pricing
d₁ and d₂
The two critical intermediate values in Black-Scholes. d₁ measures moneyness adjusted for drift, d₂ adjusts for volatility over time.
Formula
d₁ = [ln(S/K) + (r + σ²/2) · T] / (σ · √T) d₂ = d₁ - σ · √T
Variables
- S
- Current stock price
- K
- Strike price
- r
- Risk-free interest rate
- σ
- Volatility (annualized)
- T
- Time to expiration (years)
- ln
- Natural logarithm
Worked Example
Inputs
- S
- $105
- K
- $100
- σ
- 25%
- r
- 4%
- T
- 0.5
Calculation Steps
- 1
ln(105/100) = 0.04879 - 2
(0.04 + 0.0625/2) × 0.5 = 0.03563 - 3
σ√T = 0.25 × √0.5 = 0.17678 - 4
d₁ = (0.04879 + 0.03563) / 0.17678 = 0.4777 - 5
d₂ = 0.4777 - 0.17678 = 0.3009
Result: d₁ ≈ 0.478, d₂ ≈ 0.301
Intuition
d₁ tells you how many standard deviations the log-forward price is above the strike. d₂ is the same but from the perspective of the strike. The gap (σ√T) represents total uncertainty over the option life.