Random Walk Model

What is a random walk model?

A random walk is a mathematical formalization of a path that consists of a succession of random steps. The idea beyond the random walk is that the next step of a magnitude is equal to the previous state plus the change of magnitude. The change is a random variable (it does not depend on the previous state).

It comes from the classical problem of obtaining the travelled distance of a drunk person. A more detailled description of the problem can be obtained here.

The random walk models are a part of stochastic calculus. It is very important because it can be used to model systems that behave randomly (like the stock market).

Example of a gaussian random walk in one dimension (the change of magnitude follows a normal distribution, and the magnitude can change in only one direction).

Equations of the model:
$X(t)=X(t-1)+\Delta X$
$\Delta X=N(0,1)$
$S(0)=30$

Chart of ten different simulations:

Excel file: Gaussian Random Walk (I recommend to play by changing the values)

As each increment is an independent normal random variable, at the end of the period T, the possible values follow the normal distribution $N(\mu T, \sqrt{\sigma} T)$ (uncertainty is proportional to $\sqrt{T}$).

There are more types of random walk than the gaussian random walk, depending on which type of random variable is the change of the magnitude (example, a lognormally distributed variable).

What is the random walk hypothesis?

From Wikipedia: the random walk hypothesis is a financial theory stating that stock market prices evolve according to a random walk and thus cannot be predicted. It is consistent with the efficient-market hypothesis.

Basically, it means that the return of a stock can be modelled as a random variable characterized by its mean and standard deviation. In fact, the continously compounded return rate is modelled as a normal random variable.

To know more about the Random Walk model, please visit the following related posts:

Monte Carlo simulation and Random Walk (I)

Monte Carlo simulation and Random Walk (II)