Calculating Monthly Realized Variance from Daily Returns

# Compute monthly realized variance from daily data


# 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?

## Answer by Pleb (score 0, accepted)

https://quant.stackexchange.com/a/73940

### 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.