Measuring volatility

The objective of this chapter is to examine the ARCH family of volatility models and
its use in risk analysis and measurement. An overview of unconditional and conditional
volatility models is provided. The former is based on constant volatilities while
the latter uses all information available to produce current (or up-do-date) volatility
estimates. Unconditional models are based on rigorous assumptions about the
distributional properties of security returns while the conditional models are less
rigorous and treat unconditional models as a special case. In order to simplify
the VaR calculations unconditional models make strong assumptions about the
distributional properties of financial time series. However, the convenience of these
assumptions is offset by the overwhelming evidence found in the empirical distribution
of security returns, e.g. fat tails and volatility clusters. VaR calculations based
on assumptions that do not hold, underpredict uncommonly large (but possible)
losses.
In this chapter we will argue that one particular type of conditional model (ARCH/
GARCH family) provides more accurate measures of risk because it captures the
volatility clusters present in the majority of security returns. A comprehensive
review of the conditional heteroskedastic models is provided. This is followed by an
application of the models for use in risk management. This shows how the use of
historical returns of portfolio components and current portfolio weights can generate
accurate estimates of current risk for a portfolio of traded securities. Finally, the
properties of the GARCH family of models are treated rigorously in the Appendix.