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Lognormal Distribution for Stock Price Returns

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A lognormal distribution is used as the standard model stock price returns in financial economics. In this article we will go in the depth to why this is so.

Statistical Distributions

Before talking about stock price return as lognormal distribution, I want to give a quick overview of normal and lognormal distributions.

Note: this is not a complete tutorial on normal and lognormal distributions but I do talk about the properties that are important for the purpose of this article.

Normal Disribution (Gaussian Distribution)

So, what does it mean when someone says that the data follows normal distribution. An easy way of explaining this is that if you have n number of elements and if you plot all of these elements on a graph you will the shape of the data similar to the following figure:

You might be wondering why we are interested in a set of data that plots in the shape of a bell curve. The reason is that there are a few properties associated with this kind of data that can help us in making statistical inferences about the the dataset itself (e.g. 1 sigma data, 2 sigma and 3 sigma data). So, it is safe to say that any function that conforms to normal distribution (looks like a bell curve when plotted) also inherits the properties of the normal distribution.

You can download the excel file that creates a normal distribution graph in Appendix A.

Lognormal Distribution

In simple terms: a lognormal distribution is the result of a function that produces output data to look like the following figure:

Important property of this distribution is that it does not take values less than 0. But how do we get this shape? A lognormal distribution is very much what the name suggest "lognormal". I explain this as follows: Imagine that you have a function that is the exponent of some input variable. The input variable itself is a normal distribution function.

e.g.  $y=k.e^{x}$

Now, if we take a natural log of this function then we end up with a normal distribution. Why? because taking a natural log on an exponent function returns you the input variable and we have already stated that the input variable is a normal distribution.

You can download the excel file that creates a normal distribution graph in Appendix A.

Stock Price Analysis

Now, lets look at why lognormal distribution is the distribution of choice when pricing assets. There are 3 important observations that we have to recognize in order to understand this:

1. The price of a stock at time $t_1$ is dependent on 2 variables: the stock price at $t_0$ and rate of return $(r)$ for the interval $[t_0, t_1]$:
What this means is that for 1 period of time $[t_0, t_1]$, we can get the new stock price by using the simple transformation:

past stock price x rate of return = current stock price

2. The rate of return $(r)$ follows a normal distribution:
This is an important assumption to understand. In order to completely understand this we should first think about the properties of a normal distribution, particularly: the shape of the distribution. Just by looking at a normal distribution graph, we can easily tell the farther you move away from the mean the likelihood of getting a sample becomes less hence the likelihood of getting samples close to the mean is much higher. In other words: if we have a mean rate-of-return equal to $r_0$ for time $t_1$, we will see the new rate-of-return $r_1$ at time $t_1$ very "close to" $r_0$.  This property of being "close to" gives us the idea of the rate-of-return conforming to a normal distribution.

3. For the time interval $[t_0, t_1]$ we continuously compound the return:
A very important concept in finance is compounding the return over an interval of time.

What I do want to quickly go over is that a continuously compounded rate of return is expressed using the mathematical concept of exponent $e^{(x)}$. So, if have have a rate of return = $r$, then the continuously compounded rate of return is: $e^{(r)}$.

Example

Suppose you are following a stock and you are interested in estimating the price of the stock after a time interval $h$, and this time interval could be anything from an hour to a year... Let's have the following variables:

• initial stock price = $S_0$
• end stock price = $S_1$
• rate of return = $r$

We can have the following equation to get the stock end price:

 $S_1 = S_0 . e^{(r)}$ [1]

Where did $e$ come from? $e$ is showing that the rate of return $r$ is continuously compounded.

Now, let's talk about why it is said that asset prices (stock price in this example) follow a lognormal distribution. From the example equation [1] above we can see that the rate of return is an input to the exponent function and $S_0$ is a constant. If we take the natural log of the the expression then we are left with a normal distribution, why?...because we know that the rate of return follows a normal distribution and also because taking the natural log of an exponent function leaves you with the input (i.e $r$ in this example).

0 #8 AMIR 2012-11-28 20:36
thanks,it was very good for me.

I think you should mention that returns being normally distributed is only a model, taken from brownian motion /wiener process in Physics...

0 #6 Omkar 2012-04-30 05:37
Thanks a ton for this explanation, crisp and clear.

0 #5 MathsManiac 2012-04-16 09:11

0 #4 Estefanos Montecarlo 2012-03-04 02:26
Good one, simple and to the point!

+1 #3 Sam 2011-11-14 17:19
That's great.

0 #2 Anar 2011-11-14 17:04

Very good explanation - thank you.

Other websites skip explaining why lognormal is accepted as the right distribution for stock prices. One website's explanation was simply to say that - because it is accepted as the standard by the industry!!!