HOW TO CALULATE THE VOLATILITY OF A BONANZA BASE

In finance we want to know how much profit (return) is generated on something of value to others that we buy, an asset, when we sell it. Alternatively we may want to know the return on the asset from one time to the next as we hold on to it.  There are a couple of good reasons for paying attention to returns instead of raw prices. 

First, financial markets are very close to perfectly competitive at the intra-day level where we buy into a position in a bonanza base or when we later sell on the break of the upward price trend in the markup phase of the stock’s life cycle.  Second, returns have much more attractive statistical properties such as stationarity (returns will come back to an average value) and ergodicity (the return generating system will return to states that are closely similar to previous ones over a sufficiently long period of time).  This said, here is how we perform the calculations.

The simple (net) return is defined as:

                                                           Equation 1

However, to make the measure comparable across different time periods we use the continuously compounded return also known as the log return of an asset:

         Equation 2

Where

From here we can construct the volatility measure by calculating sample standard deviations of this time series (r_t) over length of the bonanza base.  This is done as follows:

Over each bonanza base time interval, t, the sample variance of each price series is recorded as that bonanza base’s observation of the variance of log returns (n_rt):

                                Equation 3

where n = the number of observations available within the bonanza base; is the log return within the bonanza base; t indexes the tick to tick observations on r_t; and T indexes the time interval that covers the length of the bonanza base over which we are measuring volatility.  Note that the variance of log returns is a measure of the statistical dispersion of stock returns in the bonanza base, indicating how far from the expected return the returns typically are.

            From here, the standard deviation (volatility) of the log return within the bonanza base is simply the square root of the variance measure from equation 3:

                                         Equation 4

            This is how we calculate the volatility of a bonanza base where we prefer lower to higher volatility.  Sometimes people wonder why standard deviation is used to interpret volatility instead of variance.  In probability and statistics, the standard deviation is the more commonly used measure of statistical dispersion because it is defined as the square root of the variance.  This gives us a measure of dispersion that is (1) a non-negative number, and (2) has the same units as the data.  Variance only gives us a non-negative number whereas standard deviation generates a non-negative number that is also in the same units as the original price data it is calculated from.

            If stock returns are normally distributed things are simple because the mean and standard deviation fully describe the distribution.  The problem is that stock prices are actually lognormal in distribution.  For this reason we take the standard deviation of the logarithm of the stock return series to be completely accurate in our description of stock price volatility. In reality, however, we can take just the standard deviation of the stock return series with strong confidence that we are correctly estimating volatility.  This is because stock returns very rarely hit the lower barrier of zero where assumptions of normality create violations when describing a lognormal return.

                                                Equation 5

-SMB

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