The Black-Scholes model (1973) revolutionized options trading by providing a closed-form equation to price European options. It won the Nobel Prize in Economics in 1997.

The Call Price Formula: `C = S·N(d1) K·e^(rT)·N(d2)`

Where: - S = current stock price - K = strike price - r = risk-free interest rate - T = time to expiration (in years) - σ = implied volatility (annualized) - N(x) = cumulative standard normal distribution

d1 and d2: - `d1 = [ln(S/K) + (r + σ²/2)·T] / (σ·T)` - `d2 = d1 σ·T`

Numerical Example: - Stock price S = $100, Strike K = $100 (ATM) - T = 30 days = 0.0822 years, σ = 25%, r = 5% - d1 0.158, d2 0.086 - N(d1) 0.563, N(d2) 0.534 - Call price $100 × 0.563 $100 × 0.996 × 0.534 $2.98

A 30-day ATM call on a $100 stock with 25% IV costs roughly $2.98 (~3% of stock price).