---
title: "Modeling lognormal properties of stock prices"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575612/Modeling%20lognormal%20properties%20of%20stock%20prices"
format: markdown
---
The most common model for the price of a non-dividend paying stock comes from a model of the *return* of a stock, which is assumed to follow an Ito process, as follows (see, for example, [Hull (1993)](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26579303), Chapter 9 and [Wilmott (2001)](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26579303)):

  


![image](media://08fc954e-0afb-4192-832d-f3d4bcde0612)

  


> 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). The model has a lot of intuitive appeal because it reflects that the change in price of the stock is proportional to the stock price at that moment.

  


Integrating Equation 1 over time, we get:

![image](media://846261e4-003d-4755-8f8b-ad482fe64b1c)

  


> Macro (mathblock)

                                                                                                          Equation 2

  


Alternatively:

  


![image](media://7cf6f339-e2d3-41a7-980e-2c3e7d82b898)

  


> Macro (mathblock)

                                                                                                            Equation 3

  


The relationship between consecutive periods in a time series is:

  


|  |  |  |
| --- | --- | --- |
| ![image](media://d1afc31e-7999-4c78-aad6-bc929ac86f57)<br>> Macro (mathinline) | or | > Macro (mathinline)<br>![image](media://1364a50a-3959-452f-85e3-999ab689f94b) |

  


*ASIDE: If you attempted to guess the above formula from **[Central Limit Theorem](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574953/)** principles you will be surprised to see the inclusion of a* *s*<sup>*2*</sup>*/2 component in Equation 2 for the mean return. It is there because the return in each short period is a function of the stock price at that moment, and therefore cannot be simply added up in log space. The proof of the formula is outside the scope of this guide: it is a result of applying Ito's Lemma (see, for example **[Hull](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26579303)** (1993, page 208), a mathematical result crucial in **[Black and Scholes work on valuing of options](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26586589)**.*

  


From the definition of a [Lognormal distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575244/), if log(S<sub>t</sub>) is normally distributed, then S<sub>t</sub> must be lognormally distributed. Thus, under this model, **the price of a stock will be lognormal**.

  


If the stock had no variance, Equation 3 would look like this:

 

![image](media://2f6db7b7-ea24-440c-a2c8-f92e67a315b6)

 > Macro (mathinline)

 or        

![image](media://d3108aed-25f4-459c-bc49-1d5d03200319)

    > Macro (mathinline)



  


If *m* is defined as the yearly continuously compounded return, then the daily return m<sub>d</sub> is determined by solving:

  


   > Macro (mathinline)

                    i.e., simply             *m*<sub>*d*</sub> = *m*/365

 [Lognormal Random Walk](#Lognormal_random_walk) model offers an example.

  


An extension to this model is to consider [mean reversion](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575450/), which means that the stock price tends to move back towards the expected return as a function of how far away it is at any moment from that expected return. The section [Stock price with mean reversion](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575613/) extends the above model to include mean reversion.

> Macro (anchor)



The links to the Lognormal Random Walk software specific models are provided here:

<details>
<summary>Crystal Ball</summary>

> Macro (inline-media-image)

 [Lognormal_random_walk](/wiki/download/attachments/26579343/Lognormal_random_walk-CB.xlsx?version=1&modificationDate=1506113496000&cacheVersion=1&api=v2)
</details>

  


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

> Macro (inline-media-image)

 [Lognormal_random_walk](/wiki/download/attachments/26581760/Lognormal_random_walk-AtRISK.xlsx?version=1&modificationDate=1506097516000&cacheVersion=1&api=v2)
</details>

  


The Lognormal Random Walk model is also the underpinning behind the Black Scholes formula, and can therefore be [used to value call options](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26586589).

  


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