Weekly Volatility

In "Calculation of Historical Volatility of Daily Returns" I have calculated the historical volatility of the stock price of General Electric. This calculation has been the basis for a comparison with the volatility of a computer generated series with normally distributed log returns.

In a few more posts I am going to extend this comparison to returns over an increasing period of time. I will start today with considering the weekly returns that were calculated in my post "Understanding Market Volatility".

The following picture shows the distribution of the annualized weekly log returns of GE stock price, compared with the normal distribution with the same standard deviation and mean.



In the case of a computer generated price series with normally distributed log returns, increasing the period over which the returns are calculated by a factor of T, simply decreases the historical volatility by a factor of the square root of T. This is a consequence of the fact that such price series is what statisticians call a Wiener process or a Gaussian random walk.

Since this concept is quite important I will expand on it briefly here. If I use the following definitions of annualized log daily returns and of daily returns, with Y indicating the number of trading days in a year,



and if I construct the annualized log daily returns to be a normally distributed random variable with given mean mu and standard deviation sigma,



and I define the annualized log returns over m days and the returns over m days like this



I can derive



where I use the summation properties of normally distributed independent random variables.

Eq. (8) seems to imply that the stock price at day i could be derived from the stock price at day i-m following the law



In fact, this is incorrect. There is only one random walk, characterized by a given mu and sigma, which is the one generating the random daily price sequence. The random m-day returns must be derived from this random walk and should not be calculated by generating a new random walk with a sigma divided by the square root of m. I'm going to explain how to do it now.

First of all, I recall the definitions of mean and variance over a sample of k consecutive days and I apply them to the case of the m-day log returns:


Of course, the square root of (10) is the m-day historical volatility calculated using a k day sample. But I will derive the k day mean first. I expect the k day mean of the m-day log returns to be a normally distributed random variable like this


and I will solve for the factors a and b. First of all, I re-write (7) like this



where I associate a subscript to the random variable N to indicate the particular random number generated on that specific day.

Using (12) in the definition (9) I get



Expanding (13) leads to different expressions depending on the relative magnitude of k and m:



Now, using the summation property of normally distributed indipendent random variables, I find that the factor a is always equal to 1,



and that the factor b is given by different formulas depending on the relative size of k and m:



Deriving the distribution for (10) is too complicated. It is not a Chi distribution as in the case of the 1-day returns. To make a comparison I will just use a random walk that is 10 times larger than the GE price series and use the histogram of the historical volatility for this random walk as an approximation of the true probability distribution of the historical volatility.

The standard deviation of the annualized GE daily log returns that we are using is 3.848. This is what has been used to generate the random walk price series. We should espect the standard deviation of the annualized GE weekly log returns to be close to that value divided by the square root of 256/52, that is 1.748.

In fact, the mean is 0.088, the standard deviation is 1.746, the skewnesss is -0.146 and the kurtosis is 7.134.

This is the random price history with normally distributed log returns that will be used in my comparisons:



The following graphs show the mean, historical volatility, skewness and kurtosis calculated by taking 21 consecutive days of annualized GE weekly log returns and sliding this window over the whole period of time available (time goes from right to left):



These graphs show the same quantities calculated for the random walk prices:



The histograms of the mean and of the historical volatility of the random walk agree very well with the theoretical probability distributions (the normal distribution and the histogram obtained from the a random walk 10 times larger):



That is definitely not the case when I consider the histograms obtained from the GE weekly returns:



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