Delta: Three Questions, One Number
Delta answers three questions at once, and traders routinely confuse the useful two with the misleading third.
Worth reading first: Moneyness: Where the Stock Sits Relative to Your Strike
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Delta is the first greek most people meet and the one most often misused. It is not a complicated idea. The trouble is that one number gets used to answer three different questions. Two of those answers are solid. The third is a rough approximation that traders repeat as though it were exact.
Have a play with the model below, then we will take the three readings one at a time.
Delta across every stock price
The solid line is delta at your chosen expiration; the dashed line is the same strike with 120 days left. Shorten the expiration and watch the curve stiffen into a step.
- Delta at 30 days
- Delta at 120 days. Flatter, less certain
Reading one: the rate of change
This is the definition, and it is exact. Delta is how much the option's price changes when the stock moves a dollar.
Δ option price ≈ delta × Δ stock pricePer share. Multiply by 100 for the change in the contract’s dollar value.
A put with a delta of −0.30 gains about $0.30 a share, or $30 a contract, for every dollar the stock falls. A call with delta 0.65 gains $65 a contract per dollar of rise. Calls have positive delta and puts negative, because they move in opposite directions.
The important word is about. Delta is the slope of a curve, so it is only accurate for small moves. The slope itself changes as the stock moves, and how fast it changes is gamma, which is the next lesson.
Reading two: the share equivalent
Multiply delta by 100 and you get the number of shares the contract currently behaves like. A 0.30 delta call acts like 30 shares. A short 0.30 delta put also acts like being long 30 shares, because you make money when the stock rises.
This hedge ratio reading is the useful one once you hold more than one position. It lets you add up a whole portfolio of options and stock into a single directional number. Two short 0.30 delta puts plus 100 shares is not “some options and some stock”. It is 160 shares of directional risk, and that is something you can size against your account.
Reading three: the odds, roughly
You will hear that delta is the probability of finishing in the money, so a 0.30 delta put has a 30% chance of assignment. That is close enough to be useful and wrong enough to be worth knowing about.
The model's actual probability of finishing in the money is a related but different number, N(d₂) if you want the notation. Delta is N(d₁). They come apart for three reasons: volatility, time, and the fact that the price distribution is lognormal rather than symmetrical.
| Implied volatility | Put delta | Model P(ITM) | Gap |
|---|---|---|---|
| 15% | −0.15 | 13% | 2 points |
| 30% | −0.30 | 27% | 3 points |
| 60% | −0.37 | 32% | 5 points |
| 90% | −0.40 | 33% | 7 points |
A $95 put on a $100 stock, 30 days out. Delta versus the model’s actual probability of finishing in the money.
At normal volatility the approximation is fine. Being three points out on an assignment estimate will not change what you do. On a high-volatility name the gap widens enough to matter, and it always leans the same way: delta overstates the risk of assignment. Use delta as your probability and you are being a little conservative, which is a reasonable direction to be wrong in.
What makes delta move
Three things reshape the curve you were just dragging, and each is worth feeling directly in the widget.
Moneyness. Deep in the money, delta heads towards 1.00 and the option is effectively stock. Deep out of the money it heads towards zero and the option barely reacts to the stock at all. At the money it sits near 0.50.
Time. Compare the solid and dashed lines. With 120 days left the curve is gentle, because there is time for anything to happen and no strike is certain. Drag the expiration down to a week and it stiffens towards a step. Near expiry, delta is really just answering whether the contract is in the money.
Volatility. Raising IV flattens the curve for the same reason more time does. Both widen the range of ground the stock could cover, which makes every strike less certain and pulls every delta towards 0.50.
What premium sellers do with it
Delta is the standard way of expressing a strike choice. “Sell the 16 delta put” is an instruction anyone can act on, whatever the stock, the price or the volatility. “Sell the strike $5 below” is not.
A 0.16 delta short put is roughly one standard deviation out of the money, wins something like 84% of the time, and pays little. A 0.30 delta short put pays a good deal more and gets assigned far more often. Neither is right. They are different businesses, and the strike selection lesson puts real numbers on the trade-off.
What can go wrong
Treating delta as fixed. It is a snapshot. The 0.16 delta put you sold three weeks ago might be a 0.45 delta put now, carrying three times the exposure you signed up for.
Sizing by contracts instead of delta. Ten far-out contracts can carry more risk than two near ones, and the position list will not show it.
Trusting the probability reading on a volatile name. Use it as a rough and slightly conservative guide, not as an input to a model.
Key takeaways
- Delta is exactly the rate of change of the option price per $1 of stock movement.
- Multiplied by 100 it is a share equivalent, and that is the reading to size positions with.
- Delta approximates the probability of finishing in the money, and overstates it. Mildly at normal volatility, more on volatile names.
- Delta moves as moneyness, time and volatility change. Less time and lower volatility both steepen the curve.
- Choosing a strike by delta works on any stock, in a way that choosing by dollar distance does not.
Check your understanding
1. You are short one put with a delta of −0.25. The stock falls $2. Roughly what happens to the position?
2. You hold 100 shares and are short two puts at 0.20 delta each. What is your total share-equivalent exposure?
3. Delta says 0.30. How should you read that as a probability of assignment?
Record delta at entry
Premium Tracker stores the delta you sold at, so you can look back and see which strike-selection rule actually paid.
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