Geometric Brownian Motion: The Stochastic Model Behind Stock Prices and Black-Scholes

Why stock prices are modelled multiplicatively, not additively

A plain random walk — price changes by a random additive amount each step — has an obvious flaw for a stock price: it can go negative, and a 1-dollar move means something completely different to a $5 stock than to a $500 stock. What actually seems to hold, at least approximately, is that percentage returns are the roughly random, independent quantity, not absolute price changes. Geometric Brownian Motion (GBM) builds the model directly from that observation: the instantaneous percentage change in price is a deterministic drift plus Gaussian noise.

dS = μ·S·dt + σ·S·dW

  S  = price
  μ  = drift (expected return rate)
  σ  = volatility
  dW = increment of a Wiener process (Brownian motion)

Solving the stochastic differential equation

Applying Itô's lemma — the stochastic-calculus chain rule needed because dW² doesn't vanish the way an ordinary infinitesimal squared does — to ln(S) gives the closed-form solution directly, which is why GBM is tractable enough to be the default textbook model:

S(t) = S(0) · exp[ (μ − σ²/2)·t + σ·W(t) ]

Discretised for simulation, one time step Δt:
S(t+Δt) = S(t) · exp[ (μ − σ²/2)·Δt + σ·√Δt·Z ],   Z ~ N(0,1)

That −σ²/2 term is easy to miss and important: it is the Itô correction, and it means the expected price grows at rate μ while any single simulated path's typical (median) growth rate is μ−σ²/2, strictly less than μ when volatility is nonzero. This is not a simulation artefact — it is a real mathematical fact about compounding random percentage returns, sometimes called volatility drag: a stock that goes up 50% then down 50% is down 25% overall, not back to even, and higher volatility makes that gap between the arithmetic mean return and the realised compounded return larger.

Log-normal, not normal

Because ln(S(t)) is normally distributed by construction (it's ln S(0) plus a linear function of the Gaussian W(t)), the price S(t) itself is log-normally distributed — always positive (solving the plain random walk's negative-price problem outright), and right-skewed: the distribution has a long tail toward large gains and is bounded below at zero, which matches the observed asymmetry of real return distributions reasonably well over short-to-medium horizons.

The Black-Scholes connection

GBM is not just a convenient toy — it is the exact underlying price process assumed by the Black-Scholes option pricing model (Black, Scholes and Merton, 1973), which is precisely why GBM remains the default first model taught for equity prices: the same log-normal assumption that makes GBM simple to simulate is also what makes the Black-Scholes partial differential equation solvable in closed form, giving option prices as an explicit formula rather than requiring numerical solution.

Where GBM breaks down

Real markets deviate from GBM's assumptions in well-documented ways: volatility clustering (large moves tend to be followed by large moves, contradicting GBM's constant σ), fat tails (extreme moves happen far more often than a log-normal distribution predicts — the 1987 crash was a many-standard-deviation event under GBM's own assumptions), and jumps (news events cause discontinuous price gaps that a continuous diffusion process cannot produce at all). Extensions like GARCH models (time-varying volatility), jump-diffusion models (Merton, 1976) and stochastic-volatility models (Heston, 1993) each relax exactly one of these assumptions. GBM remains the starting point precisely because it isolates, in the simplest possible closed-form model, what pure random compounding does on its own — everything else is a correction layered on top.

Frequently asked questions

Try it live

Everything above runs in your browser — open Stock Price — GBM and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.