Stable distributions with applications to South African financial data.

Abstract

In recent times, researchers, analysts and statisticians have shown a keen interest in studying Extreme Value Theory (EVT), particularly with the application to mixture models in the medical and financial sectors. This study aims to validate the use of stable distributions in modelling three Johannesburg Stock Exchange (JSE) market indices, namely the All Share Index (ALSI), Banks Index and the Mining Index, as well as the United States of American Dollar (USD) to South African Rand (ZAR) exchange rate. This study leverages the unique properties of stable distributions when modelling heavy-tailed data. Nolan’s S0-parameterization stable distribution (SD) was fitted to the returns of the three FTSE/JSE indices and USD/ZAR exchange rate and a hybrid Generalized Autoregressive Conditional Heteroskedasticity (GARCH)-type model combined with stable distributions was fitted to each return series. The two-tailed mixture model of the Generalized Pareto Distribution (GPD), stable distribution, Generalized Pareto Distribution referred to as GSG, as well as the Stable-Normal-Stable (SNS) and Stable-KDE-Stable (SKS) was fitted to evaluate its relative performance in modelling financial data. Results show that the S0-parameterization SD fits the South African financial returns well. The hybrid GARCH (1,1)-SD model competes favourably with the GARCH-GPD model in estimating Value-at-Risk (VaR) for FTSE/JSE Banks Index, FTSE/JSE Mining Index and the USD/ZAR exchange rate returns. The hybrid EGARCH (1,1)-SD competes well against the GARCH-GPD model for the FTSE/JSE ALSI returns. Inconclusive results are observed for the short position of the fitted GKG models; however, in the long position, an appropriate fit of the GPD-KDE-GPD (GKG) model, where KDE is the kernel density estimator, is emphasised for all four return series. The proposed mixture models, GSG, SNS and SKS models, are found to be a good alternative in fitting South African financial data to the commonly used GPD-Normal-GPD (GNG) mixture model. The results of this study are important to financial practitioners, risk managers and researchers as the proposed mixture models add more value to the literature on the applications of extreme mixture models.