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title: "Stock price with mean reversion"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575613/Stock%20price%20with%20mean%20reversion"
format: markdown
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This section adds a [mean reversion](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575450/) sophistication to the stock price model developed in [modeling lognormal properties of stock prices](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575612/).

 

It has already been shown that the most common model for the price of a non-dividend paying stock comes from a model of the *return* of a stock, as follows:

 

![image](media://0a7c2ed6-08f1-486c-9b79-adf97a79b4eb)

> Macro (mathblock)

                                                                                                  Equation 1

 

where *DS* is the change in the stock price at some small time interval *Dt*, *m* is the proportional return of the stock (usually expressed as an annual proportion, like 10%) and *s* is the stock price volatility (approximately equal to the standard deviation of the yearly return of the stock

 

This gives the equation for the stock price at some time t as:

 

![image](media://66f6a3e1-c2cd-4b1c-a6bb-cd9b3f9e91ff)

> Macro (mathblock)

                                                                                                      Equation 2

 

where *log* means the natural log of the variable. A random variable X such that log(X) = N(m,s) has a mean E(X) of Exp(m+s2/2) – see, for example, the equation for the mean of the <u>[Lognormal (format 2) distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575245/)</u>. Comparing this with Equation 2, we therefore see that the mean of St = E[St] takes the following value:

 

![image](media://2f601848-738c-40ab-bc7a-37a1eabdaec3)

> Macro (mathblock)

                                                                                                      Equation 3

 

The section <u>[Mean reversion](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575450/)</u> described how for a variable X that, without mean reversion, would take the form:

 

![image](media://984db9c8-f9cb-48d7-a8b6-c2d4a118992f)

 

> Macro (mathblock)

 

we can introduce mean reversion from one period to the next one, as:

  

![image](media://f0db7a0f-b3a2-4e06-a29c-34a5ef209978)

             

> Macro (mathblock)

 

In our stock price model, it is the *return* of the stock that is considered the fundamental random variable, so this formula would apply to the log of the stock price. We therefore add mean reversion to Equation 2 as follows :

 

Start with:

 

![image](media://c3098ce5-cec6-4ab5-ab25-dca9fc08ede7)

> Macro (mathblock)

                                                                                         from Equation 2. Then, applying mean reversion:

 

![image](media://28f5983d-2a2d-46b8-a0ef-942ad37e8f11)

 

> Macro (mathblock)

                                                                                                   Equation 3

 

From Equation 2, we see that:

![image](media://463d3444-693b-451c-8de3-832af9617ac9)

 

> Macro (mathblock)

                                                                                                    Equation 4

 

Combining Equations 3 and 4:

 

![image](media://8ddb57a1-84ab-4f2c-96e9-bf48e89625d0)

> Macro (mathblock)

                                                                                                       Equation 5

 

Increasing values of the mean reversion parameter b reduces the spread (variance, volatility) of the stock price St with time. It also reduces the expected return.

 

Building such a model is not possible in some simulation software packages such as Crystal Ball, as a function is needed in the software to run multiple simulations. An example of the mean reversion model, can be found in the model [Stock Price with Mean Reversion](#Stock_price_with_mean_reversion)

> Macro (anchor)



The link to the Stock Price with Mean Reversion software specific model is provided here:

<details>
<summary>@Risk</summary>

> Macro (inline-media-image)

 [Stock_price_with_mean_reversion](/wiki/download/attachments/26581760/Stock_price_with_mean_reversion-AtRISK.xlsx?version=1&modificationDate=1506097527000&cacheVersion=1&api=v2)
</details>

 

 

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