Delta

简要解答

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.

深入了解

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.

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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.