GBM: dS = μS dt + σS dW gives log-normal price paths. Black-Scholes formula for a European call: C = S·N(d₁) − K·e−rT·N(d₂) where d₁,d₂ involve log(S/K), risk-free rate, and volatility σ. The model assumes constant σ — the "volatility smile" shows it isn't.