = £30.

P(daily loss above £200) = 0.08 × (1 + 0.2 × 150 ÷ 30) to the power −5 = 0.08 × 2 to the power −5 = 0.25%, about one day in 400.
The 1-in-100-day loss (99% value at risk) is about £127.
The 1-in-500-day loss is about £214, a day you would expect roughly once in two years of trading.
With 40 exceedances, these numbers carry wide error bars. A small change in ξ moves the 1-in-500 estimate a lot.

That £214 figure is worth setting your bank and daily stop-loss around, rather than your average bad day.

Where it's good

Sizing a trading bank and daily loss limits around realistic worst cases.
Estimating the chance of a catastrophic in-play position (a late goal against a big lay).
Stress testing a staking plan beyond what the backtest happened to include.
Modelling record scorelines or very large price moves after goals.

Limitations and pitfalls

By definition there is little tail data. Tens of exceedances give very uncertain ξ; treat outputs as rough ranges.
Choosing the threshold is a trade-off: too low breaks the theory, too high leaves too few points. Check stability across several thresholds.
It assumes the process is stable. If your stakes, markets or strategy changed, old losses may not describe the future.
Losses cluster: bad days often follow bad days (tilt, a bad run of markets), which the basic model ignores.
It will not predict a loss type you have never experienced, such as a platform outage or an unmatched hedge.
For most recreational punters with level stakes, simpler drawdown simulation is enough; EVT earns its keep for active traders and large banks.

How to build it

scipy.stats.genpareto (fit, sf, isf) in Python; pyextremes for threshold selection; R: evd, extRemes.
Data: at least a few hundred days of profit and loss at consistent stake sizes.
Tip: plot the mean excess over a range of thresholds and pick where the line becomes roughly straight.

Value at risk: the risk figure EVT estimates more accurately in the tail.
Drawdown analysis: sequences of losses rather than single days.
Risk of ruin: the chance of losing the whole bank.
Normal distribution: the thin-tailed model EVT corrects.