Realized Volatility and Standard Deviation Across Time Horizons
Summary
The document distinguishes realized volatility, calculated as the square root of summed squared log returns, from the standard deviation of returns, which averages squared deviations. For returns assumed to have a zero mean, the key difference is the division by the number of observations. The discussion explains that these quantities refer to different time scales: realized volatility aggregates return variation over the sampled interval, while the standard deviation describes variation per observation. Converting between horizons requires assumptions about how variance accumulates through time.
The answers also correct the log-return definition and discuss equal or unequal weighting of intraday returns. Equal weights assume returns are identically distributed; alternative weights can represent intraday volatility patterns. Overnight returns may be excluded for intraday analysis or incorporated for longer horizons, with the choice depending on the purpose. The suggested scaling to daily or annual variance assumes variance grows linearly with time, so it may be inappropriate when that assumption fails. The material is conceptual and offers no empirical comparison of these assumptions.
Key ideas
- Realized variance sums squared returns, while a per-observation variance estimate averages them.
- The distinction reflects the time horizon represented by each measure.
- Scaling variance across horizons assumes variance accumulates linearly through time.
- Equal weighting assumes returns are identically distributed, while varying weights can model intraday patterns.
- Whether to include overnight returns depends on the analysis objective.
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# Realized Volatility vs. Standard deviation of log returns
# 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 defined 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 estimate 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} $ term in the standard deviation calculation.
Can you explain the significance of this? Why does realized variance not have 1/N and how can the 2 be interpreted?
reference https://en.wikipedia.org/wiki/Realized_variance
## Answer by phdstudent (score 7)
https://quant.stackexchange.com/a/19319
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 $\sqrt{(251)}$.
In other words, your last expression is the 5-min realized volatility whereas the second to last expression is the daily realized volatility.
## Answer by Soumirai (score 1)
https://quant.stackexchange.com/a/60115
- Your definition of a log return is wrong, it is $y_i = \ln{(x_{i+1}/x_i)}$
- You have 5-min returns so you are going to first compute 5-min variance:
$Variance = \frac{1}{N}\sum_{i=1}^N y_i^2 = \sum_{i=1}^N w_iy_i^2$
Here we have $w_i = 1/N$. By doing this you assume that the $y_i$ are iid, a relatively strong assumption. The "identically distributed" means that variance is linearly distributed in time. If you don't suppose them to be identically distributed (for instance if you think that mid-day returns are less volatile than end of day returns), you can use a different weighting by changing the $w_i$.
- You may want to incorporate the overnight return in your computation as well. It can matter or not, it depends on which purpose you are computing the variance for. If it is for an intraday purpose only then you don't need it.
If you want to work with volatilities longer than intraday, you should include the overnight move with an appropriate weighting $w_i$. For instance if you assume that the distribution of that overnight return is the same as that of a 5-min intraday return (probably a very bad assumption), you can also use $w_i=1/N$.
Other example: overnight time is $24-6.5 = 17.5$ hours, or $17.5 * 60 / 5 = 210$ periods of 5 minutes, so you can consider that the overnight move consists of 210 5-min returns. For instance 209 zeros and one $y_i$. Or 210 times $y_i/210$. It will obviously increase your number of observations $N$ by 210 for each overnight return you include.
- Last, if you are not interested by 5-min variance (i.e. the variance of 5 min returns), you can scale it to obtain the variance for a different time horizon. Again this scaling is done under assumptions of how variance is distributed across time. If again we assume it is linearly distributed across time, you would have to scale your 5-min variance by 288 to get a daily variance, because there are 288 times 5-min in a day. And if you want to get a yearly variance from your daily variance, again you can multiply it by 365 (or 252, depending on how you account for non-business days...)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.