From random walks to Brownian motion

Continuous-time finance is built on Brownian motion. The bridge from the discrete random walks we observe (daily returns) to the continuous-time Brownian motion of Black-Scholes is one of the deepest moves in applied probability — and one of the most useful, because everything from the Black-Scholes formula to interest-rate term-structure models follows from the consequences of that bridge.

Symmetric random walk

Take i.i.d. ε_i ∈ {-1, +1} each with probability 1/2. Define S_n = ε_1 + ε_2 + ... + ε_n. This is a simple symmetric random walk on the integers. Its moments: E[S_n] = 0, Var(S_n) = n.

Scaling to continuous time

Take n steps per unit time. To preserve variance at scale ~ T (continuous time), each step should have variance 1/n. Rescale: S_{nT}/√n has variance T as n → ∞. The continuous-time limit is a process whose value at any time T has variance T — Brownian motion.

W(T) = lim_{n→∞} (1/√n) Σ_{i=1}^{nT} ε_i ~ N(0, T)

Donsker's invariance principle

Made rigorous, this is Donsker's theorem (1951): properly rescaled, simple random walks converge as continuous-time stochastic processes to Brownian motion. Equivalently, the path-valued central limit theorem. The convergence is not just in finite-dimensional distributions but in path space (weak convergence in C[0, T]).

From random walk to GBM

Stock prices don't have zero drift and don't have constant volatility-in-levels. Replace each step ε_i with a scaled Gaussian σ √(Δt) Z_i + μ Δt. Take Δt → 0:

log S(t + Δt) ≈ log S(t) + (μ - σ²/2) Δt + σ √(Δt) Z

In the continuous-time limit, log S(t) is Brownian motion with drift; S(t) is geometric Brownian motion (GBM):

dS = μ S dt + σ S dW

The familiar Black-Scholes-Merton model for stock prices, derived from the random walk by Donsker scaling.

Discrete tree vs continuous BM

The Cox-Ross-Rubinstein binomial tree approximates GBM with up-moves of u = e^(σ√Δt) and down-moves of d = e^(-σ√Δt), risk-neutral probabilities p = (e^(rΔt) - d)/(u - d). As Δt → 0, the tree converges to the BS PDE. This is why binomial pricing converges to Black-Scholes — it is precisely a discrete scaling of the same Donsker construction.

Markov property in the limit

Random walks are Markov — the future depends only on the present. Brownian motion inherits this: P(W(t + s) ∈ A | F_t) = P(W(t + s) ∈ A | W(t)). The Markov property is what makes PDE methods possible — option values depend only on the current price (and time), not the price history.

Why we square-root scale

If steps are independent with variance σ², n steps have variance nσ². Vol of the sum is σ√n. The square-root-of-time rule is just the CLT applied to a sum of independent variances — true for Brownian motion exactly, true approximately for daily returns that are roughly i.i.d.