Read the change in the slope
For a hypothetical call with delta 0.55 and gamma 0.04, a $1 stock rise changes delta locally toward 0.59. A $1 fall changes it toward 0.51. Gamma itself changes too; these are nearby estimates, not a rule to extend across any price range.
Long conventional calls and puts generally have positive gamma; short positions reverse the sign. Positive gamma means delta tends to increase as the stock rises and decrease as it falls.
Compare two approximations
Allowing delta to change
Read the chart data
| Stock price change · USD | Delta only (Premium change · USD/share) | Delta + gamma estimate (Premium change · USD/share) |
|---|---|---|
| -2 | -1.1 | -1.02 |
| -1 | -0.55 | -0.53 |
| 0 | 0 | 0 |
| 1 | 0.55 | 0.57 |
| 2 | 1.1 | 1.18 |
For a $2 increase, delta alone estimates 0.55 × 2 = $1.10 per share. Including the second-order term gives 1.10 + ½ × 0.04 × 4 = $1.18. For a $2 decrease, the same approximation is −1.10 + 0.08 = −$1.02. The asymmetry comes from curvature, not an invented historical return.
Understand concentrated exposure
Gamma can become large for near-the-money options as expiration approaches. A small stock move can then change directional exposure rapidly. An option being short-lived or inexpensive in dollars does not mean its position sensitivity is stable.
Suppose a short call’s long-contract delta moves from 0.50 to 0.80. One short standard contract changes from about −50 to −80 share-equivalent delta. A hedge chosen earlier may no longer offset the position. A sharp move can leave little time to respond.
Read gamma alongside time and volatility
Positive gamma is not free profit. Purchased options have an upfront cost, often negative theta and exposure to changes in IV. Realized movement, execution costs and any hedging rules determine the result. Short gamma can accompany premium collection, but adverse moves can expand losses quickly.
Use a fresh model for large scenarios, preserve funding capacity, and check exercise mechanics. This lesson explains the changing exposure; it does not propose a delta-hedging strategy or promise that repeated hedging will earn money.
With delta 0.55 and gamma 0.04, is the estimated premium gain for a $2 move simply 0.63 × 2?
Show the explanation
No. That uses an estimated ending delta for the whole move. The local second-order estimate is 0.55 × 2 + ½ × 0.04 × 2² = $1.18 per share.
Sources and editorial notes
Original explanations and hypothetical examples prepared with AI assistance. Primary reference check dates are listed below. Numerical examples are hypothetical unless explicitly identified otherwise.
- OCC / OIC: Gamma
Definition, sign and conventional units for this sensitivity. Fixed-input arithmetic, scenarios and visuals are original and explicitly local. Checked October 17, 2026.
- OCC / OIC: Delta
Definition, sign and conventional units for this sensitivity. Fixed-input arithmetic, scenarios and visuals are original and explicitly local. Checked October 17, 2026.