# Realized Volatility vs. Standard deviation of log returns

I am interested in calculating high frequency 5-minute intraday volatility. I am going to use the standard Realized volatility which is the square root of the sum of squared log returns.

Given X is the log price of a stock the log return Y is calculated as $$log \ returns = Y = \frac{x_i+1}{x_i}$$

Then the realized variance is the sum of the squared log returns:

$$Realized\ Variance = \sum\limits_{i=1}^{n} (y_{t_{i}})^2$$

$$Realized \ Volatility = \sqrt{\sum\limits_{i=1}^{n} (y_{t_{i}})^2 }$$

For 5-minute realized volatility n = 78 (there are 6.5 hours in the NYSE trading day)

Now if Y is the log returns and the mean of Y is assumed to be zero you can also calculate a standard deviation

$$standard \ deviation = \sqrt{\frac{1}{N}\sum\limits_{i=1}^{N} (y_i)^2}$$

So you can see the only difference between the Realized Volatility of Y and the standard deviation of Y is the $$\frac{1}{N}$$ termin the standard deviation calcualtion.

Can you explain the significance of this? Why does realized variance not have 1/N and how can the 2 be interpreted. Thank you.

It is all a matter of frequency. For instance if you want to get annual realized volatility you multiply your last expression by $\sqrt{(N*251)}$ or the second to last expression by $\sqrt{(251)}$.