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Probability of touch

The chance the underlying trades at a given strike at any point before expiration; roughly double the probability of expiring beyond it.

Touching is easier than finishing. Under the standard model the probability of touching a level is approximately twice the probability of ending past it, because for every path that finishes beyond the strike there is a mirror path that touches and comes back.

This is the number that matters for anyone who manages positions rather than holding to expiry. A short strike with a 20% chance of expiring in the money has roughly a 40% chance of being tested, and being tested is when the loss, the stress, and the decision to roll actually happen.

Example: XYZ at $50, short the 45-day $45 put with a 22% probability of finishing in the money. The probability of touch is about 44%. Nearly half the time you will watch XYZ trade at $45 before expiration, whatever the final outcome.

Related: probability-itm, tested-side, delta-as-probability, management-at-21-dte

See it drawn

Original diagrams for the ideas on this page. Illustrative, not real market data.

How a call option's delta changes with the underlying priceAn S-shaped curve rising from zero, passing through about a half at the strike, and flattening near one.Delta of a call option1.000.5008090110120Out of the moneyAt the moneyIn the money1.00 means it moves one-for-one with the stockdelta ≈ 0.50 at the strikeStrike 100Underlying price
Delta across the range of prices. Delta says how much a call's price moves for a one-point move in the stock. Far below the strike it is near 0 and the option barely reacts; at the strike it is about 0.50; far above it approaches 1 and tracks the stock.

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