Lesson 13 of 20 · 7 min
Gamma: the rate of change of delta
Gamma measures how quickly your exposure changes underneath you, and it explodes near expiry.
Delta describes exposure now. Gamma describes how unstable that description is. It is the change in delta per one-dollar move in the underlying — a second derivative, and the reason option positions cannot be managed with delta alone.
In the base case gamma is 0.0555. That means a one-dollar rise in the underlying should raise delta by about 0.055, from 0.533 to roughly 0.588. Check it against the model: delta at $101 is 0.587. Gamma predicted the change in the prediction, and it was right to three decimal places.
The behaviour that makes gamma important is its concentration. Gamma is largest for at-the-money contracts and it grows sharply as expiry approaches. In the base case at 30 days gamma is 0.055; at 5 days it is 0.136; at 1 day it is 0.305. So a one-dollar move in the last session changes delta by 0.30 — the position's entire exposure profile can reorganise itself in an afternoon.
The intuition is straightforward once stated. Far from expiry, an at-the-money option does not know how it will end and its delta sits near the middle. Close to expiry, delta must resolve to either 1 or 0 depending on which side of the strike the price lands, so it becomes violently sensitive to small moves. Gamma is the measure of that impending resolution.
Gamma also explains a fact from lesson 11: a long option gains more than delta predicts on the way up and loses less than delta predicts on the way down. Positive gamma curves the payoff in the holder's favour, and it is the property the premium is largely paying for.
Key terms
- Gamma
- The change in delta per one-dollar change in the underlying. A second derivative of price.
- Positive gamma
- The property of long options: gains accelerate and losses decelerate relative to what delta alone predicts.
- Gamma concentration
- Gamma's tendency to be largest at the money and to grow rapidly as expiry approaches, because delta must resolve to 0 or 1.
Worked example
Gamma predicting delta, and gamma exploding
Underlying 100, strike 100, rate 4%, volatility 25%. Delta and gamma at two expiries.
- 30 days: delta, gamma
- 0.533, 0.0555
- Gamma's prediction of delta at $101
- 0.588
- Actual delta at $101, 30 days
- 0.587
- 5 days: delta, gamma
- 0.513, 0.1363
- Actual delta at $101, 5 days
- 0.646
- 1 day: gamma
- 0.3048
Prediction confirmed.
A one-dollar move shifted delta by 0.13.
Five and a half times the 30-day figure.
The same one-dollar move changes exposure by 0.055 at a month out and by 0.30 on the final day. Gamma is what makes late-life positions feel unrecognisable.
Check yourself
The base-case call has a delta of 0.533 and a gamma of 0.0555. What is delta after a $1 rise, and why does the answer matter more with 5 days left than with 30?
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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.