Gamma

The gamma of an option is the second derivative—the rate of change of delta with respect to the underlying asset’s price. Gamma is always positive for long options (you own them) and always negative for short options (you sold them). Gamma is highest for at-the-money options and falls to near-zero for deep in-the-money or out-of-the-money options. Gamma quantifies the instability and rehedging cost of delta-hedged positions.

Gamma as delta acceleration

If a call option has a delta of 0.5 and a gamma of 0.05, then a $1 move in the stock increases the delta to 0.55 (for upward moves) or decreases it to 0.45 (for downward moves).

This acceleration of delta is gamma. It quantifies how quickly your directional sensitivity changes as the stock moves. High gamma means delta is unstable and requires frequent rehedging. Low gamma means delta is stable.

Gamma is always positive for owned options

Whether you own a call or put, gamma is positive. This is because option value is convex—the option benefits from large moves in either direction. A $100 strike call is worth more if the stock rises to $110 OR falls to $90 (no benefit from the fall, but lower time value) relative to just staying at $100.

For short options (sold/written), gamma is negative. You lose from large moves in either direction; you want the stock to stay near the strike.

Gamma and the Greeks’ relationship

Gamma is related to theta through a key equation: for a delta-neutral portfolio, theta + ½ × volatility² × gamma ≈ 0.

This says: if you are theta-positive (profiting from time decay), you are gamma-negative (losing from large moves). If you are gamma-positive (long volatility), you are theta-negative (losing to time decay). This trade-off is fundamental in options markets.

Gamma and rehedging costs

When you delta-hedge by buying/selling shares to neutralize delta, you are betting the stock will not move much. But when the stock does move, delta changes (due to gamma), and you must rehedge. This rehedging locks in losses.

Suppose you short a call and delta-hedge by buying shares. If the stock rises:

1. The call delta increases (you are now long-gamma-exposed).
2. You must sell shares to restore delta neutrality.
3. You sell those shares at a higher price than you bought them.
4. But now the stock falls back to the original level.
5. The call delta decreases (back toward the original).
6. You must buy shares again, now at the original (higher) price.

You have locked in a loss. The size of this loss is roughly: realized volatility² × gamma × time. This is the “gamma cost” of hedging.

Gamma concentration near the strike

At-the-money options have maximum gamma. An at-the-money call with 90 days to expiration might have a gamma of 0.02 (meaning delta increases by 0.02 per $1 move). A deep out-of-the-money call on the same stock might have a gamma of 0.001.

This concentration of gamma at the strike is why near-the-money options are riskier to hedge and why traders focus gamma management effort on contracts near the strike.

Gamma and time to expiration

For at-the-money options, gamma increases as expiration date nears. A call with 1 day to expiration has much higher gamma than the same call with 60 days to expiration (assuming same volatility). The delta is forced to swing from 0.5 (if out-of-the-money) to 1.0 (if in-the-money) over the final day, creating extreme gamma.

This is why near-expiration options are volatile and difficult to hedge.

Gamma scalping

Gamma scalping is a strategy where you hold a long gamma position (own options) and delta-hedge it. As the stock moves, delta changes, you rehedge, and you pocket the rehedging profit. This requires significant transaction costs and is typically profitable only when realized volatility exceeds implied volatility.

See also