Geometric Brownian motion: the model behind stock price simulations

By Daniel Tolenaar · Updated 1 November 2026

Geometric Brownian motion (GBM) is the standard model of a stock price in quantitative finance. It says the price grows at a steady average rate (the drift, μ) plus random shocks whose size is set by the volatility (σ), with both applied in percentage terms, so the price can never fall below zero.

GBM is the assumption underneath the Black-Scholes model, and it is what most Monte Carlo simulations of stock prices draw from.

The GBM equation

As a stochastic differential equation, GBM is:

The first term is the drift: in a small time dt the price grows by μ·dt per cent on average. The second is the noise: dW_t is the increment of a Brownian motion, a normal random variable with mean 0 and variance dt. Its solution, found with Itô’s lemma, is:

So the log of the price is normally distributed and the price itself is lognormal. The −σ²/2 term surprises most people. It appears because the exponential is convex: random ups add more than equal downs take away, so the typical (median) path grows at μ − σ²/2 while the average grows at the full μ.

Simulated GBM paths

These are twenty one-year paths starting at 100, with 8% drift, simulated day by day. Move the volatility to see the fan widen:

Simulating GBM in Python, and checking it

Because the solution is known, each day can be simulated exactly with one normal draw. This simulates 100,000 paths and checks them against the theory:

import numpy as np

S0, mu, sigma, T, steps, n_paths = 100.0, 0.08, 0.20, 1.0, 252, 100_000
dt = T / steps
rng = np.random.default_rng(seed=7)

# Exact GBM step: S(t+dt) = S(t) * exp((mu - sigma^2/2) dt + sigma sqrt(dt) Z)
Z = rng.standard_normal((n_paths, steps))
log_steps = (mu - 0.5 * sigma**2) * dt + sigma * np.sqrt(dt) * Z
S = S0 * np.exp(np.cumsum(log_steps, axis=1))
ST = S[:, -1]

print(f"mean of S(T)      = {ST.mean():.2f}   (theory {S0 * np.exp(mu * T):.2f})")
print(f"median of S(T)    = {np.median(ST):.2f}   (theory {S0 * np.exp((mu - 0.5 * sigma**2) * T):.2f})")
print(f"share ending below 100 = {(ST < S0).mean():.1%}")

mean of S(T)      = 108.15   (theory 108.33)
median of S(T)    = 106.02   (theory 106.18)
share ending below 100 = 38.5%

The simulated mean and median land within a fraction of a per cent of e^μT and e^{(μ − σ²/2)T}. And although the stock is expected to gain 8%, about 38% of paths end below where they started. Theory gives N(−(μ − σ²/2)√T/σ) = N(−0.3) ≈ 38.2%.

Where GBM falls short

• Constant volatility. Real volatility clusters: calm periods and turbulent ones. GARCH and stochastic-volatility models (such as Heston) let σ move.
• No jumps. GBM moves continuously, but prices gap on news. Merton’s jump-diffusion adds sudden jumps.
• Thin tails. Daily returns have fatter tails than a normal distribution, so GBM understates extreme days. See Value at Risk for what that does to risk numbers.

Even so, GBM is the right first model: simple enough to solve exactly, and close enough that the volatility traders quote, implied volatility, is defined through it.