Option Gamma

Option gamma is the estimated change in an option's delta for a $1 change in the price of the underlying security, with other inputs held constant. FINRA defines it as "the rate of change in an option's delta based on a $1 change in the price of the underlying security," and the Options Industry Council puts it the same way: "How Delta is expected to change given a $1 move in the underlying is called Gamma." Delta tells an option holder how much the price should move now; gamma tells them how much that answer will change once the stock has moved. It is one of the options Greeks, and the council's glossary classes it as a second-order one because it measures the change in another Greek rather than in the option's price.

Using gamma to update delta. The arithmetic depends on the direction of the move. When the stock rises $1, a long call's delta increases by about the gamma; when it falls $1, delta decreases by about the gamma. A call with a delta of 0.60 and a gamma of 0.04 should show a delta of about 0.64 after a rise and about 0.56 after a fall. For puts, whose deltas are negative, a rise in the stock moves delta toward 0 and a fall moves it toward −1. Since a standard contract covers 100 shares, a gamma of 0.04 means the position's share-equivalent exposure changes by about four shares for each $1 the stock moves.

The sign, and who it helps. The council states that "long options, either calls or puts, always yield positive Gamma," that "short calls and short puts will have negative Gamma," and that stock positions have no gamma at all because their delta "is always 1.00 (long) or -1.00 (short) and will not change." Positive gamma is why an option buyer's exposure grows as the stock moves in their favor and shrinks as it moves against them: the gains on a move in the right direction come at an increasing rate, and the losses on a move in the wrong direction at a decreasing one. For a writer the same mechanics run in reverse, so exposure builds against the position as the stock moves the wrong way.

Where gamma is largest. FINRA notes that gamma is typically highest for at-the-money options near expiration, and the council adds that it is "highest when the Delta is in the .40-.60 range" and usually at its lowest as delta approaches 0 or 1. A near-dated at-the-money option has more gamma than a longer-dated option at the same strike, because its delta has little time left to settle at 0 or 1. Deep in-the-money and far out-of-the-money options carry little gamma, since their deltas are already near an endpoint. The extreme case is an option close to expiration with the stock right at the strike, where small moves swing delta sharply; that is the territory of 0DTE options, contracts traded on their last day.

Implied volatility. Gamma also depends on how much movement the market expects. The council notes that "as implied volatility decreases, Gamma of at-the-money calls and puts increases," explaining that low implied volatility options "will have a more dramatic change in Delta when the underlying moves." Like every Greek, gamma comes from a pricing model and changes as the stock price, time and implied volatility change.

Gamma is quoted per share, for a $1 move in the underlying. Add it to delta after a $1 rise, subtract it after a $1 fall, and use the average of the old and new deltas to estimate the price change over the move.

Take an example, with illustrative figures. A stock trades at $142. A call with a $140 strike and 21 days to expiration is quoted at $5.30, with a delta of 0.60 and a gamma of 0.04.

• If the stock rises $1 to $143, the new delta should be about 0.60 + 0.04 = 0.64. Using the average delta over the move, (0.60 + 0.64) ÷ 2 = 0.62, the call should be worth about $5.30 + $0.62 = $5.92, or $62 more on one 100-share contract.
• If the stock falls $1 to $141, the new delta should be about 0.60 − 0.04 = 0.56, and the average delta is (0.60 + 0.56) ÷ 2 = 0.58, so the call should be worth about $5.30 − $0.58 = $4.72, a loss of $58 a contract.
• The same $1 move gains $62 one way and loses $58 the other. That $4 difference is the holder's positive gamma at work.
• The contract's share-equivalent exposure moves from 60 shares to about 64 after the rise and about 56 after the fall.

Compare a call at the same strike with a year to run. Its gamma might be about 0.01, so the same $1 move changes its delta by only about 0.01. A writer of the 21-day call has a delta of −0.60 and a gamma of −0.04, so after a $1 rise their delta is about −0.64 and each further dollar costs them more than the last.

Pros

• Shows how stable delta is, which delta alone cannot tell.
• Explains why an option buyer's gains can run faster than their losses for the same size of move.
• Flags the positions, usually short-dated at-the-money options, whose exposure can change fastest.

Cons

• It is a second model estimate on top of delta and is itself valid only for small moves.
• It changes quickly near expiration, so a reading can be stale within minutes when it matters most.
• Its size depends on the stock's price level, so gamma figures are hard to compare across different stocks.
• For a writer, negative gamma means exposure builds against the position as the stock moves the wrong way.