Definition
Published in 1973 by Fischer Black, Myron Scholes, and Robert Merton (Nobel Prize 1997), the Black-Scholes model derives option prices from five inputs using continuous-time mathematics. Despite assumptions of constant volatility, no dividends, and European exercise, it remains the benchmark for option valuation. The volatility smile and skew patterns reveal where the model's assumptions break down.
functions Formula
C = S·N(d₁) − K·e^(-rT)·N(d₂)
d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T), d₂ = d₁ − σ√T
lightbulb Example
Stock at $100, strike $100, 1 year to expiration, volatility 20%, rate 5%. d₁ = 0.35, d₂ = 0.15. C = $100×N(0.35) − $95.12×N(0.15) = $10.45. The theoretical call option price is $10.45.
verified_user Key Points
- Nobel Prize-winning option pricing formula
- Assumes constant volatility and European exercise
- Foundation for all modern options pricing
- Volatility smile shows where assumptions fail