Stochastic modelling of volatility, leverage effects, long-memory and extremal dependence of financial markets

Abstract

This research focused on using a three-stage simulation approach verified empirically for improved volatility modelling, relevant for robust risk management. The first two stages of the study are focused on developing simulation procedures in volatility modelling using two autoregressive models that involve the family of Generalised Autoregressive Conditional Heteroscedasticity (fGARCH) and the Generalised Autoregressive Score (GAS) models. The empirical evaluations of the simulation experiments in the first two stages were carried out using the S&P SA Bond Index data. The third stage of the study is used to estimate six essential features (or stylised facts) of financial return volatility that are relevant for valuable insights into risk assessment and investment decision-making. These features include pronounced persistence, mean reversion, leverage effect or volatility asymmetry, conditional skewness, conditional fat-tailedness, and the long-memory behaviour of volatility decomposition into long-term and short-term components. Specifically, in the first stage of the thesis, the study proposed a simulation framework using the Monte Carlo simulation (MCS) resources of the fGARCH models to determine a suitable conditional distribution for the error term to model the persistence of volatility. In the process of developing this framework, this study also proposed a new true-parameter-recovery measure which is used as a proxy of the coverage probability to accurately calculate the performance of the simulation experiment. In the second stage of the thesis, the study built on the developed framework (in the first stage) to propose a simulation structure through the GAS model for selecting an optimal error distribution for volatility modelling. The investigations at this stage proceeded by using both the fat-tails of distributions and √N consistency simulation experiments to show that the GAS model with a lower unconditional shape parameter (ν∗) value of 4.1 can be used to generate an appropriate simulated dataset that properly reflects the behaviour of financial returns data relevant for modelling volatility. This dynamic structure is intended to help interested users on MCS experiments utilising the GAS model for reliable volatility persistence calculations in finance and other areas. The simulation frameworks and procedures in the first two stages can be a useful guide to scientific practitioners and upcoming researchers on the relevant simulation steps to determine a suitable error distribution for volatility modelling. In the third stage, this research comparably applied three dynamic observation-driven models consisting of the fGARCH, GAS and Beta-Skew-t-EGARCH models to estimate the stated six essential features (or characteristics) of volatility, relevant for robust investment decisions and risk evaluation in the S&P Indian stock market. To begin with, the study comparatively used the robust fGARCH and GAS models to estimate the magnitude and dynamics of the persistence in conditional volatility using the returns from the Indian market index. Next, the study comparatively used the one- and two-component Beta-Skew-t-EGARCH models to estimate other features of the return volatility that include leverage effect or asymmetry, skewness, fat-tails, and the long-memory behaviour of volatility decomposition into long-term and short-term components. Specifically, we used both the one- and two-component models to estimate leverage effects, fat-tails, and skewness in the returns. The study further used a parametric model through the ARFIMA-FIGARCH models, and three semi-parametric approaches via the log periodogram estimator of Geweke and Porter-Hudak (GPH), the local Whittle estimator, and the exact local Whittle estimator to estimate and determine the presence of long memory in the returns and the return volatility, i.e., squared returns and absolute values of returns. The results of the estimations indicate that the daily returns, squared returns, and absolute returns exhibit long memory, hence, shocks decay at a slower rate. Furthermore, we used the two-component Beta-Skew-t-EGARCH model to investigate the long-memory decomposition of volatility into long-term and short-term components. Through this two-component model, the study found the existence of both long-run and short-run components of volatility in the persistence process, but the response to the effect of shocks in the short-run is higher than in the long-run volatility. This implies that higher volatility in the process is mostly due to the short-run volatility increase. Hence, through the applications of these models using the S&P Indian index, the study shows that the Indian market returns are characterised by the six volatility features. The empirical and simulation outcomes of the experiments in the third stage are used to offer both long-term and short-term suggestions to rational investors, government, and market managers for relevant assessment of the market investment risk.