Delta: hedge ratio and probability proxy
Lesson 8 · about 10 min
The Greeks are sensitivities: each one answers "if this input changes by one unit, how much does my option price change?" Delta is the first because it is the biggest, the most useful, and the one that doubles as a rough probability.
The definition
Delta is the change in the option's price for a $1 change in the stock, holding everything else constant.
- Calls have delta between 0 and +1.00.
- Puts have delta between −1.00 and 0.
- Traders usually quote it without the decimal: a "0.35 delta call" is a "35-delta call".
XYZ at $50. The $50 call has a delta of 0.52 and trades at $1.90. XYZ ticks up to $51. The call should move to roughly $1.90 + $0.52 = $2.42. Per contract, that is a $52 gain on a $1 move in the stock.
The $50 put has delta −0.48. On the same $1 rise it loses about $0.48, going from $1.85 to about $1.37.
Delta across the chain
| Strike | Call delta | Put delta | Note |
|---|---|---|---|
| 44 | 0.92 | −0.08 | Deep ITM call, far OTM put |
| 47 | 0.78 | −0.22 | |
| 50 | 0.52 | −0.48 | ATM; slightly above 0.50 for the call because of drift and skew |
| 53 | 0.27 | −0.73 | |
| 56 | 0.10 | −0.90 | Far OTM call, deep ITM put |
Two patterns. Call delta plus the absolute value of put delta at the same strike is always 1.00 (a consequence of put-call parity). And delta moves from near zero far out of the money to near one deep in the money, passing through about 0.50 at the money.
Delta as a hedge ratio
Delta tells you how much stock the option behaves like. A 0.52-delta call on 100 shares behaves, for small moves, like 52 shares. That is the origin of the word "hedge ratio": a market maker who sells you that call and wants to be flat buys 52 shares against it.
For you, the same arithmetic gives a position's net delta, which is the single most useful summary of directional exposure:
| Position | Delta each | Contracts | Share-equivalent |
|---|---|---|---|
| Long 2 × $50 call | +0.52 | 2 | +104 |
| Short 1 × $53 call | −0.27 | 1 | −27 |
| Long 1 × $47 put | −0.22 | 1 | −22 |
| Net | +55 shares |
That book, whatever it is called, is currently long about 55 shares of XYZ. A $2 drop costs roughly $110. If you would not be comfortable being long 55 shares outright, you should not be comfortable with this option position either. Netting your deltas turns a collection of contracts into a number you can size against.
Delta as a probability proxy
Delta is also, roughly, the probability that the option finishes in the money. A 27-delta call has about a 27% chance of expiring with the stock above $53; a 90-delta put has about a 90% chance of expiring in the money. The approximation is not exact (the true figure is a related quantity that differs by a volatility term, and skew distorts it further), but it is close enough to be the way most traders think about strike selection.
This turns the chain into a probability table. Selling a 16-delta put means selling something with about a one-in-six chance of finishing in the money. Buying a 10-delta call means buying something with about a one-in-ten chance of paying off at all. Nothing about delta tells you the size of the payoff when it does happen, which is why "high probability" trades (Module 7) can still have poor expectancy.
Key idea: Delta is how many shares your option behaves like, and roughly the chance it finishes in the money. Net delta across a position is the number to size against.
How delta changes
Delta is not fixed. Three things move it:
- The stock moves. As XYZ rises, the $50 call's delta rises toward 1.00 and the $50 put's delta shrinks toward 0. The rate of that change is gamma, the next lesson.
- Time passes. With less time, there is less chance of crossing the strike, so OTM deltas fall toward 0 and ITM deltas rise toward 1.00. ATM delta stays near 0.50 until the very end, then snaps to 0 or 1 at expiration.
- Volatility changes. Higher volatility pushes all deltas toward 0.50 (more uncertainty about where the stock will end up); lower volatility pushes them toward the extremes.
| Days to expiry | $53 call delta (XYZ at $50) | $47 call delta |
|---|---|---|
| 90 | 0.38 | 0.66 |
| 30 | 0.27 | 0.78 |
| 7 | 0.12 | 0.92 |
| 1 | 0.02 | 0.99 |
The OTM call loses its delta as time runs out; the ITM call becomes a share of stock.
Dollar delta
For comparing positions across different stocks, multiply by the stock price: dollar delta = delta × 100 × stock price. A 0.52 delta call on a $50 stock is $2,600 of exposure; a 0.52 delta call on a $500 stock is $26,000. Same delta, ten times the money at risk per percentage move.
Try it: Write down every option position you hold or are considering. Compute the net delta in share-equivalents and the dollar delta. Ask whether you would hold that much stock. If not, the position is too big regardless of what the premium looked like.
Recap
- Delta is the option's price change per $1 move in the stock; calls run 0 to +1.00, puts 0 to −1.00.
- At the same strike, call delta + |put delta| = 1.00.
- Delta is the share-equivalent of a position; net delta across a book is what you should size against.
- Delta is a rough probability of finishing in the money, but it says nothing about payoff size.
- Delta moves with the stock (gamma), with time (OTM toward 0, ITM toward 1) and with volatility (toward 0.50 when volatility rises).