Business, Legal & Accounting Glossary
The Random Walk Theory basically states that choosing stocks randomly has just as good a potential return as any other method of choosing stocks.
Random Walk Theory is the theory that stock price changes have the same distribution and are independent of each other, so the past movement or trend of a stock price or market cannot be used to predict its future movement.
The random walk theory states that in an efficient market, market prices follow a random path and are therefore unpredictable. Random walk theory says that these market prices are not influenced by their past movements, so it is impossible to predict (with accuracy) which direction the market will move. Random walk theory was espoused by Louis Bachelier, a French mathematician, in 1900 and likens the unpredictable movements of market prices to the unpredictable walk of a drunk. If random walk theory is correct, technical analysis won’t work. According to “A Random Walk Down Wall Street,” by Burton Malkiel, random walk theory renders analysis inaccurate. It also states that large transaction costs will outweigh any profits. Random walk theory typically applies to short-term movements. Even proponents of random walk theory acknowledge that long-term movements have generally followed an upward trend.
Although Random Walk Theory is an important part of investment theory, it is best applied in the sense that the next short-term movement of price in any investment is as likely to be up as down. But to reduce this concept to the idea that you are better off selecting investments by throwing darts at a stock list is an oversimplification.
Fundamentalist investors believe that in the long term the value of a share of stock is determined by the prospects for future earnings. Well-managed companies that lead their industries are likely to continue doing better than their competitors.
Hence in spite of Random Walk Theory, it is still a good idea to research a stock before you invest.
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
| tW = tR – t–1R | [1] |
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.
To help you cite our definitions in your bibliography, here is the proper citation layout for the three major formatting styles, with all of the relevant information filled in.
Definitions for Random Walk Theory are sourced/syndicated and enhanced from:
This glossary post was last updated: 30th December, 2021 | 0 Views.