Monthly Volatility

Today I am going to look at the monthly volatility. The histogram of GE monthly log returns is presented here compared with the normal distribution with the same standard deviation:



Since the standard deviation of GE daily log returns is 3.848, I should expect a standard deviation of the GE monthly log returns to be close to 3.848 times the square root of 12/256, that is 0.840. In fact, the mean is 0.086, the standard deviation is 0.780, the skewness is -0.226 and the kurtosis is 4.615.

I generate 21 day mean, historical volatility, skewness and kurtosis as usual by calculating these quantities for a sample of 21 consecutive days and sliding this sample over the entire range of available data.



I calculate the same quantities for a log-normal random walk with the same standard deviation and mean as the GE data.



The mean and historical volatility histograms of the random walk are, of course, in good agreement with the theoretical distributions (the "Chi distribution" is actually the histogram obtained from a random walk 10 times larger):



The histograms of the GE data are less in disagreement than in case of the weekly returns:



Categories: stock options, volatility
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