Option Trading Blog

If you ever heard the term “Log Normal Returns” and you wondered what it means, this post is for you. I’d also throw in a graph just for you.

Why assume anything to begin with?

Good question. Before I tackle the question on why it is assumed (in most cases) that returns are log-normally distributed, we have to question ourselves what good does it bring assuming something in the first place? the short answer is that it makes life a lot easier to develop a model to work with. Life is too complex to model it with equations. We have to make some assumptions to have something to work with. Hopefully it will all make sense at the end.

Why not assume returns are normally distributed?

Another good question. A normal distribution is defined for all values, including negative. This means that there is a (very small) probability that a stock will have a negative price which can’t happen in the real world( i.e -100% return). That is why we assume returns are log-normally distributed. We assume that the log of returns is normally distributed. This is good because if the log is normally distributed, the stock price can’t be negative. This is due to the properties of the exponential function. Neat trick not?

Is it even valid to assume returns are normally distributed?

Very problematic questions. Some take the stance that log returns are NOT normally distributed. There is also the question of time intervals. The following graph is an histogram of the log returns in the MAOF 25, which is the index in the Israeli Exchange of the biggest 25 companies:

This is taken 7 years back. It looks “sorta” normal. We can determine statistically if it is actually a normal distribution or not by something that is called the Jarque Bera Test. I did just that and found out that this is not a normal distribution. And indeed, it looks like almost all of the log returns are centered around 0. Not enough dispersion. But if we only look 3 years backward, the distribution looks like the following:

Which looks more like a normal distribution. The Jarque Bera Test confirms this. But what do we believe? taking 15 years back gives us an even weirder distribution.

Conclusion

This post reviewed the necessity for log-normal distribution and also showed that assumptions in models are needed but not always reflect what happens in reality.