Delta and Gamma: An Option's Speed and Acceleration

Delta: the option's speed

The slope of the option price curve at any point is the Delta: when the stock rises $1, a call option rises about Delta dollars. Out of the money it's close to 0, in the money close to 1, and near the money about 0.5.

Gamma: the change in speed

Gamma measures how fast Delta itself changes. Gamma is biggest for at-the-money options close to expiration: the stock moves a little and the option goes from “barely moving” to “moving one-for-one with the stock”. This is where options get their “nonlinearity”.

Delta converts into shares

1 contract = 100 shares, so for small stock moves, 1 Call with a Delta of 0.5 gains or loses about as much as 50 shares. Add up your positions' Deltas to 0 and you're Delta neutral.

Key takeaways

• Delta ≈ the change in the option price when the stock rises $1. Calls range from 0 to 1, puts from −1 to 0.
• Gamma is how fast Delta changes. At-the-money options close to expiration have the highest Gamma.
• High Gamma means P&L changes sharply: buyers enjoy the acceleration, sellers suffer it.
• Delta × 100 = how many shares 1 contract is equivalent to. When it all adds up to 0, you're Delta neutral.

Try it yourself

You can do these exercises in this lesson's interactive simulation:

1. Slowly drag the stock from $70 to $130: the blue tangent gets steeper and steeper. What does Delta go from and to?
2. Put the stock at $100 and drag the days to expiration from 180 to 2: what shape does the Gamma curve below take?
3. Look at “After stock +$1” on the right: how far is the actual price from the Delta estimate? Does the gap grow as Gamma grows?
4. Click “Deep in the money”, “Deep out of the money” and “At the money · 2 days” in turn: what's “1 contract ≈ how many shares” for each? Switch to Put and look again: did the numbers turn negative?

Quiz and explanations

1. A call has Delta = 0.6 and the stock rises $2. Roughly what happens to the option price?

Answer: Up $1.2.0.6 × $2 = $1.2 (only an approximation; Gamma makes the real change slightly bigger).

2. Which option usually has the highest Gamma?

Answer: At the money, a few days to expiration.At the money and close to expiration, Delta flips quickly between 0 and 1.

3. You sold 10 calls with a Delta of 0.30. To get Delta neutral with stock, you should…

Answer: Buy 300 shares.The short calls' Delta = −0.30 × 10 contracts × 100 = −300 shares, so buying 300 shares offsets it.

Terms in this lesson

• Delta: Roughly how much an option's price changes when the underlying rises $1; near 1 deep in the money, near 0 far out of the money.
• Delta neutral (Delta Neutral): A portfolio whose total Delta is 0, so small moves in the underlying barely change its value.
• Gamma: How much Delta changes when the underlying moves $1; the higher the Gamma, the more often a Delta hedge needs adjusting.
• volatility (Volatility): How wildly the underlying's price swings; the bigger the swings, the more expensive options get.