Geometric Brownian motion
Quantitative Finance glossary · Stochastic Calculus & Continuous Time Finance
Definition
Stochastic process for stock prices: dS_t = μ S_t,dt + σ S_t,dW_t, with closed-form solution S_t = S_0 exp!left[(μ - 1/2σ^2)t + σ W_tright] — log-prices are Brownian motion with drift. Implies log-normal S_t, non-negative prices, and constant proportional volatility — the last assumption being conspicuously violated by the observed implied-vol skew. The Itô-to-Stratonovich drift correction -1/2σ^2 is the textbook 'GBM drift-trap' for new quants.
Geometric Brownian motion in plain English
Geometric Brownian motion is the classic textbook recipe for a randomly wandering stock price: each instant it drifts upward a bit on average while also getting nudged by unpredictable noise, with both the drift and the noise scaled to the *current* price. Scaling to the current price is what keeps it realistic — a $500 stock swings in bigger dollar amounts than a $5 stock, but by similar *percentages*, and the price can never fall below zero. The solution S_t = S_0 exp[(μ - 1/2σ^2)t + σ W_t] says the *logarithm* of the price does a simple random walk with a steady tilt. It's the finance equivalent of tracking a drunk person walking home: a general direction plus constant random staggering — a great first approximation, even though real markets have wilder swings than the model allows.
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