In 1827 the botanist Robert Brown pointed his microscope at pollen grains suspended in water and watched them jiggle — ceaselessly, erratically, with no current to blame and no sign of life driving them. In 1900, in Paris, a doctoral student named Louis Bachelier put the same jiggle to a very different use: he modelled the prices on the Paris Bourse with it and derived the first mathematical theory of option pricing — five years before Einstein wrote the physics paper that finally explained the pollen. Two worlds, one motion.

That motion, made rigorous, is Brownian motion — also called the Wiener process and written (W_t)_{t \ge 0}. It is randomness distilled to four axioms: the canonical continuous-time stochastic process, the continuous cousin of the random walk, and the single object on which essentially all of quantitative finance stands. The noise that drives stock prices in the Black–Scholes world, the diffusions behind interest-rate models, the shocks in every Monte Carlo risk engine — all are built from W_t. Learn this one process deeply and the rest of the course is variations on it.

The four defining properties

Brownian motion is pinned down completely by four properties — nothing else is assumed, and everything else we will ever prove about it follows from these:

• Starts at zero: W_0 = 0.
• Independent increments: over disjoint time intervals the increments W_{t_2} - W_{t_1},\, W_{t_4} - W_{t_3}, \dots are mutually independent — what the process does next does not depend on its past. This is the mathematical form of "the market has no memory".
• Gaussian stationary increments: for s < t, W_t - W_s \sim N(0,\, t - s), a normal with mean 0 and variance equal to the elapsed time — the law depends only on the gap t - s, not on where it sits. A week of noise in March is statistically identical to a week of noise in November.
• Continuous paths: almost every sample path t \mapsto W_t(\omega) is continuous (no jumps).

Taking s = 0 gives the headline fact W_t \sim N(0, t), so \mathbb{E}[W_t] = 0 and \operatorname{Var}(W_t) = t. The spread grows like \sqrt{t}: uncertainty fans out with time — but slower than time itself, a point we will make precise below.

One path of the noise

The axioms are austere; the object they describe is anything but. Below is a single sample path of W_t over t \in [0,1], built as a running sum of many tiny independent normal increments \Delta W \sim N(0, \Delta t). Study its texture: it is continuous — you can trace it without lifting your pen — yet violently jagged at every scale, and it shows no preferred direction. It looks, not coincidentally, exactly like an intraday price chart with the drift stripped out. Refresh to draw a fresh path — a different \omega — and notice that while every path is different, the statistics (the typical spread, the roughness) are always the same.

Worked examples: computing with the axioms

The four properties are not just a definition to memorise — they are a calculator. Three computations every quant does in their head:

Example 1 — P(W_2 > 0). By the Gaussian-increments property with s = 0, W_2 \sim N(0, 2): a normal centred at zero. A centred normal is symmetric about 0, and being a continuous distribution it puts zero probability on the single point \{0\}, so the two halves split the probability evenly:

Nothing about the time 2 mattered: P(W_t > 0) = \tfrac{1}{2} for every t > 0. Pure noise is equally likely to be above or below where it started, at any horizon.

Example 2 — the distribution of W_3 - W_1. This is an increment over the interval [1, 3], of length 3 - 1 = 2, so directly from the axiom

Note it has the same law as W_2 — that is stationarity of increments in action: only the length of the window matters, not where it sits. Its standard deviation is \sqrt{2} \approx 1.41.

Example 3 — P(W_4 > 2). Here we standardise. W_4 \sim N(0, 4) has standard deviation \sqrt{4} = 2, so Z = W_4 / 2 \sim N(0, 1) and

A one-standard-deviation move — about a one-in-six event. Every "chance the price ends above the strike" calculation in this course is a dressed-up version of this three-line computation.

Mean and variance, derived from the axioms

Let us not take those numbers on faith — every one of them falls straight out of the four properties, with no skipped steps. Start with a single time t > 0 and write W_t as an increment off the start:

which is allowed because W_0 = 0 by the first property. The increment W_t - W_0 runs over the interval [0, t] of length t - 0 = t, so by the Gaussian-increments property

Reading the mean and variance straight off this normal law:

So the variance at time t is exactly t — nothing more was needed than "starts at zero" plus "increments are N(0, \text{gap})".

The covariance: \operatorname{Cov}(W_s, W_t) = \min(s, t)

Values at two different times are linked through the history they share. Fix two times and, without loss of generality, label them so that s < t. The idea is to split the later value into "what had happened by s" plus "what happened afterwards", because those two parts are independent. Write

a plain algebraic identity. Now expand the covariance using its bilinearity (covariance is linear in each argument):

Take the two terms in turn. The first is just a variance,

by the result we just derived. For the second term, notice that W_s = W_s - W_0 is the increment over [0, s] and W_t - W_s is the increment over the disjoint interval [s, t]. By the independent-increments property these two increments are independent, and the covariance of independent random variables is zero:

Adding the two pieces back together,

We assumed s < t, so s = \min(s, t); had we labelled them the other way the same argument would give t = \min(s, t). Either way the answer is the smaller of the two times:

The \sqrt{t} law: risk grows slower than time

\operatorname{Var}(W_t) = t looks innocent, but its square root is one of the most consequential facts in finance. The typical size of the noise — its standard deviation — is \sqrt{t}, so quadrupling the horizon only doubles the spread:

To make the uncertainty 10\times bigger you must wait 100\times longer. Watch the law in the figure below: a bundle of independent Brownian paths, fanning out inside the curves \pm\sqrt{t} (one standard deviation — about 68\% of paths inside at any fixed time) and \pm 2\sqrt{t} (about 95\%). The envelope is a square-root, not a straight line — steep at first, then flattening.

This is exactly why traders quote volatility scaled by the square root of time. An annualised volatility of 16\% corresponds to a daily volatility of roughly 16\% / \sqrt{252} \approx 1\% (there are about 252 trading days in a year), and a "10-day value-at-risk" is the 1-day figure scaled by \sqrt{10} \approx 3.16, not by 10. Whenever you see a \sqrt{T} in a pricing or risk formula, this axiom is where it came from.

Brownian motion is a martingale

A process is a martingale when its best forecast of any future value, given everything seen so far, is simply its current value — the mathematical definition of a fair game. Brownian motion is the prototype, and the proof is two lines of the same split-the-increment trick we used for the covariance. Fix s < t and let \mathcal{F}_s stand for the history of the path up to time s. Split:

The first term is W_s exactly because W_s is already known at time s — conditioning on the history leaves it alone. For the second term, the increment W_t - W_s is independent of everything in \mathcal{F}_s (that is the independent-increments axiom), so conditioning changes nothing and the conditional expectation collapses to the plain one:

since the increment is N(0, t-s), a centred normal. Therefore

Knowing the whole past of the noise gives you no edge in predicting where it goes next — the expected future position is exactly where you stand. This fair-game property is the seed of the deepest idea in the course: modern pricing theory says that (suitably discounted, under the right probability measure) asset prices themselves must be martingales, or free money would exist. When you meet that principle, remember it started here.

Self-similarity: zoom in, same statistics

Take a Brownian motion, speed up its clock by a factor c > 0, and shrink the result by \sqrt{c}:

Run the new process through the four axioms. It starts at B_0 = W_0/\sqrt{c} = 0. Its increments over disjoint intervals come from increments of W over disjoint intervals, so they are independent. Each increment is Gaussian with

and its paths are continuous because W's are. All four axioms hold: B is again a standard Brownian motion. This is self-similarity: Brownian motion has no natural time scale. Zoom into any stretch of the sample path above — milliseconds or decades — and after rescaling the axes by the square-root law you see a curve with identical statistics. The path is a statistical fractal, equally jagged at every magnification, which is why the figure looks like a stock chart whether the axis reads minutes or years. (Real markets break this perfect scaling — volatility clusters, jumps — and the Brownian benchmark is exactly what makes that deviation measurable.)

A note on its history

The name honours the botanist Robert Brown, who in 1827 watched pollen grains jitter erratically in water. The mathematics that turned that jitter into a rigorous object was supplied by Norbert Wiener in the 1920s — which is why the process bears his name too. Remarkably, it had already been used to model the stock market in 1900 by Louis Bachelier, five years before Einstein's celebrated physical account — the first time chance was given a continuous, time-evolving shape in finance.