Gamma

Gamma: Rate of Change of Delta

Gamma (Γ) measures how much delta changes for a $1 move in the underlying:

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

Black-Scholes Gamma

Gamma is the same for both calls and puts (put-call parity):

$\Gamma = \frac{N'(d_1)}{S \cdot \sigma \cdot \sqrt{T}}$

where $N'(x)$ is the standard normal PDF:

$N'(x) = \frac{e^{-x^2/2}}{\sqrt{2\pi}}$

Interpretation

• Gamma measures the curvature of the option price with respect to the stock price
• High gamma means delta changes rapidly — the hedge needs frequent rebalancing
• Gamma is highest for ATM options near expiry
• Gamma is always positive for long options (calls and puts)

Why Gamma Matters

If you hold a delta-neutral portfolio and the stock makes a large move, gamma tells you how much your delta has drifted. A high-gamma position benefits more from large moves (volatility) but requires more frequent rebalancing.

Relationship to Theta

Long gamma (long options) tends to come with negative theta — you pay for the convexity through time decay.