Geometric Brownian Motion

You can now sample from any distribution you like (lesson 2). But a single random draw is a single dot — a one-off coin flip. A price is a story that unfolds over time: tomorrow built on today, the day after built on tomorrow. To simulate a stock, you don’t want one draw — you want a path, hundreds of draws chained nose-to-tail so each day inherits where the last one left off. The model that does this chaining, the workhorse under almost every Monte Carlo engine in finance, is Geometric Brownian Motion (GBM). This lesson is about the engine itself: how one random number per step rolls into a whole fan of plausible futures.

The random walk, and why the naive version is broken

Analogy. Picture a drunkard leaving the pub. Each step he lurches a random amount left or right. Where he ends up after a hundred steps is a random walk — the sum of a hundred little random shocks. Prices feel the same: tomorrow’s price is today’s price plus some random jostle.

The tempting first model writes that literally. Tomorrow’s price equals today’s plus a random shock measured in dollars:

$S_{t+1} = S_t + \sigma\, Z, \quad Z \sim N(0,1)$

This is an additive (arithmetic) random walk, and for prices it is broken in two ways.

First, it can go negative. A normal shock has no floor — string together enough bad draws and $S_t$ marches straight through zero into negative territory. A stock can’t cost −$4. The model permits an impossible state.

Second, a fixed dollar shock is nonsense across price levels. A $1 wiggle is a yawn on a$1000 stock (0.1%) and an earthquake on a $1 penny stock (100%). Real markets don’t move in fixed dollars — they move in percentages. A blue chip and a penny stock can both have 2% days; they will never have the same dollar days. The additive walk bakes in the wrong unit.

GBM’s fix: walk in log returns, not dollars

GBM repairs both flaws with one move: don’t add a random number to the price — multiply the price by a random factor. Equivalently, do the random walk on the log return rather than on the price itself. Model the percentage change, and two things fall out for free.

Because the change is multiplicative, the price can only ever be scaled up or down — never pushed below zero. Multiply $100 by anything positive and you stay positive. And because the move is a percentage, a 2$5 or $5000. Both problems vanish at once.

This connects straight to the lognormal prices you met in options pricing. If each step’s log return is normal, and you add up many normal log returns, the total log return is normal too — which means the price (the exponential of that total) is lognormal. A lognormal variable can’t be negative and is right-skewed: a long tail of big winners on the upside, a floor at zero on the downside. That is exactly the shape of real long-horizon stock returns, and it’s the shape GBM produces by construction.

Drift versus diffusion: the discrete step

Here is the engine itself — the formula you simulate one step at a time:

$S_{t+\Delta t} = S_t \exp\!\left(\left(\mu - \tfrac{1}{2}\sigma^2\right)\Delta t + \sigma\sqrt{\Delta t}\,Z\right),\quad Z \sim N(0,1)$

Don’t be spooked — every piece is something you already know. Take them one at a time:

• $S_t$ — today’s price, where the step starts.
• $\mu$ — the drift: the expected log-growth rate per year (say 10% for a stock you expect to grow). This is the deterministic pull, the direction the price leans on average.
• $\sigma$ — the volatility: the annual standard deviation of returns, your old friend. This scales the random jostle.
• $\Delta t$ — the time step, in years. For a daily simulation, $\Delta t = 1/252$ (252 trading days per year). Smaller step = finer-grained path.
• $Z$ — a fresh standard normal draw, a brand-new $N(0,1)$ sample at every single step. This is precisely the lesson-2 sampling, now chained: one draw per day, 252 draws for a year.
• $\sqrt{\Delta t}$ — the random part scales with the square root of time, because variance (not standard deviation) adds linearly. Over a step a quarter as long, the random move shrinks only by a half, not a quarter.

The exponent splits cleanly into two parts. The drift term $\left(\mu - \tfrac{1}{2}\sigma^2\right)\Delta t$ is deterministic — it’s the same every step, the steady lean. The diffusion term $\sigma\sqrt{\Delta t}\,Z$ is random — it’s the dice-roll, the wandering. Drift is where the price wants to go; diffusion is the noise that knocks it around on the way. Every GBM step is a tug-of-war between the two.

The mysterious −½σ² term

You probably eyed that $-\tfrac{1}{2}\sigma^2$ sitting next to the drift and wondered where it crawled in from. It’s called the Itô correction, but forget the calculus — the intuition is the genuinely useful part, and you already met it as variance drag.

Analogy. A stock drops 50% one day and gains 50% the next. Are you back where you started? No — you’re at $100 \to 50 \to 75$. You’re down 25%, despite the average daily return being a cheerful $(-50\% + 50\%)/2 = 0\%$. Volatility quietly ate your money. The arithmetic mean return (the simple average) overstates what you actually compound; the geometric mean (what your wealth really grows at) is always lower, and the gap between them widens with volatility.

That gap is exactly $\tfrac{1}{2}\sigma^2$. The drift you feel in your compounded wealth isn’t the raw $\mu$ — it’s $\mu - \tfrac{1}{2}\sigma^2$. The correction term docks your growth rate by half the variance to account for the drag that bouncing around inflicts on compounding. Leave it out and your simulated paths drift systematically too high: you’d be claiming the price grows at its arithmetic average, which volatility makes impossible to actually achieve.

The consequence. Two stocks with the same expected return $\mu$ but different volatility do not grow your money at the same rate. The more volatile one suffers more drag and compounds slower. This is why, in a moment, you’ll see the median GBM path drift below the mean: the average is dragged up by a few lucky exponential winners, but the typical path lags, taxed by its own variance.

One step, worked end to end

Let’s actually turn the crank once. Take a stock with:

• $S_0 = 100$ (starting price)
• $\mu = 10\%$ per year $= 0.10$ (drift)
• $\sigma = 20\%$ per year $= 0.20$ (volatility)
• $\Delta t = 1/252 \approx 0.003968$ (one trading day)

Step 1 — the drift per step. The deterministic part of the exponent is $\left(\mu - \tfrac{1}{2}\sigma^2\right)\Delta t$. First the variance-drag-adjusted drift: $\mu - \tfrac{1}{2}\sigma^2 = 0.10 - \tfrac{1}{2}(0.20)^2 = 0.10 - 0.02 = 0.08$ per year. Per day: $0.08 \times \tfrac{1}{252} \approx 0.000317$. Tiny — a daily lean of about +0.0317%.

Step 2 — the diffusion magnitude per step. The random part scales by $\sigma\sqrt{\Delta t} = 0.20 \times \sqrt{1/252} = 0.20 \times 0.06300 \approx 0.01260$. So a one-standard-deviation day is about a 1.26% jostle — forty times larger than the drift. On any single day, randomness utterly dominates the gentle drift. (That’s why one day tells you nothing and a year tells you a lot: drift accumulates, noise partly cancels.)

Step 3 — draw a Z and take the step. Suppose today’s normal draw lands at $Z = +1.0$ (a one-sigma up day). The exponent is:

$0.000317 + 0.01260 \times 1.0 = 0.01292$

Exponentiate to get the growth factor: $\exp(0.01292) \approx 1.01300$. Multiply onto the price:

$S_{t+\Delta t} = 100 \times 1.01300 = 101.30$

A one-sigma-up day nudged $100 to about$101.30. Had $Z$ come in at $-1.0$ instead, the factor would be $\exp(0.000317 - 0.01260) \approx \exp(-0.01228) \approx 0.98780$, landing the price at about $98.78.

Step 4 — roll it 252 times. Now chain it: feed $101.30$ back in as the new $S_t$, draw a fresh $Z$, take another step. Do this 252 times, each with its own independent normal draw, and you’ve simulated one full year — one complete price path, one possible future for the stock. The drift quietly accumulates across the year; the noise wanders up and down but its ups and downs partly cancel, so over 252 steps the drift finally becomes visible under the jitter.

Why one path is useless and a thousand is gold

Here is the trap. You run GBM once, get a path that ends at $134, and conclude "the stock will be worth$134.” Wrong. That single path is one draw from an infinite cloud of possible futures — a single roll of a 252-sided dice rolled 252 times over. Run it again with fresh random draws and you’ll land somewhere completely different. One path tells you almost nothing.

The power comes from simulating thousands of paths. Each is a different possible future; together they trace out the entire distribution of outcomes. Now you can ask real questions: what’s the median ending price? The 5th percentile (a bad year)? The chance of ending below $80? None of those is answerable from one path; all of them fall out of ten thousand.

And the shape of that fan is the lognormal you were promised. It’s right-skewed: a handful of paths catch a long winning streak and rocket far above the crowd (no ceiling on the upside), while the worst paths are floored at zero. Those few exponential winners drag the mean ending price above the median — the typical path lands lower than the average, because the average is inflated by a small number of moonshots. The mean is not the most likely outcome; in a right-skewed world, most paths land below average.

The island below is that fan. Each thread is one GBM path starting from the same price; the cloud they form is the outcome distribution. Drag the drift to tilt the whole cloud up or down, and crank the volatility to watch the fan widen — and the right-skew exaggerate — as variance drag pushes the typical path further below the lucky outliers.

Pitfalls: GBM is a baseline, not the truth

GBM is gorgeous and it is wrong — usefully wrong, the way a frictionless plane is useful in physics, but wrong in ways that matter when money is on the line. Two assumptions are doing heavy lifting:

Constant volatility. GBM uses a single fixed $\sigma$ forever. Real markets have volatility clustering — calm stretches and stormy stretches arrive in clumps (big moves beget big moves). A model with one constant $\sigma$ can’t capture that rhythm; it sprinkles its volatility evenly, like a weather forecast that predicts the same drizzle every day of the year.

Independent normal increments. GBM assumes each day’s $Z$ is an independent, normal draw. Reality has fat tails (crashes happen far more often than a normal allows — the 1987 single-day move was a “25-sigma” event a normal model says should never occur in the universe’s lifetime) and jumps (prices gap on news without trading through the intervening levels). GBM’s smooth, continuous, thin-tailed paths simply can’t gap or crash the way real prices do.

So treat GBM as the honest baseline: the simplest model that gets positivity, percentage moves, and lognormal shape right — and the foundation everything fancier is built on top of. When its assumptions chafe, quants reach for richer engines: jump-diffusion (bolt sudden gaps onto GBM), stochastic volatility (let $\sigma$ itself wander randomly, e.g. the Heston model), and others you’ll meet later. GBM isn’t the destination; it’s the launchpad.

Putting it together

Geometric Brownian Motion is the engine under simulated prices. It walks not the dollar price but the log return, so moves are multiplicative percentages — which keeps prices positive and makes the price distribution lognormal. Each discrete step, $S_{t+\Delta t} = S_t \exp\!\left(\left(\mu - \tfrac{1}{2}\sigma^2\right)\Delta t + \sigma\sqrt{\Delta t}\,Z\right)$, is a tug-of-war between a deterministic drift and a random diffusion driven by a fresh standard normal $Z$. The $-\tfrac{1}{2}\sigma^2$ term is variance drag made honest: it docks growth so paths compound at the realistic geometric rate. Chain 252 of these and you get one path — a single possible future. Roll thousands and you get the whole right-skewed fan, the outcome distribution where a few exponential winners lift the mean above the median. And remember GBM is a baseline, not gospel: constant vol and normal increments miss the clustering, fat tails, and jumps that fancier models add.

Next up — Simulating Portfolio Outcomes — we point this engine at a real goal. Feed GBM your retirement contributions and a multi-decade horizon, and the fan of paths becomes a cone of outcomes: the spread of pots you might retire on, from the unlucky 5th percentile to the lucky 95th. Same engine, now answering the question that actually keeps people up at night — will I have enough?