Brownian Motion

You already know the random walk: a drunkard staggering left or right, one discrete step at a time. Now do something that sounds destructive but turns out to be the most important move in all of mathematical finance — shrink the steps. Make each step smaller, take more of them, squeeze the gaps between them toward zero. In the limit, the jagged staircase stops being a sequence of jumps and becomes a single unbroken curve that wiggles at every scale: Brownian motion, also called the Wiener process, written $W_t$. It is the continuous-time object under the Black–Scholes formula, under every diffusion model, under the geometric Brownian motion you met simulating prices. This lesson builds it from the random walk up.

From random walk to Brownian motion

Analogy. Imagine filming the drunkard’s walk and then speeding the film up while zooming out. Each individual lurch becomes too small to see, but the overall drift of his position is still there, smoothed into a continuous trembling line. Brownian motion is that limit made exact.

Here is the construction. Start with a symmetric $\pm 1$ random walk: at each tick you add $+1$ or $-1$ with equal probability. To fit $N$ steps into a time interval of length $t$, set the gap between steps to $t/N$ and scale each step by $\sqrt{t/N}$. Define

$W_t^{(N)} = \sqrt{\tfrac{t}{N}}\,\sum_{k=1}^{N} \xi_k, \qquad \xi_k = \pm 1 \text{ each with probability } \tfrac12.$

As $N \to \infty$, this jagged sum converges to Brownian motion $W_t$. Two facts make the limit work, and both are worth understanding precisely.

Why $\sqrt{t/N}$ and no other scaling? This is the whole trick. The steps $\xi_k$ are independent with variance $1$ each, so the unscaled sum of $N$ of them has variance $N$. Multiplying by $\sqrt{t/N}$ multiplies the variance by $(\sqrt{t/N})^2 = t/N$. So:

$\operatorname{Var}\!\big(W_t^{(N)}\big) = \frac{t}{N}\cdot N = t.$

The $N$‘s cancel exactly. The variance lands on $t$ — finite and nonzero — for every $N$, and stays there in the limit. Try any other scaling and you break it: scale by $t/N$ (too small) and the variance collapses to $0$ as $N \to \infty$ (the limit is the boring constant $0$); scale by $\sqrt{t}/\,^4\!\sqrt{N}$ or anything growing faster and the variance blows up to $\infty$. Only the square-root scaling keeps a live, finite random object in the limit. That is why diffusion lives at the $\sqrt{\text{time}}$ scale and nowhere else.

Why the limit is Gaussian. Each $W_t^{(N)}$ is a scaled sum of $N$ independent identically distributed pieces. The Central Limit Theorem says any such sum, once standardized, converges to a normal distribution — regardless of the shape of the individual steps (here, a coin flip). So the limiting increment isn’t just some random variable with variance $t$; it is specifically $\mathrm{Normal}(0, t)$. The CLT is the reason Brownian motion is Gaussian and not, say, uniform.

The island below makes the limit visible. Each thread is one random walk launched from zero; together they form a spreading cloud, and the dashed funnel is the $\pm\sqrt{t}$ envelope — the standard deviation widening like the square root of time. More, smaller steps would smooth each thread into a continuous Brownian path; the cloud’s shape is already the limit. Drag the drift slider to lean the whole cloud (we’ll add drift formally at the end).

The defining properties

Rather than re-deriving Brownian motion every time, we pin it down by four axioms. A process $W_t$ is standard Brownian motion if:

1. Starts at zero: $W_0 = 0$.
2. Independent increments: for non-overlapping time intervals, the changes are statistically independent. What the path does between $0$ and $1$ tells you nothing about what it does between $1$ and $2$. The future increment doesn’t care about the past — there is no memory.
3. Gaussian increments: $W_{t+s} - W_t \sim \mathrm{Normal}(0,\, s)$. The increment over a gap of length $s$ is normal, mean zero, and variance equal to the length of the gap. Bigger gap, proportionally bigger variance.
4. Continuous paths: $t \mapsto W_t$ is a continuous function — no teleporting, no gaps.

Worked example — the $\sqrt{t}$ law, now exact. Property 3 with $t=0$ gives $W_t = W_t - W_0 \sim \mathrm{Normal}(0, t)$. So

$\operatorname{Var}(W_t) = t, \qquad \operatorname{sd}(W_t) = \sqrt{t}.$

This is the same $\sqrt{t}$ spread you saw in the random walk’s $\pm\sqrt{t}$ funnel — but now it’s an exact identity, not an approximation. After $1$ year the standard deviation is $1$; after $4$ years it is $\sqrt{4}=2$; to double the spread you need four times the horizon.

Worked example — a probability. Since $W_1 \sim \mathrm{Normal}(0,1)$, its standard deviation is exactly $1$. The chance the path lands within one standard deviation of the start at time $1$ is the familiar normal figure:

$P(|W_1| < 1) = P(-1 < W_1 < 1) \approx 0.68.$

And at time $t=4$, $W_4 \sim \mathrm{Normal}(0,4)$ has standard deviation $2$, so $P(|W_4| < 2) \approx 0.68$ as well — one standard deviation always captures about $68\%$, the window just widens as $\sqrt{t}$.

Continuous but nowhere differentiable

Here is the property that makes Brownian motion genuinely strange. Its paths are continuous — you can draw one without lifting your pen — yet they are nowhere differentiable: at no single instant does the curve have a well-defined slope. There is no $dW/dt$. Anywhere.

Analogy — the fractal coastline. Measure a coastline with a kilometre-long ruler and you get one length. Switch to a metre-long ruler and you find more crinkles, so the length grows. Switch to a centimetre ruler and it grows again — the coast is equally jagged at every zoom level, never smoothing into a clean tangent. A Brownian path is exactly like that in time: zoom into any interval and it looks just as wild and wiggly as the whole, no matter how far you magnify. It is statistically self-similar — a fractal curve.

Why no slope? A derivative is the limit of $\big(W_{t+h} - W_t\big)/h$ as $h \to 0$. But that increment has standard deviation $\sqrt{h}$, so the ratio has typical size $\sqrt{h}/h = 1/\sqrt{h}$, which blows up to infinity as $h \to 0$. The path moves too much over short intervals — like $\sqrt{h}$, not like $h$ — for any finite slope to exist.

Why it matters. Ordinary calculus is built on derivatives and the chain rule, and both assume your functions are smooth. Brownian motion has neither a derivative nor smoothness, so you cannot differentiate a function of $W_t$ with the rules you learned in school. This is precisely the gap that Itô calculus fills — a new calculus for non-differentiable, $\sqrt{t}$-scaled paths. (That’s the next lesson, and the property below is its seed.)

Quadratic variation = t

This is the single weirdest fact about Brownian motion, and the one that finance cannot live without. Chop $[0,t]$ into $n$ equal pieces and add up the squared increments:

$\sum_{k=1}^{n} \big(W_{t_k} - W_{t_{k-1}}\big)^2 \xrightarrow[n\to\infty]{} t.$

The sum of squared wiggles converges to $t$ — a fixed, non-random number — not to zero. People summarize this as the slogan "$(dW)^2 = dt$": an infinitesimal Brownian increment, when squared, behaves like the time step itself.

Worked intuition. Split $[0,t]$ into $n$ pieces each of width $\Delta t = t/n$. Each increment has variance $\Delta t$, so on average its square is about $\Delta t$. There are $n = t/\Delta t$ of them, so the sum of squares is roughly

$\frac{t}{\Delta t}\cdot \Delta t = t.$

And the randomness in that sum washes out as $n$ grows (you’re averaging many independent squared terms), so it doesn’t just average $t$ — it converges to $t$.

Contrast with a smooth function. Take any differentiable $f$. Its increment over a step $\Delta t$ is about $f'(t)\,\Delta t$, so its square is of order $(\Delta t)^2$. Summing $n = t/\Delta t$ of those gives $(t/\Delta t)\cdot(\Delta t)^2 = t\,\Delta t \to 0$. Smooth functions have zero quadratic variation. Brownian motion does not — that nonzero $t$ is the mathematical fingerprint of roughness, the thing no smooth curve can produce.

Brownian motion with drift (arithmetic BM)

Standard Brownian motion goes nowhere on average — its mean is zero forever. To model something with a trend, bolt a straight line onto it. Arithmetic Brownian motion is

$X_t = \mu t + \sigma W_t,$

a deterministic line $\mu t$ plus scaled Brownian noise $\sigma W_t$. Here $\mu$ is the drift (the average rate of climb) and $\sigma$ is the volatility (how violently the noise jostles). Its distribution follows immediately from the Brownian facts:

$X_t \sim \mathrm{Normal}\big(\mu t,\ \sigma^2 t\big).$

The mean is $\mu t$ (the line), and the variance is $\sigma^2 t$ — the Brownian variance $t$ scaled by $\sigma^2$.

Worked example. Let $\mu = 2$ per year, $\sigma = 3$ per year, and look at $t = 4$ years.

• Mean: $\mu t = 2 \times 4 = 8$.
• Variance: $\sigma^2 t = 9 \times 4 = 36$, so the standard deviation is $\sqrt{36} = 6$.
• So $X_4 \sim \mathrm{Normal}(8, 36)$. A one-standard-deviation band runs from $8 - 6 = 2$ up to $8 + 6 = 14$, and roughly $68\%$ of outcomes fall inside it.
• Notice $P(X_4 < 0)$ is not zero: that’s about $1.33$ standard deviations below the mean, a chance of roughly $9\%$. The process can — and sometimes does — go negative.

That last point is the whole pitfall.

Putting it together

Brownian motion is the scaling limit of a random walk: shrink the $\pm 1$ steps by $\sqrt{t/N}$, take $N \to \infty$, and the Central Limit Theorem hands you a continuous Gaussian process $W_t$. It is defined by four axioms — start at zero, independent increments, $\mathrm{Normal}(0,s)$ increments over a gap $s$, continuous paths — from which $\operatorname{Var}(W_t)=t$ and the $\sqrt{t}$ spread fall out. Its paths are continuous but nowhere differentiable (too rough for any slope, which is why Itô calculus exists), and their quadratic variation equals $t$ — the "$(dW)^2 = dt$" fact that seeds the $\tfrac12\sigma^2$ Itô term. Add a trend and you get arithmetic Brownian motion $X_t = \mu t + \sigma W_t \sim \mathrm{Normal}(\mu t, \sigma^2 t)$ — useful, but it can go negative and adds fixed-dollar noise, which is why prices use geometric BM instead.

Next up — Itô’s Lemma — we build the calculus that the nowhere-differentiable path forced on us. The "$(dW)^2 = dt$" fact from quadratic variation becomes a concrete rule, an extra $\tfrac12\sigma^2$ term appears in every chain-rule calculation, and out of it falls geometric Brownian motion — the same engine you simulated, now derived from first principles.