Financial Risk Modelling and Portfolio Optimization with R

6

Suitable distributions for returns

6.1 Preliminaries

It was shown in Chapter 4 that risk measures like VaR and ES are quantile values located in the left tail of a distribution. Given the stylized facts of empirical return series, it would therefore suffice to capture the tail probabilities adequately. This is the subject of extreme value theory, which will be covered in the next chapter. However, the need often arises to model not just the tail behaviour of the losses, but the entire return distribution. This need arises when, for example, returns have to be sampled for Monte Carlo type applications. Therefore, the topic of this chapter is the presentation of distribution classes that allow returns to be modelled in their entirety, thereby acknowledging the stylized facts. A desired distribution should be capable of mirroring not only heavy-tail behaviour but also asymmetries. In particular, the classes of the generalized hyperbolic distribution (GHD) and its special cases, namely the hyperbolic (HYP) and normal inverse Gaussian distributions (NIG), as well as the generalized lambda distribution (GLD) will be introduced in Sections 2 and 3. A synopsis of available packages follows in Sections 4 and 11, and the chapter ends with applications of the GHD, HYP, NIG and GLD to financial market data.

6.2 The generalized hyperbolic distribution

The GHD was introduced into the literature ...