Stripping the Margin Out of a Quoted Price

Here's a problem that shows up in three places I care about: recovering a risk-neutral density from an option chain, recovering a fair mid from a wide bid-ask, and recovering a fair probability from a bookmaker's two-sided price.

They're the same problem. An observable spread has to be decomposed into a fair value plus a margin whose distribution across states is unobserved. The betting version is the cleanest laboratory for it, because the labels are hard, immediate, unambiguous and free — the game ends, and you find out.

So I ran it there. 5,280 NFL games, 2006–2025, free public data.

Four estimators, not five

The standard toolkit has five ways to strip margin out of a two-sided quote:

• Multiplicative — divide each raw implied probability by the booksum. Equal proportional margin on every outcome. This is what every free no-vig calculator does.
• Additive — subtract the excess evenly. Equal absolute margin, which loads relatively harder on longshots.
• Power — raise each raw probability to a power, solve for the exponent that makes them sum to one.
• Shin — solve for an implied proportion of informed traders. This one has an economic story: margin as protection against informed flow.
• Worst-case — a deliberately conservative bound. A screening floor, not a fair value.

First result, and it's a clean one: Shin and additive are numerically identical on two-way markets. Maximum absolute difference across 5,280 games: 1.93 × 10⁻¹³. That's floating-point noise, not disagreement.

Anyone treating them as two independent methods is double-counting one opinion. There are four distinct estimators here, not five.

On mainlines, the choice doesn't matter at all

This reverses what I expected. I went in assuming the multiplicative method — the default everywhere, chosen because it's a one-liner — would be measurably biased. Ten thousand five hundred and sixty side-observations, cluster-bootstrapped by game:

Every confidence interval contains 1.00. Brier scores identical to four decimal places. Bin by bin, the recovered probability sits inside the 95% binomial interval for realised frequency in every single bin from 0 to 1.

The null could not be rejected. That's a useful negative result, not a failed experiment: it means a screen built on mainline prices is trustworthy, and any edge it flags is not an artifact of how you stripped the margin.

The reason turns out to be structural, and it leads directly to the finding.

The error law

De-vig error is not a property of the estimator. It's the mismatch between how the margin was loaded into the price and how you strip it out. Simulating a quote where margin goes in one way and comes out another:

(worst-case error across the four estimators, in percentage points of probability)

The law that falls out:

de-vig error ∝ (margin) × |p − 0.5|

At p = 0.50 every estimator agrees exactly, at every margin level. All of the disagreement is asymmetry, amplified by margin.

That's a far more useful rule than "pick the right method," because it tells you where you are allowed to trust a recovered fair value at all. The NFL moneyline result is clean precisely because that market is cheap — 2.65% hold — and mostly near pick'em. The method choice was invisible because the conditions made it invisible, not because the estimators agree in general.

How much flagged edge is phantom

Translate the error into money. Take the most common real-world mismatch — margin loaded additively, stripped proportionally — and ask what share of a flagged +3% edge is specification error rather than edge:

On a 10% longshot in a high-margin market, a screen showing +3% expected value is measuring mostly its own specification error. And that is exactly where any screen points you, because that's where raw numbers look most mispriced. The tool's error term and the tool's output are maximised in the same place.

The rule I'd actually use

Not "pick the best estimator." The right framing is shrinkage with segment-varying intensity. If your edge estimate carries estimation-error variance, you shrink toward zero in proportion to it — and the key point is that this variance is not constant across the book. It's about 0.35 percentage points near pick'em and around 5 points on longshots.

So shrinkage should be near 1 on symmetric, liquid, low-margin quotes and near 0 on asymmetric, high-margin ones. Concretely:

• Trust symmetric, low-margin quotes directly. No shrinkage needed.
• Compute all four methods and take the worst case as your fair value anywhere margin exceeds 5% or |p − 0.5| exceeds 0.15. If it still shows an edge against the most conservative estimator, the edge is structural rather than specification error.
• Never screen far-from-even outcomes on a single method. The phantom fraction exceeds the edge.
• Shrink by the method spread, not by a constant.
• Prefer symmetric markets. They're the only place where fair value is estimator-independent.

And one thing that generalises past this entirely: more data will not fix specification error. Estimation error shrinks with sample size. Specification error is a property of the model, and it sits there at the same magnitude no matter how many observations you collect. If your dominant error term is specification, collecting a bigger dataset is the wrong response.

Back to option chains

The transfer is direct. Wide-spread, far-out-of-the-money options carry exactly this pathology: an observable two-sided quote, an unobservable split between fair value and market-maker margin, and a spread that widens precisely where the payoff is most convex. The standard fix — take the mid — is the multiplicative estimator, applied silently and without anyone checking whether it's the right one for that quote's asymmetry.

The general result is worth keeping: when you decompose an observed price into a fair value plus an unobserved margin, your specification error vanishes at the symmetric point and grows with the product of spread width and asymmetry.

Near the money, take the mid and stop worrying. Far from it, in a wide market, your fair value is a modelling opinion — and you should be treating it like one.