Why equity-index divergences persist

The event contract (a digital / range)

A binary claim that pays a fixed amount (commonly $1) if the index settles inside the stated range (or at/above the stated level) at expiration and nothing otherwise. Cash-settled to the venue's stated settlement value and time, fully collateralized. Its price is, directly, the market-implied probability of the event.

The realized-volatility estimate (not tradeable)

A statistical estimate of the settlement distribution — not a traded instrument. The index's recent daily returns give an annualized realized volatility; a lognormal model scaled to the time remaining turns that into a probability for the range. There is nothing to hold: it settles to nothing, because it is a measurement placed next to the market price rather than a position.

The Scopes Fair Value for a range event is the probability the index finishes inside it under a lognormal model whose volatility is the index's realized volatility over a trailing window matched to the horizon; for an "above K" event it is the probability of finishing above K under the same model.

Realized volatility is the annualized standard deviation of recent daily log returns — a backward-looking measure of how much the index HAS moved. It is not read from option prices, so it does not inherit the risk premia and skew embedded in the options surface. That independence is the point: the fair value is a statement about the distribution implied by realized movement, held up against the distribution the market is pricing.

The estimate is not the truth. Realized volatility is a sample, the window length is a choice, and scaling a short window to a longer horizon assumes a flat volatility term structure the real one rarely has. The fair value is therefore a model estimate with its own uncertainty, not an identity.

The variance risk premium

Option prices — and, tracking them, the contract's traded price — embed a premium for bearing volatility risk, so the market's implied distribution is typically wider than the one realized volatility alone produces. The market therefore assigns more probability to ranges away from the current level than the realized-volatility fair value does. This is a structural, persistent difference between an implied and a statistical distribution, not a disagreement about the outcome.

Volatility term structure

Realized volatility measured over a trailing window and scaled to the horizon assumes today's pace of movement continues at the same rate. The market prices a term structure — near-dated and far-dated volatility differ, and movement clusters around known dates (data releases, meetings). Over longer horizons the scaled realized estimate and the market's horizon-specific view diverge for reasons of calendar, not disagreement.

Estimation window

A short window reacts quickly but is noisy; a long window is steady but stale. The fair value matches the window roughly to the horizon, but any single window is one estimate of a moving quantity. Two reasonable windows return two fair values, most visibly after a change in regime — so a gap smaller than the spread between windows sits inside the estimate's own uncertainty.

Settlement specification

The contract settles to a specific reference value and time; the realized-volatility model is agnostic about the exact settlement procedure. Near a range boundary, small differences in the settlement definition move the contract's probability in ways the model does not track — so part of a gap can be the settlement basis rather than a difference in view.

Implied versus realized. The market's distribution is risk-neutral — it prices in the premium participants charge for bearing volatility. The realized-volatility fair value is closer to a real-world estimate of movement. A gap between them is expected on average, and it leans one way: implied wider than realized. The desk's value is in showing that difference and its size, not in claiming the market is wrong.

What the calibration record adds. Because the gap has a known direction on average, the interesting question is not that it exists but whether, and where, settlements land closer to the statistical distribution than to the market's — which is measured over many resolved events on the calibration record, not asserted from any one gap.

The realized-volatility fair value is not a tradeable instrument. There is no option to buy, no spread to hold, no second venue to settle against — it is a measurement, so no position turns the gap into anything. That is the first reason the gap is not a trade: one side of it does not trade.

The gap also has a name. Most of it is the variance risk premium — the market pricing a wider distribution than realized movement — plus estimation and term-structure differences. A premium that is compensation for bearing risk is the price of that risk, not something to be captured; what remains after it is estimation noise, bounded by the choice of window.

That is the structural reason an equity-index divergence persists: it is the standing difference between an implied distribution and a statistical one, not a gap waiting to close. Whether settlements fall nearer one distribution or the other is what the calibration record measures, over many resolved cases rather than one.