Geometric Brownian Motion

One seed. A thousand futures.

Each green thread is a Monte-Carlo price path. They fan out from S₀ = 100; their endpoints pile into the lognormal curve on the right.

E[S_T] theory—S₀·e^{μT}

mean (empirical)—n = 0

σ(S_T) theory—spread

% paths > S₀—ended up

The SDE · dS = μS dt + σS dW

μ drift, annualised0.08

σ volatility, annualised0.20

T horizon, years1.00

N paths300

Δt steps / year

How to use

Drag μ (drift), σ (volatility), T (horizon) and N (paths). The paths animate out from left to right; their terminal values land in the histogram on the right edge, and the dashed curve is the theoretical lognormal PDF. Hit New draw for fresh randomness, Replay fan to watch them grow again. Toggle Lin / Log scale.

The math

Geometric Brownian Motion is the SDE

dS = μ·S·dt + σ·S·dW

with closed-form solution

S_t = S₀ · exp((μ − σ²/2)·t + σ·W_t).

Discretised as S_{t+Δt} = S_t · exp((μ − σ²/2)·Δt + σ·√Δt · Z) with Z ~ N(0,1). The −σ²/2 correction is the variance drag: higher volatility lowers the typical (median) terminal price even when the expected value increases.

Theoretical moments at horizon T:

E[S_T] = S₀ · exp(μ·T)
Var[S_T] = S₀² · exp(2μT) · (exp(σ²T) − 1).

What to notice

The terminal distribution is lognormal, not normal. It is right-skewed: a few paths drift far up, most cluster lower. The dashed curve on the right histogram is the theoretical lognormal PDF — empirical bins should match it closely as N grows.

Even with positive drift, a substantial fraction of paths end below S₀. The "expected return" and the "expected log return" diverge by exactly σ²/2 per year — the difference between the mean and the median of the lognormal.

Toggle log-scale: trajectories become straight-line random walks around the deterministic drift line.

Why this is the foundation

GBM is the assumed price process behind Black-Scholes option pricing, Merton's portfolio problem, and most undergraduate quantitative finance. Every more realistic model (jump diffusion, stochastic volatility, regime switching, Heston, SABR) is built as an extension or refinement of this baseline.

Real returns deviate from GBM in three documented ways: fat tails (extreme moves more frequent than lognormal predicts), volatility clustering (calm and stormy periods cluster), and leverage effect (volatility rises when prices fall). GBM has none of these — yet it remains the reference point.

Citations

Hull, J.C. Options, Futures, and Other Derivatives, 11th ed., Ch. 14 (Wiener processes and Itô's lemma).
Shreve, S. Stochastic Calculus for Finance II: Continuous-Time Models, Ch. 3.
Samuelson, P. (1965). "Proof That Properly Anticipated Prices Fluctuate Randomly." Industrial Management Review.
Bachelier, L. (1900). "Théorie de la spéculation." (Original arithmetic Brownian motion model; predates Samuelson's geometric variant by 65 years.)

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