Realized variance

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Realized variance or realised variance (RV, see spelling differences) is the sum of squared returns. For instance the RV can be the sum of squared daily returns for a particular month, which would yield a measure of price variation over this month. More commonly, the realized variance is computed as the sum of squared intraday returns for a particular day.

The realized variance is useful because it provides a relatively accurate measure of volatility[1] which is useful for many purposes, including volatility forecasting and forecast evaluation.

Related quantities[edit]

Unlike the variance the realized variance is a random quantity.

The realized volatility is the square root of the realized variance, or the square root of the RV multiplied by a suitable constant to bring the measure of volatility to an annualized scale. For instance, if the RV is computed as the sum of squared daily returns for some month, then an annualized realized volatility is given by .

Properties under ideal conditions[edit]

Under ideal circumstances the RV consistently estimates the quadratic variation of the price process that the returns are computed from.[2] Ole E. Barndorff-Nielsen and Neil Shephard (2002), Journal of the Royal Statistical Society, Series B, 63, 2002, 253–280.

For instance suppose that the price process is given by the stochastic integral

where is a standard Brownian motion, and is some (possibly random) process for which the integrated variance,

is well defined.

The realized variance based on intraday returns is given by where the intraday returns may be defined by

Then it has been shown that, as the realized variance converges to IV in probability. Moreover, the RV also converges in distribution in the sense that

is approximately distributed as a standard normal random variables when is large.

Properties when prices are measured with noise[edit]

When prices are measured with noise the RV may not estimate the desired quantity.[3] This problem motivated the development of a wide range of robust realized measures of volatility, such as the realized kernel estimator.[4]

See also[edit]

Notes[edit]

  1. ^ Andersen, Torben G.; Bollerslev, Tim (1998). "Answering the sceptics: yes standard volatility models do provide accurate forecasts". International Economic Review. 39: 885–905. doi:10.2307/2527343. 
  2. ^ Barndorff-Nielsen, Ole E.; Shephard, Neil (May 2002). "Econometric analysis of realised volatility and its use in estimating stochastic volatility models". Journal of the Royal Statistical Society, Series B. 64: 253–280. doi:10.1111/1467-9868.00336. 
  3. ^ Hansen, Peter Reinhard; Lunde, Asger (April 2006). "Realized variance and market microstructure noise". Journal of Business and Economic Statistics. 24: 127–218. doi:10.1198/073500106000000071. 
  4. ^ Barndorff-Nielsen, Ole E.; Hansen, Peter Reinhard; Lunde, Asger; Shephard, Neil (November 2008). "Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise". Econometrica. 76: 1481–1536. doi:10.3982/ECTA6495.