Option Gamma

Options
Updated Apr 2026 • Has calculator

The rate of change in an option's delta for a $1 move in the underlying stock.

What is Gamma?

Gamma (Γ) measures how fast delta changes as the stock price moves. A high gamma means delta is very sensitive to price moves — the option behaves non-linearly. Gamma is identical for calls and puts with the same inputs, is always positive, and peaks for at-the-money options near expiry. Portfolio managers use gamma to measure the hedging risk of a delta-neutral position.

Formula

Γ = φ(d₁) / (S · σ · √T)

Worked Example

Worked example — Apple Inc. (AAPL) — ATM option, representative Q1 2024

Representative Q1 2024 market conditions

Step 1  Stock price (S): $185, Strike (K): $185, σ = 28%, T = 0.25 yrs
Step 2  d₁ = 0.163, φ(0.163) = 0.393 (standard normal PDF)
Step 3  Γ = 0.393 / (185 × 0.28 × 0.50) = 0.393 / 25.9 ≈ 0.0152
Step 4  → If AAPL moves $1, delta changes by ±0.0152
Step 5  → Gamma is highest for ATM options approaching expiry (pin risk)

Source: Hull, J.C. — Options, Futures, and Other Derivatives, 11th ed., Ch. 19 (2024-01-15)

Calculate Gamma

Current market price of the underlying stock

Option strike price

Annual risk-free rate

Time to expiration in years

Annualised implied volatility

Gamma

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Not investment advice.

How to Interpret Gamma

< 0.01
Low gamma — deep ITM/OTM or long-dated option
0.01 – 0.03
Moderate gamma — typical for ATM, 30–90 day options
0.03 – 0.1
High gamma — ATM near expiry, significant pin risk
> 0.1
Very high gamma — very short-dated ATM option

📚 Options Basics — Complete the path

  1. Delta (Call)
  2. Gamma
  3. Theta (Call)
  4. Vega
  5. BS Call
Sources
  1. Hull, J.C. — Options, Futures, and Other Derivatives, 11th ed.
  2. Investopedia — Gamma