Geometric Brownian Motion

Geometric Brownian motion (GBM) is the standard model for stock prices. It says percentage changes (not absolute changes) follow Brownian motion: $$dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$$ Solution (via Ito's lemma applied to $\ln S_t$): $$S_t = S_0 \exp\left[\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right]$$ So $\ln(S_t / S_0) \sim N\left((\mu - \sigma^$$\frac{2}{2}$)t,\ \sigma^2 t\right) — log-returns are normally distributed. Intuition: A stock that goes up 10% then down 10% doesn't return to its starting price (it's at 99%). GBM captures this asymmetry naturally — it models multiplicative noise. The $-\sigma^$$\frac{2}{2}$ correction (the "Ito correction") accounts for the fact that the geometric mean of log-normal returns is less than the arithmetic mean. Key properties: - $S_t > 0$ always (prices can't go negative) - $E[S_t] = S_0 e^{\mu t}$ (expected price grows exponentially) - Returns over any interval are log-normally distributed - Volatility scales with $\sqrt{t}$ Concrete example: Stock at \$100, $\mu = 8\%$, $\sigma = 20\%$. After 1 year: - Expected price: $100 e^{0.08} = 108.33$ dollars - Median price: $100 e^{0.08 - 0.02} = 106.18$ dollars (lower due to Ito correction!) - 95% range: $100 e^{0.06 \pm 1.96 \times 0.20} \approx [72, 157]$ dollars When to use: The Black-Scholes model assumes GBM. It's the baseline model for option pricing, risk management, and Monte Carlo simulation. In interviews: "model this stock price" almost always means GBM. Alternative approach: Real stock returns have fat tails and volatility clustering. Jump-diffusion and stochastic volatility models extend GBM for better realism.