Geometric Brownian Motion (GBM)
A continuous-time random growth model where relative price changes are normal and prices stay strictly positive.
GBM assumes dS/S = μ dt + σ dW, which means the logarithm of the price follows a Brownian motion with drift; over any horizon, prices are lognormally distributed.
It implies strictly positive prices and a simple relationship between drift, volatility, and the distribution of returns.
Its main role is as a tractable baseline for modelling and derivations like Black–Scholes; real markets show jumps, volatility clustering, and fat tails.
Engineering analogy: GBM is a linear stochastic differential equation in log-space — simple enough to solve exactly, but only an approximation of the true system.
Geometric Brownian Motion (GBM) sample paths. Prices stay strictly positive. The drift μ tilts the centre of the fan; the volatility σ controls how wide and skewed the fan becomes.
The thin grey lines are individual GBM paths. The blue line is the median across paths; the orange dashed line is the expected path S₀ e^{μt}. The shaded band shows the middle 50% of outcomes at each time.
Notice that the mean at T sits above the median: GBM is lognormal, so the distribution of S_T is skewed. In log-space, log(S_T / S₀) is approximately normal with mean (μ − ½σ²)T and std σ√T – exactly what the log-return stats show.
As σ increases, the fan widens dramatically: extreme paths go far above the drift curve, even if the median stays close to it. For control and risk work, this is why GBM is a useful baseline signal model for log-returns, but it underestimates real-world jump risk and fat tails in markets.