Gamma
Second derivative of option price with respect to the underlying — the rate of change of delta. Identical for European calls and puts on the same strike, expiration, and underlying.
When to use: Use to gauge how unstable a delta-hedge will be. High gamma means delta changes quickly, requiring frequent rehedging; low gamma means a stable hedge.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| Gamma | Gamma | Change in delta per $1 change in the underlying | integer |
| S | Stock Price | Current price of the underlying | $ |
| Strike | Strike Price | Exercise price of the option contract | $ |
| Rf | Risk-Free Rate | Continuously compounded annual risk-free rate as a decimal | % |
| Sigma | Volatility | Annualized volatility of the underlying as a decimal (e.g. 0.25 for 25%) | % |
| T | Time to Expiration | Time to expiration in years (e.g. 0.25 for 3 months) | years |
Real-Life Examples
Example 1: ATM Gamma, 30% Vol, 90 Days
Stock at $100, $100-strike 90-day option, r = 4%, σ = 30%.
Given
Step-by-Step
A $1 stock move shifts delta by ~0.026. ATM options have peak gamma — a small move in spot meaningfully changes the hedge ratio.
Frequently Asked Questions
Because put-call parity ties their deltas by a constant (Δ_call − Δ_put = 1), so their derivatives with respect to S are identical. Gamma is a property of the strike and expiration, not the option type.
For ATM options near expiration. Deep ITM and deep OTM options have low gamma because their deltas are already near 1 or 0 and rarely move much; ATM options have unstable deltas that swing rapidly with small moves.
The standard normal probability density function evaluated at d1: φ(x) = exp(−x²/2)/√(2π).