Probabilities from option prices

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Option prices carry information about how likely the market considers different outcomes. A call that costs almost nothing implies the market sees little chance the stock reaches that strike; an expensive one implies the opposite.

Options Band converts implied volatility into explicit figures: the estimated chance that a stock finishes above or below a chosen level by a chosen date. The calculation uses a lognormal model with zero drift, described at the probability methodology page.

These are always called estimates, never predictions. They describe what current option prices imply under a specific model — a starting point for reading the market, not a statement about what will happen.

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The model treats the stock's future price as lognormally distributed around today's price, with the width of the distribution set by implied volatility and time. Two inputs do nearly all the work: how volatile the market thinks the stock is, and how long it has to move.

A worked example with round numbers:

• Stock price: $100
• Level of interest: $110
• Implied volatility: 30%
• Time: 1 year
• Width of the distribution: σ√t = 0.30 × 1 = 0.30
• Distance in log terms: ln(110 / 100) ≈ 0.095

Under zero drift the model centers the distribution so the expected price equals today's price, which places the median slightly below $100. Standardizing the distance — 0.095 plus a small adjustment of σ²t/2 = 0.045, divided by 0.30 — gives about 0.47 standard deviations. A normal table turns that into an estimated 32-in-100 chance of finishing above $110, and roughly a 68-in-100 chance of finishing at or below it.

Halve the volatility to 15% and the estimate falls to roughly 24 in 100; shorten the time and it falls further. The estimates move continuously with option prices, which is the point: they are a readout of the market, not an opinion layered on top. How past estimates have scored against actual outcomes is tracked in the historical database.

Facoltativo: il dettaglio formale non necessario per le verifiche

Formally, Options Band estimates the probability that a lognormal price with volatility σ and zero drift finishes beyond a level K at horizon t:

P(finish above K) = N( −( ln(K / S) + σ²t / 2 ) / ( σ√t ) )

where S is the current price and N is the standard normal distribution function. Zero drift is a deliberate simplification: it avoids embedding any return assumption and keeps the figure a pure volatility readout. The full assumptions are at the probability methodology page. Limitations follow directly: a single volatility input flattens the skew, the lognormal shape understates tail outcomes, and the figures are model quantities rather than observed frequencies — which is why they are labeled estimates and scored after the fact.

Two misconceptions recur. First, that delta equals the probability of finishing in the money — the two are related but not identical, and both are model outputs. Second, that a single outcome can validate or refute an estimate — a 70-in-100 estimate that misses was not necessarily wrong, which is why scoring only means something across the many snapshots recorded in the historical database.