The SDE
A GBM satisfies the stochastic differential equation:
\[dS_t = \mu S_t\,dt + \sigma S_t\,dW_t\]where $\mu$ is the drift (expected return), $\sigma$ is volatility, and $W_t$ is a standard Wiener process.
Solving via Itô’s lemma
Apply Itô’s lemma to $X_t = \log S_t$. Since $f(S) = \log S$, $f’ = 1/S$, $f’’ = -1/S^2$:
\[dX_t = \left(\mu - \frac{\sigma^2}{2}\right)dt + \sigma\,dW_t\]This is an arithmetic Brownian motion. Integrating:
\[S_t = S_0\exp\!\left[\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right]\]The $-\sigma^2/2$ Itô correction is crucial — it accounts for the convexity of the exponential.
Distribution of returns
Log-returns over horizon $T$:
\[\log\frac{S_T}{S_0} \sim \mathcal{N}\!\left(\left(\mu - \frac{\sigma^2}{2}\right)T,\; \sigma^2 T\right)\]So $S_T$ is log-normally distributed. The expected price $\mathbb{E}[S_T] = S_0 e^{\mu T}$ grows at the drift rate, not the median $S_0 e^{(\mu-\sigma^2/2)T}$.
Limitations
GBM assumes constant volatility and normally distributed log-returns. Empirical asset returns exhibit:
- Fat tails: extreme moves occur far more often than the normal predicts
- Volatility clustering: periods of high volatility tend to persist
- Leverage effect: volatility rises when prices fall
These motivate extensions: stochastic volatility models (Heston), jump-diffusion (Merton), and rough volatility.