Realized variance or realised variance (RV, see
spelling differences
Despite the various English dialects spoken from country to country and within different regions of the same country, there are only slight regional variations in English orthography, the two most notable variations being British and American ...
) 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
which is useful for many purposes, including volatility forecasting and forecast evaluation.
Related quantities
Unlike the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
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
Under ideal circumstances the RV consistently estimates the quadratic variation of the price process that the returns are computed from.
Ole E. Barndorff-Nielsen and Neil Shephard (2002),
Journal of the Royal Statistical Society
The ''Journal of the Royal Statistical Society'' is a peer-reviewed scientific journal of statistics. It comprises three series and is published by Wiley for the Royal Statistical Society.
History
The Statistical Society of London was founded ...
, Series B, 63, 2002, 253–280.
For instance suppose that the price process
is given by the
stochastic integral
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...
:
where
is a standard
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
, 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 probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to ...
in the sense that
:
is approximately distributed as a standard normal random variables when
is large.
Properties when prices are measured with noise
When prices are measured with noise the RV may not estimate the desired quantity.
This problem motivated the development of a wide range of robust realized measures of volatility, such as the
realized kernel The realized kernel (RK) is an estimator of volatility. The estimator is typically computed with high frequency return data, such as second-by-second returns. Unlike the realized variance, the realized kernel is a robust estimator of volatility, in ...
estimator.
See also
*
Volatility (finance)
In finance, volatility (usually denoted by ''σ'') is the degree of variation of a trading price series over time, usually measured by the standard deviation of logarithmic returns.
Historic volatility measures a time series of past market pric ...
Notes
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Mathematical finance