Standard Brownian motion is the building block of every continuous-time model in finance — from Black-Scholes to HJM to stochastic volatility. Quant research and trading interviews test whether you can state its properties precisely and apply Itô's lemma fluently.

What is Standard Brownian Motion?

A standard Brownian motion (also called a Wiener process) W(t) is a continuous-time stochastic process defined by four properties. Memorize these — interviewers ask for them verbatim.

W(0) = 0

The process starts at zero almost surely. This is the initial condition.

Independent increments

For any 0 ≤ s < t ≤ u < v, the increments W(t) − W(s) and W(v) − W(u) are independent. The past trajectory carries no information about future moves.

Normal increments with variance = elapsed time

For any 0 ≤ s < t, the increment W(t) − W(s) ~ N(0, t − s). The standard deviation grows as √(t − s) — slower than linear, which is what keeps paths continuous.

Continuous sample paths

With probability 1, the function t ↦ W(t) is continuous. Despite this, the paths are nowhere differentiable — a key fact that makes ordinary calculus fail and forces us to use Itô calculus instead.

Why It Matters for Quant Interviews

Brownian motion is foundational because almost every quantitative finance model is built on top of it:

Black-Scholes — stock price modeled as geometric Brownian motion: dS = μS dt + σS dW. The option price emerges from replicating the payoff with a delta hedge.
Interest rate models — Vasicek, CIR, and Hull-White all drive rates with a Brownian term. The HJM framework uses Brownian motion to model the entire forward rate curve.
Stochastic volatility — Heston and SABR use two correlated Brownian motions (one for the asset, one for variance), which introduces the concept of correlation dW₁ dW₂ = ρ dt.

Being able to say "the volatility term in Black-Scholes comes from the σ dW part of the SDE, and the ½σ² correction in log returns comes from Itô's lemma" is the level of fluency interviewers want.

Key Results to Know

Zero Drift Property

E[W(t)] = 0

Brownian motion has zero drift by definition.

Linear Variance Growth

Var[W(t)] = t

Variance grows linearly with time.

Second Moment

E[W(t)²] = t

Follows directly since E[W(t)] = 0 and Var[W(t)] = t.

Covariance Structure

E[W(s)·W(t)] = min(s, t)

Covariance of Brownian motion at two times.

[W, W]_t = t (quadratic variation)
The sum of squared increments converges to t — this is why dW² = dt in Itô calculus.

propertyLong-term Behavior

W(t)/t → 0 as t → ∞
Despite paths being unbounded, the time-average converges to zero (law of large numbers analog).

The quadratic variation result — dW² = dt — is the single most important mechanical fact in Itô calculus. It is what generates the extra correction term in Itô's lemma.

Itô's Lemma

Itô's lemma is the stochastic chain rule. Given an Itô process dX = μ dt + σ dW and a smooth function f(t, X), the differential of f is:

theoremItô's Lemma (scalar form)

df = (∂f/∂t + μ·∂f/∂x + ½σ²·∂²f/∂x²) dt + σ·∂f/∂x dW

∂f/∂t — ordinary time derivative

μ·∂f/∂x — drift term from dX (same as ordinary calculus)

½σ²·∂²f/∂x² — the Itô correction term, arising because dW² = dt (not zero)

σ·∂f/∂x dW — the stochastic diffusion term

The Itô correction term ½σ²·∂²f/∂x² is what separates stochastic calculus from ordinary calculus. In ordinary calculus, second-order terms like (dx)² vanish. In Itô calculus, dW² = dt is first-order, so the second derivative term survives.

Example — log of a GBM: Let X = S where dS = μS dt + σS dW. Apply Itô's lemma to f(S) = ln(S). Then ∂f/∂S = 1/S and ∂²f/∂S² = −1/S². Substituting:

d(ln S) = (μ − ½σ²) dt + σ dW
The −½σ² correction is why log returns are lower than the arithmetic drift μ. Forgetting this term is one of the most common interview mistakes.

Common Interview Questions

definitionWhat is E[W(t)²]?

Answer: t

Since E[W(t)] = 0 and Var[W(t)] = t, we get E[W(t)²] = Var[W(t)] + E[W(t)]² = t. A clean one-liner — but know why.

definitionWhat is E[e^(W(t))]?

Answer: e^(t/2)

Since W(t) ~ N(0, t), this is the moment generating function of a normal: E[e^(aZ)] = e^(a²σ²/2) with a = 1, σ² = t. Result: e^(t/2).

definitionDerive the SDE for a stock price under GBM

Solution:
Assume ln(S) follows an arithmetic Brownian motion: d(ln S) = α dt + σ dW. Apply Itô's lemma in reverse (or directly to f = eˣ) to get dS = (α + ½σ²)S dt + σS dW. Rename μ = α + ½σ².

How Itô's lemma leads to Black-Scholes

Model dS = μS dt + σS dW. Let V(t, S) be the option price. By Itô's lemma, dV has a dW term. Construct a portfolio Π = V − ΔS that eliminates dW by choosing Δ = ∂V/∂S. The resulting riskless portfolio must earn the risk-free rate, giving the Black-Scholes PDE:

∂V/∂t + ½σ²S²·∂²V/∂S² + rS·∂V/∂S − rV = 0

Test Your Knowledge

Test your stochastic calculus knowledge

Practice Brownian motion, Itô's lemma, and Black-Scholes derivation questions from top quant firms.

Stochastic Calculus Questions →Black-Scholes Explained →