In order to understand why Lognormal distribution is suitable for stock price returns, let’s first think about stock prices, the following graph is a stock price chart over time:T.

**Section A:
**

# Section B:

Why is z(H) normally distributed?

We have to recognize some assumptions in order to realize that z(H) is normally distributed. We know that z(H) is made up of T sub intervals and for each interval we can say that these sub intervals are:

- Independently distributed

a. This is because the stock price change (and hence return) over a sub interval is not dependent on the previous price change (return). - Identically distributed

a. With the interval sufficiently small stock return appear to have the same distribution.

The first two assumptions tell that stock price follows a random walk (see: http://en.wikipedia.org/wiki/Random_walk) and random walk is associated with “efficient market” theory. - The expected continuously compounded return is:

a. Here, is the size of the sub interval and is the return per unit time and is independent of the interval length of the subinterval . The expected return will increase/decrease based on the size of . - The variance of the continuously compounded return is:

a. Here, is the size of the sub-interval and is the variance per unit time and is independent of the length of the interval length of the subinterval h.

Based on these 4 assumptions we can say that the returns are normally distributed. We can also say that is normally distributed. Why? Because of central limit theorem (see: Central Limit Theorem). The central limit theorem says that the summation of a large number of independent distributions results in a normally distributed variable .

Now, that we have understood that is normally distributed, please read "Section A" again and to understand why stock return is lognormally distributed.

## Comments

+++thatsafunnypic.com+++

RSS feed for comments to this post