Realized variance

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 ${\displaystyle {\sqrt {252\times RV}}}$.

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 ${\displaystyle P_{t}=\exp {(p_{t})}}$ is given by the stochastic integral

${\displaystyle p_{t}=p_{0}+\int _{0}^{t}\sigma _{s}dB_{s},}$

where ${\displaystyle B_{s}}$ is a standard Brownian motion, and ${\displaystyle \sigma _{s}}$ is some (possibly random) process for which the integrated variance,

${\displaystyle IV=\int _{0}^{t}\sigma _{s}^{2}ds,}$

is well defined.

The realized variance based on ${\displaystyle n}$ intraday returns is given by ${\displaystyle RV^{(n)}=\sum _{i=1}^{n}r_{i,n}^{2},}$ where the intraday returns may be defined by

${\displaystyle r_{i,n}=p_{\frac {it}{n}}-p_{\frac {(i-1)t}{n}},\qquad i=1,\ldots ,n.}$

Then it has been shown that, as ${\displaystyle n\rightarrow \infty }$ the realized variance converges to IV in probability. Moreover, the RV also converges in distribution in the sense that

${\displaystyle {\sqrt {n}}{\frac {RV^{(n)}-IV}{\sqrt {2t\int _{0}^{t}\sigma _{s}^{4}ds}}},}$

is approximately distributed as a standard normal random variables when ${\displaystyle n}$ is large.

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]

• Volatility (finance)