pricing
Black-Scholes Call
The cornerstone of options pricing. Calculates the theoretical fair value of a European call option based on five inputs.
Formula
C = S · N(d₁) - K · e⁻ʳᵀ · N(d₂)
Variables
- C
- Call option price
- S
- Current stock price
- K
- Strike price
- r
- Risk-free interest rate
- T
- Time to expiration (years)
- N(x)
- Cumulative normal distribution
- d₁, d₂
- See d₁ and d₂ formulas
Worked Example
Inputs
- S
- $100
- K
- $100
- σ
- 20%
- r
- 5%
- T
- 0.25 (3 months)
Calculation Steps
- 1
d₁ = [ln(100/100) + (0.05 + 0.20²/2) × 0.25] / (0.20 × √0.25) - 2
d₁ = [0 + (0.05 + 0.02) × 0.25] / 0.10 = 0.0175 / 0.10 = 0.175 - 3
d₂ = 0.175 - 0.20 × √0.25 = 0.175 - 0.10 = 0.075 - 4
N(0.175) ≈ 0.5695, N(0.075) ≈ 0.5299 - 5
C = 100 × 0.5695 - 100 × e⁻⁰·⁰¹²⁵ × 0.5299 = 56.95 - 52.33
Result: C ≈ $4.61
Intuition
Think of N(d₁) as the probability-weighted share equivalent and N(d₂) as the probability the option finishes in-the-money. The formula is essentially: "what you expect to receive minus what you expect to pay."