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Extending Realized Variance Across Multiple Trading Days

Article Quant Q&A · Author: Nel
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Summary

The document asks how to extend high-frequency realized volatility from a single day to a multi-day horizon. Its proposed method is to calculate daily realized variance from intraday returns and add the daily variances across the target period. Equivalently, the sampling interval can be expressed in units of the longer horizon, with the number of observations adjusted to cover that period. The response treats this as the straightforward multiperiod extension when the goal is to aggregate realized variance.

The example uses five-minute observations and a five-day horizon, and reports that the resulting annualized volatility is in a similar range to two range-based estimators. That comparison is an informal consistency check, not a general validation. A key limitation is that simply summing intraday realized variances omits price jumps between market close and the next opening. Such overnight moves may behave differently from jumps during trading hours, so the method may need adjustment when close-to-open risk matters. The sampling convention also depends on the assumed trading-session length and on using a compatible time unit.

Key ideas

  • Daily realized variance can be aggregated across days by summing the daily estimates.
  • A multiperiod calculation requires expressing the sampling interval and observation count in units consistent with the longer horizon.
  • Annualized volatility from the example is compared with range-based estimators as a rough reasonableness check.
  • Summing intraday estimates alone omits jumps between market close and the next opening.
  • The appropriate sampling interval depends on the assumed trading-session length and time convention.

Tags

VolatilityStatisticsHigh-frequency trading
Full text
# How to extend Realized Volatilty to multiple periods


# How to extend Realized Volatilty to multiple periods












I'm trying to calculate 5-day realized volatility (as proxy for integrated volatility) using 5-min frequency data.

I'm working from the paper

CORRECTING THE ERRORS: VOLATILITY FORECAST EVALUATION USING HIGH-FREQUENCY DATA AND REALIZED VOLATILITIES by Andersen et al (2005)

I'm able to use $$ RV_t(h) \equiv \sum_{i=1}^{1/h} r^{(h)2}_{t-1+ih} $$ with $t = 1$ day and $h=81$ for 81 5-min samples per day to get the daily realized variance.

In the paper, p.282, footnote 4

> For notational simplicity, we focus our discussion on one-period return and volatility measures, but the general results and associated measurement error adjustment extend in a straightforward manner to the multiperiod case

I would be very grateful if someone could point out that straightforward extension to me.

Thanks

Edit I'm going with the simple approach of adding the variances for 5 days, i.e. $h=81 * 5$ Annualized volatility calculated from this is in the same range as the Yang-Zang & Parkinsons estimators

Any input would still be appreciated.

## Answer by Diego F Medina (score 1, accepted)

https://quant.stackexchange.com/a/35754

I think your approach for adding the Realized Variances is correct. The focus on one period returns is simply because it eases notation. The only thing you would need to change is the units of time you are working with. This is $t=1$ (e.g. week) means five days and under that notation you would just need to change your $h$ to the new appropriate value $h/5$. This is equivalent to just adding the Realized Variances.

However, I believe the value of the time step $h$ for $5$ minutes is initially (considering 8 hours of trading) would be: $$h=\frac{5}{8*60}\approx0.0104$$ So that as your cited paper says:

> [...] $1/h$ is assumed to be an integer [...]

For the $5$ days case: $$h=\frac{1}{8*60}\approx0.0021$$

Note that this extension to multiple periods does not take into account jumps that occur between closure and market opening (ignored by simple adding the daily $RV$'s) which have shown different behaviours compared to the jumps during trading hours. For a detail discussion on how to include close-open jumps see for example Yang&Zhang.

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.