Chapter 3 · Stochastic Processes in Finance

Geometric Brownian Motion

Randomness with a direction — and a sense of proportion.
The big idea

If Brownian Motion is a drunken walk where someone wanders aimlessly around a room, then Geometric Brownian Motion is that same person walking on a moving walkway at an airport. They still wobble left and right unpredictably — but the floor itself is carrying them forward. The wobble is random. The walkway is not.

Moving walkway analogy for GBM — drift carries the walker forward while they wobble randomly Airport moving walkway with a figure wobbling side to side while being carried steadily forward THE MOVING WALKWAY — Drift (μ) Random wobble = Diffusion (σ) General drift forward = Drift (μ)

The path is jagged (random wobble), but the walkway ensures a steady forward trend.

Why plain brownian motion wasn't enough

Standard Brownian Motion had a critical flaw for finance: it could push a stock price below zero — which is impossible in reality. It also assumed that a $2 move meant the same thing whether a stock was worth $10 or $1,000. That's clearly wrong. GBM fixes both problems elegantly by switching from dollars to percentages.

Standard Brownian Motion

The price changed by ± $2.

A $10 stock could drop to –$5. A $1,000 stock barely notices a $2 move. The math treats both the same — which makes no sense.

Geometric Brownian Motion

The price changed by ± 2%.

A $10 stock moves $0.20. A $1,000 stock moves $20. The moves are proportional to the price — which is how markets actually work.

The two ingredients

GBM breaks every price movement into exactly two distinct forces working simultaneously:

Ingredient 1
The Drift (μ)
The moving walkway. This is the predictable part — if a stock historically grows 8% a year, the drift is that steady forward pull. Even when the price wiggles, the drift keeps the general trajectory biased in one direction.
Ingredient 2
The Diffusion (σ)
The drunken wobble. This is the random Brownian Motion part — the volatility and noise that makes the path jagged instead of a straight line. Bigger σ means wilder swings.
Chart showing drift line vs actual GBM path — drift is smooth, actual path is jagged around it A time-series chart with a smooth upward drift line in amber and a jagged GBM path in purple around it Drift (μ) — the walkway Actual GBM path — μ + σ $100 $184 Time →

The amber dashed line is pure drift — the purple jagged path is what actually happens when you add diffusion.

The formula, decoded

In research papers and textbooks, GBM is written as a single compact equation. It looks intimidating — but every piece of it maps directly back to our airport walkway story:

dSt = μStdt + σStdWt
dSt
The tiny change in price right now
μSt dt
The Drift — expected return × current price × time
σSt dWt
The Shock — volatility × current price × random jostle

Notice that both the drift and the shock are multiplied by the current price (St). That's what makes it "geometric" — every move is a percentage of where you already are, not a fixed dollar amount.

The volume twist — when σ isn't constant

In advanced models, the wobble (σ) itself can change based on trading volume. Think of it this way: if millions of people are in the pool hitting the beach ball, all those forces cancel each other out and the ball barely moves. But if only one person is in the pool, a single push sends it flying. The same logic applies to stocks:

Two price paths showing high-volume low-volatility vs low-volume high-volatility Side-by-side charts: left shows a smooth path with high volume, right shows a wildly jagged path with low volume High Volume — σ is small Low Volume — σ is huge Many trades = forces cancel out One trade can move price 5% (Inertia from high liquidity) (Illiquidity amplifies moves)

High volume dampens volatility — low volume amplifies it. Same drift, very different diffusion.

Try it — build your own GBM path

Adjust the drift (μ) and volatility (σ) and see how GBM paths change. Notice how drift pulls the average upward while volatility controls the wildness of the wobble.

10%
20%
Path 1
Path 2
Path 3
Drift only

In short: Brownian Motion is pure randomness. Geometric Brownian Motion is randomness with a trend, where every jump is a percentage of the current price. This keeps prices positive, makes moves proportional, and is the most common way hedge funds model normal stock behavior. Two ingredients — drift and diffusion — are all you need to describe the motion of markets.

Next Chapter: Quadratic Variation →