Random Walk

Explained: arithmetic random walk arithmetic random walk with drift geometric random walk random walk simple random walk

A random walk is a simple type of discrete stochastic process whose increments form a white noise. Since a white noise has zero mean, a random walk is a martingale. Let's formalize this.

If you have not already done so, see the notation conventions documentation. A discrete univariate stochastic process R is called a random walk if its increments

form a white noise. Because there are different types of white noises, there are different types of random walks. A simple random walk—what probability theorists generally call a random walk—is one whose increments form a strong white noise whose terms only take on the values 1 or –1, each with probability 0.5. A realization of a simple white noise is indicated in Exhibit 1 along with the corresponding realization of the white noise of its increments.

Simple Random Walk Exhibit 1

The top graph indicates a realization of a simple white noise. The bottom graph indicates the corresponding realization of the white noise of its increments.

The top graph of Exhibit 1 illustrates how a simple random walk takes random "steps" up or down, which is what motivated the name "random walk."

In finance, an arithmetic random walk is a random walk with increments that are a Gaussian white noise. This can be represented as

tR – t–1R = tN

[2]

where the ^tN are independent and identically distributed standard normal random variables, and is constant. If a constant drift term is added, this becomes an arithmetic random walk with drift:

tR – t–1R = + tN

[3]

For modeling the behavior of certain asset prices, such as stock prices, arithmetic random walks have a number of limitations. They can take on negative values, which is an impossibility for many asset's prices. Also, asset price fluctuations tend to be proportional to those prices. For example, a 50 dollar stock might experience daily price fluctuations on the order of one dollar while a 200 dollar stock might experience daily price fluctuations on the order of four dollars. This is not reflected by arithmetic random walks, whose standard deviations don't increase with the value of the process. For these reasons, geometric random walks often provide superior modeling of asset prices over time.

A geometric random walk is technically not a random walk, at least according to the general definition given above. It is a strictly positive stochastic process whose log returns follow a Gaussian white noise. This can be expressed as

log( tR / t–1R ) = + tN

[4]

where, again, the ^tN are independent and identically distributed standard normal random variables, and is constant.

All the above concepts generalize naturally to multivariate random walks.

Related Internal Links

ARCH A category of conditionally heteroskedastic stochastic processes. autoregressive moving-average process A type of stochastic process. autoregressive process A type of stochastic process. Brownian motion A simple continuous stochastic process that is widely used in physics and finance for modeling random behavior that evolves over time. heteroskedasticity A condition where a stochastic process has non-constant second moments. joint normal distribution A multivariate distribution with normal marginal distributions. martingale A type of stochastic process that has zero drift. moving-average process A type of stochastic process. normal distribution Perhaps the most important probability distribution for probability and statistics. random walk hypothesis Financial model based on the empirical observation that stock and commodity prices behave like a random walk. stochastic volatility model A category of conditionally heteroskedastic stochastic processes. time series and stochastic processes An introductory article. volatility A metric of  variability in a stochastic process. volatility clustering A property of some stochastic processes that they experience periods of high and low variance. white noise A simple form of stochastic process. Sponsored Links Ads by Contingency Analysis

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