Financial Risk Modelling and Portfolio Optimization with R

8

Modelling volatility

8.1 Preliminaries

The previous two chapters introduced quantitative methods for risk modelling in the case of non-normally distributed returns, that is, the extreme value theory and the generalized hyperbolic and generalized lambda distribution classes. The first method addresses the tail modelling of a return process, whereas the second focuses on adequately capturing the entire distribution. With respect to the value-at-risk and expected shortfall risk measures it was assumed that the financial market returns are i.i.d. Hence, these risk measures are unconditional in the sense that these measures do not depend on prior information. As already shown in Chapter 3, volatility clustering is one of the stylized facts of financial market returns. Given this stylized fact, the assumption of i.i.d. returns is clearly violated. Therefore, this chapter introduces a model class that takes volatility clustering explicitly into account. As will be shown, conditional risk measures can be deduced from these models. Here the phenomenon of volatility clustering directly feeds into the derived risk measures for future periods in time.

8.2 The class of ARCH models

The class of autocorrelated conditional heteroscedastic (ARCH) models was introduced in the seminal paper by Engle (1982). This type of model has since been modified and extended in several ways. The articles by Engle and Bollerslev (1986), Bollerslev et al. (1992) and Bera and Higgins (1993) provide an overview ...