Gamma Explained: The Rate of Change of Delta

Before this article: Delta Hedging: How Market Makers Cover Directional Risk.

What Gamma measures

Delta tells you how much an option's value moves when the underlying moves. Gamma tells you how fast that Delta changes. It's literally the second derivative of the option's value with respect to the underlying's price, or the first derivative of Delta.

$$ \Gamma = \frac{\partial \Delta}{\partial S} = \frac{\partial^2 V}{\partial S^2} $$

Where:

• $\Gamma$ = the option's Gamma.
• $\Delta$ = the option's Delta.
• $S$ = the underlying asset's price.
• $V$ = the option's value (price).

Key idea: if Delta is speed, Gamma is acceleration. A Delta that changes fast means the dealer has to rebalance their hedge much more often, and that's what connects Gamma to the market's actual price movement.

Where Gamma is highest

Gamma isn't constant across strikes. It's highest near the current price (ATM) and decreases the further an option is, whether ITM or OTM.

flowchart LR
    OTM["OTM<br/>Low Gamma"] --> ATM["ATM<br/>High Gamma"] --> ITM["ITM<br/>Low Gamma"]

Note: Gamma also grows as expiration gets closer. A 0DTE option carries far more intense Gamma than one with weeks left: that behavior gets its own dedicated article later in this guide.

Delta vs Gamma, side by side

Numeric example

Example: an ATM call on the underlying with a 0.50 Delta and 0.02 Gamma. If the underlying rises 10 points, the new Delta is approximately:

$$0.50 + (0.02 \times 10) = 0.70$$

The dealer who sold that call now needs to hedge a higher Delta than before, not because they changed their mind, but because Gamma's mechanical formula requires it. Illustrative figures to explain the mechanics.

Why this creates two market regimes

This is where the framework becomes actionable. Depending on whether dealers are, in aggregate, long or short Gamma, their hedging either brakes the price move or accelerates it. That defines two regimes you'll see in detail later in this guide:

• Long Gamma Regime: dealers buy when price falls and sell when it rises. Their hedging dampens the move: the market tends to trade in a range.
• Short Gamma Regime: dealers sell when price falls and buy when it rises. Their hedging amplifies the move: the market tends toward trend and volatility.

Next step

You now know what Gamma measures and why it creates two distinct regimes. The next article covers why the dealer doesn't hedge just once, but keeps rebalancing all day long: Gamma Hedging.

Continue with: Gamma Hedging: Why Dealers Adjust Their Coverage All Day Long (/docs/gamma-hedging-dynamic-coverage, next wave)