Brownian motion: the defining properties and what follows

Brownian motion is defined by three short properties, and everything else — the covariance, the non-differentiability, the scaling law — is derived from them in a line or two. Interviewers ask for the derivations rather than the facts, because deriving them is what shows the definition is actually understood.

Definition 1.1

The definition

Standard Brownian motion, $W_0 = 0; \quad W_t - W_s \sim N(0, t-s); \quad \text{increments independent}$ — Plus continuity of paths. Three properties and a regularity condition, from which the entire theory follows.

Fact	Derivation
$\mathbb{E}[W_t] = 0$	From $W_t = W_t - W_0 \sim N(0,t)$
$\mathrm{Var}(W_t) = t$	Same
$\mathrm{Cov}(W_s, W_t) = \min(s,t)$	Split the later one into the earlier plus an independent increment
$W_{ct} \overset{d}{=} \sqrt{c}\,W_t$	Both sides are normal with variance $ct$
$-W_t$ is a Brownian motion	Symmetry of the normal
$tW_{1/t}$ is a Brownian motion	Check the covariance: $\min(s,t)$ again

Table 1.2 · What follows immediately. The covariance derivation is the one to have ready: write

W_t = W_s + (W_t - W_s)

, note the second term is independent of

W_s

and mean zero, and the answer is

\mathbb{E}[W_s^2] = s

.

Why the square root of time appears everywhere. The scaling property says that speeding up time by $c$ is the same as scaling space by $\sqrt{c}$ : Brownian motion is self-similar with exponent one half. That single symmetry is the reason volatility scales as $\sqrt{t}$ , why an at-the-money option costs about $0.4S\sigma\sqrt{T}$ , and why a random walk after $n$ steps is typically $\sqrt{n}$ from the origin. When you see a square root of time anywhere in finance, this is where it came from.

Proposition 1.3

The paths are continuous and nowhere differentiable

Over an interval $\Delta$ , the motion typically moves $\sqrt{\Delta}$ , so the difference quotient is $\sqrt{\Delta}/\Delta = 1/\sqrt{\Delta}$ , which blows up as $\Delta \to 0$ . The paths are continuous because $\sqrt{\Delta} \to 0$ , and non-differentiable because the ratio does not — both facts come from the same exponent.

Holds when

Total variation is infinite, which is why ordinary Riemann–Stieltjes integration against $dW$ fails and the Itô integral is needed.
Quadratic variation is finite and equals $t$ , which is the substitute that makes the theory work.
Real price paths are not literally Brownian, but the roughness is qualitatively right.

Figure 1.4 · Variance grows like $t$ , size like $\sqrt{t}$ . Both start together and separate immediately. Every scaling result in the subject is one of these two lines: volatility annualises by

\sqrt{t}

, variance by

t

, and confusing which is which is how a daily number becomes a wrong annual one.

Brownian motion: the defining properties and what follows

After this lesson you should be able to

The definition

The paths are continuous and nowhere differentiable

The rest of this lesson is in Premium