An option has a price today, and that price moves when the stock moves. Delta says by how much. It is the first number a trader checks on any position, because it answers the question that matters most: how much stock am I really long or short right now? Every other Greek in this module describes how delta itself changes, so it is worth getting delta clear first.
Delta as price sensitivity
Delta is the change in an option's price for a $1 move in the stock. Say SPY is at $750 and the $750 call has a delta of 0.55. If SPY rises to $751, the call gains about 55 cents per share, which is $55 on one contract. If SPY falls to $749, the call loses about the same. A put moves the other way, so put deltas are negative: the $750 put might show minus 0.45, losing 45 cents on that same $1 rise.
Call deltas run from 0 to 1. Put deltas run from 0 to minus 1. At the same strike and expiration, the call delta and the put delta differ by one, near enough: a 0.55 call sits next to a minus 0.45 put.
Delta as a hedge ratio
One contract covers 100 shares, so a 0.55 delta call behaves like 55 shares of SPY. That number is called the hedge ratio. If you own the call and want no exposure to the next $1 move, you sell 55 shares against it. Own ten contracts and you sell 550. A position with zero net delta is delta neutral. Market makers do this all day: they take on the option and hedge the stock, so what they are left holding is not a view on direction but a view on movement and time.
Delta as a rough probability
Delta also works as a quick estimate of the chance the option finishes in the money. A 0.25 delta call has roughly a one in four chance of expiring above its strike; a 0.90 delta call is very likely to. It is an approximation: the true model probability is a slightly different number, and delta runs a little above it by an amount that grows with volatility and time. It is close enough that traders name strikes by delta. The 25 delta put means the same thing on any stock at any price.
A call with 60 days left. Raise volatility or days and the gap between the two lines widens. At the strike, a 0.50 delta call is slightly less than a coin flip to finish in the money.
How delta changes
Delta depends on where the strike sits against the stock, called moneyness, and on how much time is left.
- Moneyness. A deep in-the-money call is nearly certain to be exercised, so it moves one for one with the stock: delta near 1. A far out-of-the-money call barely reacts: delta near 0. At the money sits close to 0.50. Plot delta against strike and you get an S-shaped curve.
- Time. With months to go, the curve is gentle: a $5 move changes delta a little. In the last days before expiration, the curve turns into a step. An at-the-money option snaps toward 0 or 1 with small moves in the stock, because there is no time left for it to come back.
Higher implied volatility has the same effect as more time: it flattens the curve, so every strike looks a bit more like a coin flip.
The S-curve steepens into a step as expiry approaches. Hover to read delta at any price; switch to puts to see the same curve shifted down by one.
The maths: delta, probability and the put-call link
In the Black-Scholes model a call's delta is e^(-qT) N(d1) and the model's probability that it finishes in the money is N(d2), where d2 = d1 - v√T. Because d1 is always larger than d2, delta always sits above that probability, and the gap grows with v√T: more time or more volatility widens it. For a one-month option at 30% volatility, v√T is about 0.09, so a 0.50 delta call has roughly a 46% chance of finishing in the money.
Put-call parity ties the two sides together. A call minus a put at the same strike is a forward, whose delta is e^(-qT), so call delta minus put delta equals e^(-qT). With no dividend that is exactly one, which is why a 0.55 call sits beside a minus 0.45 put.
Your turn
Open the SPY chain on the monthly expiration. Read the call and put deltas at the money, note that they differ by one, then scan down the put side to find the strike whose delta is nearest minus 0.25.