Delta

Delta measures how much an option’s price changes for a one-unit move in the underlying. A call has a delta between 0 and 1, a put between −1 and 0. Multiplied by the contract size, it gives the equivalent position in the underlying, which is what a trader hedges.

How it works

delta = change in option price for a 1-point move in the underlying. An at-the-money option has a delta close to 0.50 in absolute terms. Deep in-the-money options approach 1 and behave like the underlying itself. Far out-of-the-money options approach 0. Delta moves with spot, time and volatility, and gamma measures how fast it changes. Traders often read delta as a rough probability of finishing in the money. The shortcut is useful but imprecise: for a call under Black-Scholes, the risk-neutral probability is N(d2), while delta is N(d1).

Example

Hypothetical: a trader holds 10 call contracts with a multiplier of 100 and a delta of 0.40. The position delta is 10 × 100 × 0.40 = 400 shares. To be delta neutral, the trader sells 400 shares. If the stock then rises by $0.50, the calls gain about 400 × $0.50 = $200 and the short stock loses $200.

On the desk

Delta depends on the volatility model. On a skewed surface, the delta that comes out of Black-Scholes at a fixed implied vol differs from the delta implied by how the surface actually moves with spot. Desks that ignore this end up systematically over- or under-hedged.