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
Delta is the first greek everyone meets and the one most often misused. The confusion is not that it is complicated — it is that a single number gets used to answer three different questions, and two of those answers are solid while the third is a convenient lie that traders repeat anyway.
Move 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 one 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 per share — $30 per contract — for every dollar the stock falls. A call with delta 0.65 gains $65 per contract per dollar of rise. Calls have positive delta, puts negative, because they move in opposite directions.
The word doing quiet work there is approximately. Delta is the slope of a curve, and it is only accurate for small moves, because the slope itself changes as the stock moves. How fast it changes is gamma, 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 of stock. A short 0.30 delta put also acts like being long 30 shares — you profit when the stock rises.
This hedge ratio reading is the practical one for anyone running more than one position. It lets you add up an entire portfolio of options and stock into a single directional exposure. 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 a number you can actually size against your account.
Reading three: the odds, approximately, with caveats
You will hear that delta is the probability of the option finishing in the money. A 0.30 delta put has “a 30% chance of assignment”. This is close enough to be useful and wrong enough to be worth knowing about.
The genuine model quantity for probability of finishing in the money is a related but different number (N(d₂), for anyone who wants the notation). Delta is N(d₁). They diverge for three reasons: volatility, time, and the fact that the underlying 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 ordinary volatility the approximation is fine — being three points out on an assignment estimate will not change a decision. On a high-volatility name it drifts far enough to matter, and it always drifts in the same direction: delta overstates the risk of assignment. If you use delta as your probability, you are being slightly conservative, which is a tolerable place to be wrong.
What makes delta move
Three things reshape the curve you just dragged, and each one is worth feeling directly in the widget.
Moneyness. Deep in the money, delta approaches 1.00 — the option is effectively stock. Deep out of the money it approaches zero — the option barely notices 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: there is plenty of time for anything to happen, so no strike is very certain. Drag the expiration down to a week and the curve stiffens toward a step function — near expiry, delta is really just answering “is this in the money or not?”
Volatility. Raising IV flattens the curve, for the same reason more time does. Both increase the amount of ground the stock could plausibly cover, which makes every strike less certain and pulls every delta toward 0.50.
What premium sellers actually do with it
Delta is the industry’s standard unit for expressing strike selection. “Sell the 16 delta put” is a complete, portable instruction — it works across any stock at any price with any volatility, which “sell the strike $5 below” does 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 accordingly little. A 0.30 delta short put pays substantially more and gets assigned far more often. Neither is correct; they are different businesses. The strike selection lesson puts real numbers on that trade-off.
What can go wrong
Treating delta as fixed. It is a snapshot. The 0.16 delta put you sold three weeks ago may be a 0.45 delta put today, with three times the directional exposure you signed up for.
Sizing by contracts instead of delta. Ten far-out contracts can carry more risk than two near ones, and it will not be obvious from the position list.
Trusting the probability reading on a volatile name. Use it as a rough, slightly conservative guide, not as a number to build a model on.
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, but 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.
- Strike selection by delta is portable across stocks in a way that selecting by dollar distance is 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.