Ekonometriese modelle in finansiële risiko

Abstract

This paper provides an overview of the contributions by prof JH Venter to financial risk and volatility modelling, estimation and forecasting. Venter's research is based on the classical GARCH model which he refines in various ways. In the classical GARCH model the innovation distribution is assumed to be standard normal but recent research emphasized the need for more general distributions allowing both asymmetry (skewness) and kurtosis in the innovation distribution to obtain better fitting models. This can be achieved by variance mixtures of normal distributions. If the mixing variable is taken as unit inverse Gaussian distributed the resulting innovation distribution has the NIG-distribution whose density can be computed analytically, easing likelihood calculations. In essence this is the NIG-GARCH model for daily returns. Venter interprets the mixing variable as a latent factor due to news noise impacts that adjust the traditional GARCH volatilities to account for events occurring after market closure on the previous day. This GARCH model with NIG-distributed innovations leads to more accurate parameter estimates than the normal GARCH model. Venter concludes that it is the mixing concept and not the particular distribution choice that leads to better fitting models. This is encouraging in the sense that it is the underlying phenomenon that one is trying to model that is important, more so than the specific mathematical forms that one uses in the process, and the results should be stable when these forms are varied over reasonably possible alternatives. These models are fitted to an empirical data set and in the process of doing so, Venter finds empirical support for Merton's ICAPM. In order to obtain even more accurate parameter estimates, and since Venter expected an information gain if more data is used, he extends the above mentioned model to cater for high, low and close data, as well as full intra-day data, instead of only daily returns. This is achieved by introducing the Brownian inverse Gaussian (BIG) process, which follows naturally from the unit inverse Gaussian distribution and standard Brownian motion. This model postulates that over each trading day anew the intra-day return process follows a Brownian motion with drift and volatility that are inherited from the previous day in typical GARCH fashion but are also subject to random Inverse Gaussian (IG) distributed news noise impacts that arrive after market closure on the previous day. He calls it the BIG-GARCH model and derives the likelihood function needed to fit the model; this uses the daily returns and realized volatilities as sufficient statistics. Venter then introduces a number of new distributions related to the Inverse Gaussian distribution and also derives diagnostics that may be used to check the quality of fit. The new model produces two volatility measures, called the expected volatility and the actual volatility. Venter shows that the latter is close to the realized volatility. Fitting these models to empirical data, he finds that the accuracy of the model fit increases as one moves from the models assuming normally distributed innovations and allowing for only daily data to those assuming underlying BIG processes and allowing for full intra-day data. However, Venter encounters one problematic result, namely that there is empirical evidence of time dependence in the random impact factors. This means that the news noise processes, which is assumed to be independent over time, are indeed time dependent, as can actually be expected. In order to cater for this time dependence, Venter extends the model still further by allowing for autocorrelation in the random impact factors. The increased complexity thatthis extension introduces means that one can no longer rely on standard Maximum Likelihood methods, but have to turn to Simulated Maximum Likelihood methods, in conjunction with Efficient Importance Sampling and the Control Variate variance reduction technique, in order to obtain an approximation to the likelihood function and the parameter estimates. He finds that this time dependent model assuming an underlying BIG process and catering for full intra-day data fits generated data and empirical data very well, as long as enough intra-day data are available. Several ideas for future research are given, especially the need to explore the application of this approach to other important areas of statistical finance, such as volatility forecasting and pricing of derivatives. He finds that this time dependent model assuming an underlying BIG process and catering for full intra-day data fits generated data and empirical data very well, as long as enough intra-day data are available. Several ideas for future research are given, especially the need to explore the application of this approach to other important areas of statistical finance, such as volatility forecasting and pricing of derivatives

In hierdie artikel word 'n oorsig gegee van die bydrae wat prof JH Venter gemaak het tot finansiele risiko en volatiliteits-modellering, -beraming en -vooruitskatting. Venter gebruik die klassieke GARCH model as basis vir sy navorsing en konsentreer veral daarop om beter modelle as die normaal verdeling te vind vir die innovasie verdeling. In die verband stel hy 'n raamwerk voor waarbinne die klas van gemiddelde-normaal-mengselverdelings bespreek en evalueer word. Hy vind dat die klas van verdelings onderling min verskil ten opsigte van die akkuraatheid van modelpassing en volatiliteitsberaming, maar heelwat verbeter op die klassieke model. Wanneer intradag-data beskikbaar is, kan gerealiseerde volatiliteit gebruik word om volatiliteit te beraam. In hierdie konteks formuleer Venter 'n nuwe model wat 'n hibriede samestelling van die tradisionele GARCH-model en 'n stogastiese volatiliteitsmodel is. Hy verwys hierna as die BIG-GARCH- model en lei die aanneemlikheidsfunksie af wat nodig is vir die modelpassing. Twee volatiliteitsmaatstawwe spruit voort uit die nuwe model, naamlik verwagte volatiliteit en werklike volatiliteit. Venter wys dat laasgenoemde maatstaf baie nou verwant aan gerealiseerde volatiliteit is