Is Volatility Time Dependent? - Part II

In the previous post we have seen how increasing the time horizon of the stock returns has the effect of making the volatility statistics closer to the one that one would expect for a log-normal random walk. This leads me to suspect that there could be a memory effect that fades away when waiting longer for a return.

Another way of looking at this is to fix the time horizon of the returns to one day and calculate the historical volatility and the other statistical moments using longer time windows than 21 days.

I have already seen side by side the daily plots of GE mean, historical volatility, skewness and kurtosis and the corresponding quantities of a log-normal random walk, calculated using a windows of 21 consecutive trading days sliding over the available data:



Here I do the same with a window of 63 consecutive trading days:



Here I use 252 consecutive trading days:



And here I use 1260 consecutive trading days:


The sharp jumps visible in the skewness and kurtosis of GE once the window grows longer than a quarter are due to the effect of the 1987 stock market crash to the distribution of log-returns.

If we compare the theoretical distributions with the histograms of the mean and historical volatility obtained from the random walk as the duration of the window grows from 21 days



to 63 days



to 252 days



and then to 1260 days,



the observation that one can make is that the size of the fluctuations grow as the number of samples contained in the histograms decreases because the length of the window is increased keeping the total number of data constant.

However, if I make this comparison using the GE data I see, not so much from the mean histograms,



but especially from the historical volatility histograms,


that, as the time window gets larger, the fluctuations become much larger than what they are in the case of the random walk prices.

Categories: stock options, volatility
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