Asset Return Volatility Models

Abstract

Whereas the returns on financial assets can be assumed to be serially uncorrelated, although the issue of asset return predictability is still fiercely debated in the literature, see e.g. Campbell and MacKinlay (1999), it has become widely accepted that return volatility is highly autocorrelated. If the volatility process of financial asset returns were to be perfectly correlated, i.e. if volatility was constant over time, the variance of the underlying series could simply be calculated as

$${\sigma ^2} = \frac{1}{{T - 1}}\sum\limits_{t = 1}^T {{{\left( {yt - m} \right)}^2}} ,$$

(2.1)

where y _t is the continuously compounded return on the relevant asset at time t, T is the total number of observations and the sample mean $m = \frac{1}{T}\sum\nolimits_{t = 1}^T {yt}$ A practical question that arises is what the length of the sample should be when volatility changes over time. Conventional wisdom would suggest that the sample should consist of a large number of observations as this would improve the statistical accuracy of the volatility estimates. On the other hand, it also implies that extreme shocks to the return process that took place a relatively long time ago, and which contain little information about the current volatility level, will still have a major impact since all observations in the sample are weighted equally. An alternative could be to select a smaller sample size and use a rolling window principle, where the sample size T remains constant and the variance is recalculated with each new observation that becomes available. The problem of extreme values remains however as the variance estimate of equation (2.1) will drop considerably in value when observations relating to this event fall out of the sample. Exponentially weighted methods have been proposed to remedy this problem but the reported high autocorrelation of the volatility process has led to the development of more intricate time-varying volatility models which incorporate this feature.