View source for Finance:Realized variance

'''Realized variance''' or '''realised variance''' (RV, see [[Social:American and British English spelling differences|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<ref>{{cite journal |last1=Andersen |first1=Torben G. |last2=Bollerslev |first2=Tim |year=1998 |title=Answering the sceptics: yes standard volatility models do provide accurate forecasts|pages=885–905|journal=International Economic Review |volume=39 |issue=4 |doi=10.2307/2527343|jstor=2527343 |citeseerx=10.1.1.28.454 }}</ref>
which is useful for many purposes, including volatility forecasting and forecast evaluation.

==Related quantities==

Unlike the [[Variance|variance]] the realized variance is a random quantity.

The realized [[Finance:Volatility|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 <math>\sqrt{252 \times RV}</math>.

==Properties under ideal conditions==

Under ideal circumstances the RV consistently estimates the [[Quadratic variation|quadratic variation]] of the price process that the returns are computed from.<ref>{{cite journal |last1=Barndorff-Nielsen |first1=Ole E. |last2=Shephard |first2=Neil |author-link2=Neil Shephard  |date=May 2002 |title=Econometric analysis of realised volatility and its use in estimating stochastic volatility models|pages=253–280 |doi=10.1111/1467-9868.00336|journal=Journal of the Royal Statistical Society, Series B  |volume=64 |issue=2 |s2cid=122716443 |doi-access=free }}</ref>
Ole E. Barndorff-Nielsen and Neil Shephard (2002), [[Organization:Journal of the Royal Statistical Society|Journal of the Royal Statistical Society]], Series B, 63, 2002, 253–280.

For instance suppose that the price process <math>P_t=\exp{(p_t)}</math> is given by the stochastic integral

: <math>p_t = p_0 + \int_0^t \sigma_s dB_s ,</math>

where <math>B_s</math> is a standard [[Brownian motion]], and <math>\sigma_s</math> is some (possibly random) process for which the integrated variance,

: <math>IV = \int_0^t \sigma_s^2 ds,</math>

is well defined.

The realized variance based on <math>n</math> intraday returns is given by <math>RV^{(n)} = \sum_{i=1}^n r_{i,n}^2,</math> where the intraday returns may be defined by
: <math>r_{i,n} = p_{\frac{it}{n}}-p_{\frac{(i-1)t}{n}},\qquad i=1,\ldots,n.</math>

Then it has been shown that, as <math>n\rightarrow\infty</math> the realized variance converges to IV in probability. Moreover, the RV also converges in distribution in the sense that

: <math>\sqrt{n}\frac{RV^{(n)}-IV}{\sqrt{2t\int_0^t \sigma_s^4 ds}},</math>

is approximately distributed as a standard normal random variables when <math>n</math> is large.

==Properties when prices are measured with noise==

When prices are measured with noise the RV may not estimate the desired quantity.<ref>{{cite journal |last1=Hansen |first1=Peter Reinhard |last2=Lunde |first2=Asger |date=April 2006 |title=Realized variance and market microstructure noise |pages=127–218 |doi=10.1198/073500106000000071 |journal=Journal of Business and Economic Statistics |volume=24 |issue=2 |url=https://cdr.lib.unc.edu/downloads/0z708z56p }}</ref>
This problem motivated the development of a wide range of robust realized measures of volatility, such as the [[Finance:Realized kernel|realized kernel]] estimator.<ref>{{cite journal |last1=Barndorff-Nielsen |first1=Ole E. |last2=Hansen |first2=Peter Reinhard |last3=Lunde |first3=Asger |last4=Shephard |first4=Neil |author-link2=Peter Reinhard Hansen |author-link4=Neil Shephard |date=November 2008 |title=Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise |pages=1481–1536 |doi=10.3982/ECTA6495 |url=http://www.econometricsociety.org/abstract.asp?ref=0012-9682&vid=76&iid=6&aid=9&s=-9999 |journal=Econometrica |volume=76 |issue=6 |url-status=dead |archive-url=https://web.archive.org/web/20110726230752/http://www.econometricsociety.org/abstract.asp?ref=0012-9682&vid=76&iid=6&aid=9&s=-9999 |archive-date=2011-07-26 |citeseerx=10.1.1.566.3764 }}</ref>

==See also==
*[[Finance:Volatility|Volatility]]

==Notes==
{{Reflist}}

[[Category:Mathematical finance]]

{{Sourceattribution|Realized variance}}