Brownian Motion

Brownian motion — also called a Wiener process — is a continuous-time stochastic process that serves as the mathematical foundation for modeling randomness in finance. It was first observed physically by botanist Robert Brown in 1827, who noticed that pollen particles suspended in water moved erratically. The phenomenon was later explained mathematically by Einstein (1905) and formalized rigorously by Norbert Wiener (1923).

In finance, Brownian motion was first applied by Louis Bachelier in his 1900 PhD thesis, where he modeled stock prices as a random walk — remarkably, five years before Einstein's work. Today, Brownian motion is the building block for virtually all continuous-time financial models, including the Black-Scholes model, stochastic calculus, and risk-neutral pricing.

A standard Brownian motion {W(t), t ≥ 0} is defined by four properties:

1. W(0) = 0 — the process starts at zero.
2. Continuous paths — the function W(t) is continuous in t (no jumps).
3. Independent increments — for any times 0 ≤ s < t ≤ u < v, the increments W(t) - W(s) and W(v) - W(u) are independent.
4. Normal increments — W(t) - W(s) ~ N(0, t - s). The increment over a time interval of length (t-s) is normally distributed with mean 0 and variance (t-s).

Key consequences:

• W(t) ~ N(0, t) — at time t, the process is normally distributed with variance t. The standard deviation grows as √t.
• Markov property: The future evolution depends only on the current value, not the past path — Brownian motion is memoryless.
• Martingale property: E[W(t) | W(s)] = W(s) for s < t — there is no expected drift. Brownian motion is a "fair game."
• Nowhere differentiable: Despite being continuous, Brownian motion paths are infinitely jagged — they have no well-defined slope at any point. This is why ordinary calculus fails and stochastic calculus is needed.
• Quadratic variation: Over [0, t], the quadratic variation of W is exactly t: ∑(W(t_i) - W(t_i-1))² → t. This is the source of the (dW)² = dt rule in Ito calculus.

Brownian motion is the continuous-time limit of a random walk. Consider a random walk where at each time step Δt, the process moves up or down by Δx = σ√Δt with equal probability.

As Δt → 0 (and the number of steps increases proportionally), this discrete random walk converges to a continuous Brownian motion with volatility σ. This convergence result — known as Donsker's theorem — provides the mathematical bridge between discrete-time and continuous-time models.

Intuition: Think of stock prices being updated on a fast-ticking clock. At each tick, the price moves by a small random amount. As the ticking gets infinitely fast and the moves get infinitely small, the price path becomes a continuous Brownian motion. This is why the random walk model (good for daily returns) and the Brownian motion model (good for continuous-time pricing) are fundamentally the same model viewed at different time scales.

Raw Brownian motion can take negative values, which is unrealistic for stock prices. Geometric Brownian Motion (GBM) solves this by making the stock price follow:

dS = μS dt + σS dW

The key difference from arithmetic Brownian motion is the S multiplier — both the drift and diffusion are proportional to the current price level. This ensures that:

• Stock prices stay positive (S(t) > 0 for all t).
• Percentage returns (not dollar returns) are normally distributed.
• The stock price is log-normally distributed: ln(S_T/S₀) ~ N((μ - σ²/2)T, σ²T).

The explicit solution is: S_T = S₀ exp((μ - σ²/2)T + σW_T)

GBM is the stock price model assumed in the Black-Scholes model. Under risk-neutral pricing, μ is replaced by the risk-free rate r, and Monte Carlo simulations generate stock paths by sampling from this equation.

Limitations of GBM: It assumes constant volatility (contradicted by the volatility smile), no jumps (contradicted by real market crashes), and independent increments (contradicted by volatility clustering). These limitations motivate more advanced models like the Heston stochastic volatility model.