greeks
Gamma (Γ)
The second-order price sensitivity. Measures how fast Delta changes as the stock moves — it's the "acceleration" of the option.
Formula
Γ = φ(d₁) / (S · σ · √T)
Variables
- Γ
- Rate of change of Delta per $1 move in stock
- φ(d₁)
- Standard normal PDF at d₁
- S
- Current stock price
- σ
- Implied volatility
- T
- Time to expiration (years)
Worked Example
Inputs
- φ(d₁)
- 0.3541
- S
- $105
- σ
- 25%
- T
- 0.5
Calculation Steps
- 1
σ√T = 0.25 × √0.5 = 0.17678 - 2
Γ = 0.3541 / (105 × 0.17678) - 3
Γ = 0.3541 / 18.562
Result: Γ ≈ 0.0191 — Delta changes by ~0.019 per $1 stock move
Intuition
Gamma is highest for ATM options near expiration. This is why short ATM options near expiry are dangerous — small stock moves cause massive P&L swings. Long gamma = love big moves, short gamma = love stillness.