Overview of historical volatility models

Historical volatility is a static measure of variability of security returns around their
mean; they do not utilize current information to update their estimate. This implies
that the mean, variance and covariance of the series are not allowed to vary over
time in response to current information. This is based on the assumption that the
returns series is stationary. That is, the series of returns (and, in general, any time
series) has constant statistical moments over different periods. If the series is
stationary then the historical mean and variance are well defined and there is no
conceptual problem in computing them. However, unlike stationary time series, the
mean of the sample may become a function of its length when the series is nonstationary.
Non-stationarity, as this is referred to, implies that the historical means, variances and covariances estimates of security return are subject to error estimation.
While sample variances and covariances can be computed, it is unlikely that it
provides any information regarding the true unconditional second moments since
the latter are not well defined. The stationarity of the first two moments in security
return has been challenged for a long time and this has called into question the
historical volatility models. This is highlighted in Figure 2.1 which shows the
historical volatility of FTSE-100 Index returns over a 26- and 52-week interval.