Comparison of Historical Volatilities of Market Returns and of Normal Returns

In "Calculation of Historical Volatility of Daily Returns" I have calculated the historical volatility of GE returns as the standard deviation of 21 consecutive trading days of daily log returns and presented the statistics of the data using histograms of the mean, volatility, skewness and kurtosis of all the 21 day samples obtained by sliding this 21 day window over the entire period of time covered by the data.

In "Is the Distribution of Log Returns Normal?" I have generated a sample of returns that are normally distributed.

Now I can use these normally distributed data to generate rolling 21 day samples in the same way I did with the GE daily log returns and calculate the mean, standard deviation, skewness and kurtosis of all of these samples.

These plots show the result of the calculation:



It is worthwile to look at the plots obtained from GE daily log returns and at the plots from the normally distributed data side-by-side:



The quantity that shows the most striking difference between the real market data and the normally distributed data is, in fact, historical volatility. Moreover, the real market data have a few outliers, that is a few samples with a mean way off the range, which are missing in the normally distributed data.

In the case of the normally distributed data, one knows which probability distribution best describes the statistics of the mean and of the volatility.

Any course of statistics 101 will say that the histogram of the mean of 21 day samples should follow the normal distribution with mean equal to the mean of the original data from which the samples have been taken and with standard deviation equal to the standard deviation of the original data divided by the square root of the sample size, that is 21.

Getting the distribution of the volatility is a bit trickier. Since historical volatility is the standard deviation of the sample, and the standard deviation is the square root of the sample variance, we need the distribution of the square root of the sum of the squares of statistically independent variables. This distribution is known as the Chi distribution.

The following two plots show, respectively, the histogram of the mean of the 21 day samples of the normally distributed data compared with the corresponding normal distribution, and the histogram of the volatility of the same 21 day samples compared with the appropriate chi distribution:



At this point, we have enough information to compare the histograms of mean, volatility, skewness and kurtosis of the 21 day samples derived from the GE daily log returns with the corresponding distributions or histograms derived from the normally distributed data:



These plots confirm that not only the distribution of the 21 day sample means of the market data is not normal but that the distribution of the historical volatilities is considerably different than the one derived from normally distributed data.

Categories: stock options, volatility
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