Geometric Brownian Motion

Core Idea

Geometric Brownian motion (GBM) solves dS = μS dt + σS dW, where both drift and diffusion are proportional to the current value S. Applying Itô's formula to ln(S) reveals that log-returns are normally distributed: ln(S(t)/S(0)) = (μ - σ²/2)t + σW(t), so S(t) = S(0)exp((μ - σ²/2)t + σW(t)). The solution is always positive, lognormally distributed, and is the standard model for stock prices in mathematical finance — the foundation of Black-Scholes theory.

Explainer

Geometric Brownian motion is the multiplicative analogue of Brownian motion. Where Brownian motion adds random increments (dX = σ dW), GBM multiplies by random factors (dS/S = μ dt + σ dW, or equivalently dS = μS dt + σS dW). The proportionality of both drift and diffusion to the current level S means that percentage changes, not absolute changes, are the natural unit — a 1% move when S = 100 is a 1% move when S = 1000. This multiplicative structure is why GBM is the default model for prices, populations, and other quantities that grow proportionally.

Solving the SDE requires Itô's formula. Apply f(x) = ln(x) to S: d(ln S) = (1/S)dS + (1/2)(-1/S²)(dS)² = (μ - σ²/2)dt + σ dW. The Itô correction subtracts σ²/2 from the drift — a critical detail. Integrating: ln(S(t)) - ln(S(0)) = (μ - σ²/2)t + σW(t), so S(t) = S(0)exp((μ - σ²/2)t + σW(t)). Since W(t) ~ N(0,t), the log-return ln(S(t)/S(0)) is normally distributed, and S(t) itself is lognormally distributed with E[S(t)] = S(0)e^μt and Var(S(t)) = S(0)²e^2μt(e^σ²t - 1).

A subtle but important distinction: the median of S(t) is S(0)exp((μ - σ²/2)t), growing at rate μ - σ²/2, while the mean E[S(t)] = S(0)e^μt grows at the faster rate μ. The gap σ²/2 is a Jensen's inequality effect — the convexity of the exponential function means the average of e^X exceeds e^{average of X}. When σ is large, the median can decrease even as the mean increases. This has practical implications: a "typical" sample path of GBM grows slower than the expected value suggests, because the mean is pulled up by rare but extreme positive outcomes.

In mathematical finance, GBM is the foundation of the Black-Scholes model. Under the risk-neutral measure (obtained via Girsanov's theorem), the stock price follows dS = rS dt + σS dW̃ where r is the risk-free rate. The explicit lognormal distribution of S(T) allows closed-form pricing of European options: the Black-Scholes formula is a direct consequence of computing E[max(S(T) - K, 0)] under this lognormal distribution. While GBM's assumptions (constant μ, σ, no jumps, normal log-returns) are violated by real market data, its tractability and the intuitions it provides make it the essential starting point for all of quantitative finance.