Delta: what the option does when the stock moves a dollar

Delta is a rate of change: how much an option's price moves when the underlying stock moves by one dollar, everything else held still. A call with a delta of 0.40 gains about $0.40 when the stock gains $1 and loses about $0.40 when the stock falls $1. Puts have negative deltas; a −0.30 put gains $0.30 when the stock drops $1. That's the whole definition. Everything else people say about delta is derived from it, and the derivations come with conditions.

Delta as shares

The cleanest use of delta is as exposure. A short put with a delta of −0.30 gives you the same immediate sensitivity to the stock as owning 30 shares: a $1 move in the stock moves the option position by about $30. Add up the deltas across a portfolio, contract by contract, and you get a rough measure of how long or short you actually are. Ten short puts at −0.20 each is about 200 shares' worth of long exposure, which is worth knowing when you thought you had ten small positions.

The probability shortcut

Delta gets read as the probability that the option expires in the money. A 0.16-delta put becomes "about a 16% chance of being assigned," and traders talk about selling "the 16 delta" as shorthand for one standard deviation out. This works as a rough guide, and it's roughly right for options that aren't too far from the money on stocks that aren't too skewed. It also fails in specific ways.

It's the probability in the pricing model's world, which assumes no drift and a tidy distribution. Real stocks have skew: puts price a fatter downside than the model's symmetry implies, so a put's delta can understate the real chance of finishing in the money. The gap is bigger in single stocks than in indexes and bigger in stressed markets.

More important: delta approximates the probability of finishing in the money at expiration. It says nothing about the probability of being in the money at some point along the way, which is roughly double. A 16-delta put has about a 16% chance of expiring in the money and something closer to a 30% chance of touching the strike before then. Sellers who manage positions when the strike is threatened live with the second number, not the first.

Delta moves

Delta isn't fixed. As the stock falls toward a short put's strike, the put's delta grows in magnitude: −0.16 becomes −0.30 becomes −0.50 at the strike. The speed of that change is gamma, and it's why a short option that looked small becomes large at exactly the moment it's losing. Delta also drifts with time: an out-of-the-money option's delta shrinks toward zero as expiration approaches, an in-the-money one's grows toward 1.

Using it

Delta is for choosing strikes and measuring exposure. Selling at a target delta, 0.20 or 0.30, is a consistent way to pick strikes across stocks with different prices and volatilities, since it adjusts for both. Summing deltas tells you what the book actually looks like. Reading a 0.16 delta as "84% safe" is the mistake: it's 84% at one moment, in one model, for one outcome, and the trade lives through all the others.