Kurze Antwort

Delta measures how much an option's price is expected to change when the underlying stock moves by one dollar. A call with a delta of 0.40 gains about $0.40 per share when the stock rises $1 and gives back about $0.40 when it falls $1. Call deltas run from 0 to 1; put deltas run from 0 to −1.

Delta doubles as a rough moneyness scale. An at-the-money option sits near 0.50, a 25-delta option is well out of the money, and a deep in-the-money option approaches 1.00, moving almost dollar for dollar with the stock.

The delta values shown on Options Band come from Cboe's delayed feed, so read them as a recent snapshot rather than a live figure.

Genauere Betrachtung

Suppose a stock trades at $100 and a one-month $105 call costs $2.00 with a delta of 0.40. If the stock rises $1 to $101 and nothing else changes, the call is expected to be worth about $2.40. If the stock instead falls $1 to $99, the call is expected to slip to about $1.60. The option's change is roughly delta times the stock's change.

Because a standard contract covers 100 shares, delta also reads as share-equivalent exposure: that 0.40-delta call responds to small moves about the way 40 shares of stock would, and position deltas add up the same way across a portfolio.

Delta is not fixed. As the stock climbs toward the $105 strike, the call's delta drifts higher — perhaps to 0.55 — so the next dollar of stock movement is worth more than the last. The rate of that drift is gamma.

Delta is also the market's shared vocabulary for moneyness:

• A 50-delta option is at the money.
• A 25-delta option is well out of the money.
• A 90-delta option is deep in the money and tracks the stock closely.

Options Band's 25-delta skew metric uses this convention: it compares implied volatility at the 25-delta put and the 25-delta call, a strike-independent way of measuring how the market prices downside against upside. See volatility skew for the full picture.

Die formalen Details

Formally, delta is the first partial derivative of the option's model value with respect to the underlying price.

Δ = ∂V/∂S

Under Black-Scholes assumptions a European call's delta is N(d₁) and the matching put's is N(d₁) − 1, where N is the standard normal cumulative distribution. Delta is therefore model-dependent: two pricing models fed different volatilities will report different deltas for the same contract.

A common shortcut treats delta as the probability the option finishes in the money. It is close, but the risk-neutral probability is actually N(d₂), which for a call is smaller than N(d₁); the gap widens with higher volatility and longer time to expiration. How Options Band derives its probability figures is covered in the probability methodology.

Delta is a first-order estimate, reliable only for small moves. A $10 jump in a $100 stock is not worth ten times a $1 move, because delta itself shifts along the way. The values on this site come from Cboe's delayed feed and lag the live market.