Is The Distribution of Log Returns Normal?

The standard assumption made when evaluating option prices is that returns are log-normally distributed, that is that the logarithm of the returns follow a normal distribution. This is not the case, however.

I can easily generate a sequence of data of the same size as the GE sample that I have collected from Yahoo! and that are normally distributed. If you need to know what the normal distribution of events is, check Wikipedia.

To do that I need the mean and the standard deviation sigma of the GE sample:

mean = 0.085
sigma = 3.848

Using these values I generate normally distributed random numbers.

This chart shows the frequency histogram of the normally distributed sample:



The black curve is the normal distribution with the same mean and standard deviation used to generate the sample.

The agreement between the histogram and the theoretical distribution is very good as it should be.

Let's compare instead the histogram of the GE daily log returns with the normal distribution with the same mean and standard deviation:



The agreement is not so good. The discrepancy can be measured quantitatively by using two parameters: skewness and kurtosis.

Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point. Since the normal distribution is symmetric, the skewness of a normnal distribution is zero.

Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. That is, data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak. The kurtosis of a normal distribution is exactly 3.

The skewness and kurtosis of the normally distributed sample agree with the normal values:

normal sample skewness = -0.031
normal sample kurtosis = 3.044

However, the skewness and kurtosis of the GE daily log returns are quite different, about three times larger:

GE log return skewness = -0.09
GE log return kurtosis = 8.695

Categories: stock options, volatility
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