greeks
Delta (Δ)
The first-order price sensitivity. Measures how much the option price changes for a $1 move in the underlying stock.
Formula
Δ_call = N(d₁) Δ_put = N(d₁) - 1
Variables
- Δ
- Rate of change in option price per $1 move in stock
- N(d₁)
- Cumulative normal distribution of d₁
Worked Example
Inputs
- d₁
- 0.478
Calculation Steps
- 1
N(0.478) = 0.6837 - 2
Δ_call = 0.6837 → option gains $0.68 per $1 stock move - 3
Δ_put = 0.6837 - 1 = -0.3163
Result: Call Δ ≈ 0.68, Put Δ ≈ -0.32
Intuition
Delta also approximates the probability that the option finishes ITM. A 0.30 delta call has roughly a 30% chance of being profitable at expiration. ATM options hover near 0.50 delta.