Geometric Brownian Motion

A process $\{S_t\}$ follows geometric Brownian motion (GBM) if it solves the stochastic differential equation $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$, where $\mu$ is the drift rate, $\sigma$ is the volatility, and $W_t$ is a Wiener process. Applying Ito's lemma to $\ln S_t$ gives the explicit closed-form solution $S_t=S_0\exp \left[\left(\mu -\frac{{\sigma }^2}{2}\right)t+\sigma W_t\right]$, showing that $\ln (S_t/S_0)$ is normally distributed with mean $(\mu -{\sigma }^2/2)t$ and variance ${\sigma }^{2}t$, i.e. $S_t$ is log-normally distributed. This guarantees $S_t>0$ for all $t$ whenever $S_0>0$, precisely the property that makes GBM suitable for modeling asset prices. GBM is the process assumed for the underlying asset in the Black-Scholes-Merton option pricing model, and the appearance of $-{\sigma }^2/2$ rather than just $\mu$ in the exponent (a consequence of Ito's lemma's second-order correction term) is a frequently-cited illustration of how stochastic calculus differs from ordinary calculus.

• The $-{\sigma }^2/2$ correction term in the exact solution (absent from a naive first-guess exponential) comes directly from Ito's lemma's second-order term, and is a standard, memorable illustration of why stochastic calculus needs its own chain rule rather than borrowing ordinary calculus's.
• GBM assumes constant drift and volatility over time, a simplification widely known to be unrealistic (real markets exhibit volatility clustering and fat tails), which is exactly why extensions like stochastic volatility models and jump-diffusion models exist to relax this assumption while keeping GBM's tractable core.
• Despite its known limitations, GBM remains the default textbook starting point for derivative pricing precisely because it yields closed-form option pricing formulas (Black-Scholes), a rare case where mathematical tractability, not just realism, drove a model's widespread adoption.