Compute monthly realized variance from daily data

I am confused about the correct formula to compute monthly realized variance from daily data. What is the first sigma in the picture: sum or average? I mean, after subtracting each observation from monthly mean and then squaring each difference, should I just take the sum for each month or take the average?

• This formula hurts my eyes every time I happen upon it on this page. Who writes summations like that? Dec 3, 2022 at 22:46
• if you get the average that will give you the daily realized variance, whereas if you get the sum, as in the above formula, that will give you the monthly realized variance. Hence when you want to rescale the volatility you will have to multiply by sqrt(n) or sqrt(1/n)
– MaPy
Dec 5, 2022 at 10:51

You should take the sum:

The non-standard definition of realized variance in the paper of Moreira, A., & Muir, T. (2017), is an attempt to avoid an excessive amount of "infill" mathematical notation otherwise found in standard high-frequency econometrical literature. In essence, the entire paper is void of mathematical notation and only seek to define statistical models/methods where absolute necessary.

Reformulation:

Let $$X_t$$ be the log-price process, $$t \geq 0$$ denote the $$t$$'th month in your dataset and define $$n$$ as the "intramonth periods" (the authors assume $$n=22$$ days for each month). We can then define a sequence of partitions between each month $$t-1=t_0 < t_1 < \cdots < t_n = t$$ such that $$\sup_i t_i - t_{i-1} \rightarrow 0$$ for $$n \rightarrow \infty$$. This just implies that, as $$n$$ increases the distance between each intramonth timepoint converges to 0.

Furthermore, under the assumption of equidistant time-spacing of the intramonth periods, $$t_i = \frac{i}{n}$$ for $$i=1,\ldots,n$$, we can redefine the realized variance between $$t-1$$ and $$t$$ as follows:$$\;^{1}$$

\begin{align*} RV_t &= \sum_{i=1}^n \left( \left(X_{\frac{i}{n}} - X_{\frac{(i-1)}{n}}\right)- \frac{1}{n} \sum_{i=1}^n \left(X_{\frac{i}{n}} - X_{\frac{(i-1)}{n}}\right) \right)^2\\ &= \sum_{i=1}^n \left( R_{i,t}- \frac{1}{n} \sum_{i=1}^n R_{i,t} \right)^2\\ &= \sum_{i=1}^n \left( R_{i,t}- \bar{R}_{t} \right)^2\\ &\overset{\star}{=} \sum_{i=1}^n R_{i,t}^2 \end{align*}

where $$\bar{R}_{t} = \frac{1}{n} \sum_{i=1}^n R_{i,t}$$. The second equality is analogous to the definition of RV in the aforementioned paper and the last equality is standard for high-frequency econometrical papers dealing with intraday sampling frequencies.

Conclusively, after subtracting the monthly average ($$\bar{R}_t$$) from the intramonth periods ($$R_{i,t}$$) and then squaring the difference, you need to sum the squared differences.

1 The time-spacing between each intramonth period is then, $$\Delta_i = t_i - t_{i-1} = \frac{i}{n} - \frac{i-1}{n}=\frac{1}{n} = \frac{1}{22}$$, which is where the $$\frac{1}{22}$$ in the sum comes from.

$$\star$$ Very often high-frequency econometrical papers deal with intraday periods of different sizes. Between each day the effect of the estimated mean ($$\bar{R}_t$$) on realized variance/volatility, is very small and often negligible. As such it's safe to ignore (See this answer for details). However, this can not be said for intramonth periods.

• @NedaFathi Hey. I see that you have posted a question on this forum (and have already received a response) which is the proper way of asking for help :-). I am currently busy so I cannot give you a detailed answer. As such, review the answer you've received to your question, see if it makes sense and then accept + upvote it, if the answer helps you. Good luck!
– Pleb
Jan 13, 2023 at 19:15