This paper considers the high-moments and uses the skew generalized error distribution (SGED) to explain the financial market data which have leptokurtic, fat-tailed and skewness. And we compare performance with the commonly used symmetrical distribution model such as normal distribution, student¡¦s t distribution and generalized error distribution (GED). To research when returns of asset have leptokurtic and fat-tailed phenomena, what model has better predictive power for volatility forecasting? The empirical procedure is as follows: First step, make the descriptive statistics of raw data, and know that the GARCH effect should be considered, followed by selecting the optimal order of ARMA-GARCH. The second steps, make the parameter estimations of full-sample, and pick up the best model. Finally, forecast out-of-sample volatility for 1-day, 2-day, 5-day, 10-day and 20-day respectively, not only use different loss function to measure the performance, but also use DM test to compare the relative predictive power of the models under the different error distribution.

Based on Ph. D. Thesis--University of Southampton, 2002. / Includes bibliographical references.

Based on Ph. D. Thesis--University of Southampton, 2002. / Includes bibliographical references.

The 2008/9 financial crisis intensified the search for realistic return models, that capture real market movements. The assumed underlying statistical distribution of financial returns plays a crucial role in the evaluation of risk measures, and pricing of financial instruments. In this dissertation, we discuss an empirical study on the evaluation of the traditional portfolio risk measures, and option pricing under the hybrid Brownian motion model, developed by Shaw and Schofield. Under this model, we derive probability density functions that have a fat-tailed property, such that “25-sigma” or worse events are more probable. We then estimate Value-at-Risk (VaR) and Expected Shortfall (ES) using four equity stocks listed on the Johannesburg Stock Exchange, including the FTSE/JSE Top 40 index. We apply the historical method and Variance-Covariance method (VC) in the valuation of VaR. Under the VC method, we adopt the GARCH(1,1) model to deal with the volatility clustering phenomenon. We backtest the VaR results and discuss our findings for each probability density function. Furthermore, we apply the hybrid model to price European style options. We compare the pricing performance of the hybrid model to the classical Black-Scholes model. / Dissertation (MSc)--University of Pretoria, 2017. / National Research Fund (NRF), University of Pretoria Postgraduate bursary and the General Studentship bursary / Mathematics and Applied Mathematics / MSc / Unrestricted

Value-at-Risk (VaR) model fails to predict financial risk accurately especially during financial crises. This is mainly due to the model’s inability to calibrate new market information and the fact that the risk measure is characterised by poor tail risk quantification. An alternative approach which comprises of the Expected Shortfall measure and the Lognormal Jump-Diffusion (LJD) model has been developed to address the aforementioned shortcomings of VaR. This model is called the Simulated-Expected-Shortfall (SES) model. The Maximum Likelihood Estimation (MLE) approach is used in determining the parameters of the LJD model since it’s more reliable and authenticable when compared to other nonconventional parameters estimation approaches mentioned in other literature studies. These parameters are then plugged into the LJD model, which is simulated multiple times in generating the new loss dataset used in the developed model. This SES model is statistically conservative when compared to peers which means it’s more reliable in predicting financial risk especially during a financial crisis. / Statistics / M.Sc. (Statistics)

The current study empirically analyzes the extreme behavior and the impact of deregulation policies as well as financial turmoil on the extreme behavior of changes of Taiwan short term interest rate. A better knowledge of short-term interest rate properties, such as heavy tails, asymmetry, and uneven tail fatness between right and left tails, provide an insight to the extreme behavior of short-term interest rate as well as a more accurate estimation of interest risk. The predicting performances of filtered and unfiltered VaR (Value at risk) models are also examined to suggest the proper models for management of interest rate risk. By applying Extreme Value theory (EVT), tail behavior is analyzed and tested and the VaR based on parametric and non-parametric EVT models are calculated.The empirical findings show that, first, the distribution of change of rate are heavy-tailed indicating that the actual risk would be underestimated based on normality assumption. Second, the unconditional distribution is consistent with the heavier-tailed distributions such as ARCH process or Student¡¦t. Third, the right tail of distribution of change of rate are significantly heavier than the left one pointing out that the probabilities and magnitudes of rise in rate could be higher than those of drop in rate. Fourth, the amount of tail-fatness in tail of distribution of change of rate increase after 1999 and the vital factors to cause structural break in tail index are the interest rate policies taken by central bank of Taiwan instead of the deregulation policies in money market. Fifth, based on the two break points found in tail index of right and left tail, long sample of CP rates should not be treated as samples from a single distribution. Sixth, the dependent and heteroscedastic properties of data series should be considered in applying EVT to improve accuracy of VaR forecasts. Finally, EVT models predict VaR accurately before 2001 and the benchmark model, HS and GARCH, generally are superior to EVT models after 2001. Among EVT models, MRE and CHE are relative consistent and reliable in VaR prediction.

Value-at-Risk (VaR) model fails to predict financial risk accurately especially during financial crises. This is mainly due to the model’s inability to calibrate new market information and the fact that the risk measure is characterised by poor tail risk quantification. An alternative approach which comprises of the Expected Shortfall measure and the Lognormal Jump-Diffusion (LJD) model has been developed to address the aforementioned shortcomings of VaR. This model is called the Simulated-Expected-Shortfall (SES) model. The Maximum Likelihood Estimation (MLE) approach is used in determining the parameters of the LJD model since it’s more reliable and authenticable when compared to other nonconventional parameters estimation approaches mentioned in other literature studies. These parameters are then plugged into the LJD model, which is simulated multiple times in generating the new loss dataset used in the developed model. This SES model is statistically conservative when compared to peers which means it’s more reliable in predicting financial risk especially during a financial crisis. / Statistics / M.Sc. (Statistics)