Brownian Motion

Brownian motion is a random process that evolves continuously in time, moving by small independent steps whose sizes follow the normal distribution. In mathematics it is also called the Wiener process. In finance it is the basic model for the unpredictable part of price movements and the foundation of modern option pricing theory.

The name comes from the botanist Robert Brown, who in 1827 described the restless movement of tiny particles suspended in water. The idea reached finance through Louis Bachelier, whose 1900 thesis used it to model prices on the Paris exchange, five years before Albert Einstein's physical explanation of Brown's observation. Norbert Wiener gave the process a rigorous mathematical basis in the 1920s. It is the most important example of a stochastic process in continuous time.

A standard Brownian motion, written $W_t$, is defined by four properties.

• It starts at zero. $W_0 = 0$.
• Its increments are independent. The change over one interval tells nothing about the change over any later, non-overlapping interval.
• Its increments are normal. The change from time $s$ to time $t$ has a normal distribution with mean zero and variance $t - s$.
• Its paths are continuous. The process never jumps, although its path is so jagged that it has no defined slope at any point.

The third property carries the key practical rule: uncertainty grows with the square root of time. Variance is proportional to the length of the interval, so the standard deviation is proportional to its square root. Brownian motion is the limit of a random walk as the steps become smaller and more frequent, a result that follows from the central limit theorem.

Plain Brownian motion can become negative, which a share price cannot. The standard model for prices is therefore geometric Brownian motion, introduced to finance by Paul Samuelson in the 1960s, in which the percentage change in price, not the price itself, is driven by Brownian motion:

• $S_t$ is the asset price at time $t$.
• $\mu$ is the drift, the average rate of return per unit of time.
• $\sigma$ is the volatility, the standard deviation of returns per unit of time.
• $dW_t$ is the increment of a standard Brownian motion.

The equation has the solution:

Under this model, log returns are normal and the price itself follows a lognormal distribution, so it stays positive.

The square-root rule converts volatility between time horizons:

where $T$ is the horizon in years. Suppose a share has annual volatility of 20 percent and there are 252 trading days in a year. The volatility of one day's return is 20 × √(1 ÷ 252) = 20 ÷ 15.87 = 1.26 percent. Over one month of 21 trading days it is 20 × √(21 ÷ 252) = 20 × 0.2887 = 5.77 percent. The monthly figure is about 4.6 times the daily one, not 21 times, because independent moves partly cancel.

Geometric Brownian motion is the price assumption behind the Black-Scholes model. Because the price path is continuous, an option can in theory be hedged perfectly by constant small trades in the underlying, and that argument fixes the option's value. The same assumption supports the usual conventions for quoting volatility in annual terms.

The process is also the engine of Monte Carlo simulation. To value a complex derivative or to estimate value at risk, a computer draws many random paths from the model and measures the outcome on each. The independence of increments matches the weak form of the efficient market hypothesis, under which past price changes do not help to predict future ones. Models of interest rates, credit and commodities are built from the same ingredient, with extra terms that pull the variable toward a long-run level.

Real returns differ from the model in ways that matter for risk. Large moves occur far more often than a normal distribution allows, a feature known as fat tails and measured by kurtosis. Volatility is not constant: calm and turbulent periods cluster. Prices sometimes jump on news, with no trades at the prices in between. Each of these breaks an assumption of Brownian motion, and each is visible in option markets, where implied volatility varies by strike instead of being flat as the model would require.

Extensions address the gaps. Jump-diffusion models add sudden moves, stochastic volatility models let $\sigma$ follow its own random process, and GARCH models describe volatility clustering in discrete time. Brownian motion remains the reference case because it is simple, well understood and close enough for many purposes over short horizons, but risk estimates that rely on it tend to understate the chance of extreme losses.