Delta
The rate of change in an option's price for a $1 move in the underlying. Ranges from 0 to 1 for calls and −1 to 0 for puts.
Δ = ∂V / ∂S (change in option value per $1 change in spot)
Delta measures how much an option's premium changes when the underlying moves by $1. A call with delta 0.50 gains approximately $0.50 per share (or $50 per contract) if the underlying rises $1.
Delta also approximates the probability that an option expires in the money — a 0.20 delta OTM call has roughly a 20% chance of finishing ITM. Deep ITM options approach delta ±1; far OTM options approach 0.
Delta is dynamic: it increases as the option moves ITM (gamma effect) and decreases as it moves OTM. A delta-neutral portfolio is hedged against small directional moves.
On the desk
A 0.25 delta call can behave more like 0.40 after a fast rally. Size that assumed a stable delta will be wrong in the way that matters.