###### Option Greeks

The Greeks are a group of mathematical derivatives applied to help manage or... **Read More**

A time series is said to follow a **random walk process** if the predicted value of the series in one period is equivalent to the **value** of the series in the **previous period** **plus** a** random error**.

A simple random walk process can be expressed as follows:

$$\text{x}_{\text{t}}=\text{x}_{\text{t}-1}+\epsilon_{\text{t}}$$

Where:

- \(\text{x}_{\text{t}}\) = Best prediction tomorrow.
- \(\text{x}_{\text{t}-1}\) = Best value today.
- \(\epsilon_{\text{t}}\) = Random error term.

- An AR(1) time series with \(\beta_{0}=0\) and \(\beta_{1}=1\) is a random walk. This is because the best prediction for tomorrow is the best value today plus a random error term.
- The expected value of the
**error term**\(\epsilon_{\text{t}}\) is equal to zero. - The variance of the residuals is constant.
**Random walk with drift**: A time series follows a random walk with drift if it has a non-zero constant intercept term. It is expressed as:

$$\text{x}_{\text{t}}=\text{b}_{0}+\text{b}_{1}\text{x}_{\text{t}-1}+\epsilon_{\text{t}}$$

Where:

- \(\text{b}_{0}\) = Constant drift
- \(\text{b}_{1}\) = 1

Note that a random walk is expressed as:

$$\text{x}_{\text{t}}=\text{b}_{0}+\text{b}_{1}\text{x}_{\text{t}-1}+\epsilon_{\text{t}}$$

- \(\text{b}_{0}=0\) for a random walk without drift;
- \(\text{b}_{0}\neq 0\) for a random walk with drift; and
- \(\text{b}_{1}=1\) for a random walk with or without drift.

A random walk has an **undefined mean reversion level**. If has a mean-reverting level, i.e., \(\text{x}_{\text{t}}=\text{b}_{0}+\text{b}_{1}\text{x}_{\text{t}},\) then \(\text{x}_{\text{t}}=\frac{\text{b}_{0}}{1-\text{b}_{1}}\). However, in a random walk, \(\text{b}_{0}=0\) and \(\text{b}_{1}=1\), so, \(\text{x}_{\text{t}}=\frac{0}{1-1}=0\).

A random walk is **not covariance stationary.** The covariance stationary property suggests that the mean and variance terms of a time series remain constant over time. However, the variance of a random walk process does not have an upper bound. As \(t\) increases, the variance grows with no upper bound. This implies that we cannot use standard regression analysis on a time series that appears to be a random walk.

## Question

The

most accuratestatement about a random walk is that it:

- Has a finite mean-reverting level.
- Has an undefined mean-reverting level.
- Is covariance stationary.
## Solution

The correct answer is B.A random walk process has an undefined mean-reverting level, and thus it is not covariance stationary.

Reading 5: Time Series Analysis

*LOS 5 (i) Describe characteristics of random walk processes and contrast them to covariance stationary processes.*