Includes bibliographical references (leaves 52-55). / As the title, "An Alternative Model for Multivariate Stable Distributions", depicts, this thesis draws from the methodology of [J36] and derives an alternative to the sub-Gaussian alpha-stable distribution as another model for multivariate stable data without using the spectral measure as a dependence structure. From our investigation, firstly, we echo that the assumption of "Gaussianity" must be rejected, as a model for, particularly, high frequency financial data based on evidence from the Johannesburg Stock Exchange (JSE). Secondly, the introduced technique adequately models bivariate return data far better than the Gaussian model. We argue that unlike the sub-Gaussian stable and the model involving a spectral measure this technique is not subject to estimation of a joint index of stability, as such it may remain a superior alternative in empirical stable distribution theory. Thirdly, we confirm that the Gaussian Value-at-Risk and Conditional Value-at-Risk measures are more optimistic and misleading while their stable counterparts are more informative and reasonable. Fourthly, our results confirm that stable distributions are more appropriate for portfolio optimization than the Gaussian framework.

Three prominent analytical approximations for pricing basket options,by Levy (1992), Ju (2002) and Deelstra et aI. (2004), are tested for performance and accuracy. Sensitivity analysis shows that all three have greater errors in high volatility and long maturity environments, while Deelstra has weaknesses with small correlation and baskets with few stocks. Deelstra and Levy show tendencies to underprice and overprice respectively, while Ju's errors are more consistently around the true price. A mathematical understanding of the three techniques is also developed.

Includes bibliographical references (leaves 137-138). / This work investigates the modelling of non-stationary sample extremes using a roughness penalty approach, in which smoothed natural cubic splines are fitted to the location and scale parameters of the generalized extreme value distribution and the distribution of the r largest order statistics. Estimation is performed by implementing a Fisher scoring algorithm to maximize the penalized log-likelihood function. The approach provides a flexible framework for exploring smooth trends in sample extremes, with the benefit of balancing the trade-off between 'smoothness' and adherence to the underlying data by simply changing the smoothing parameter. To evaluate the overall performance of the extreme value theory methodology in smoothing extremes a simulation study was performed.

Includes bibliographical references (leaves 130-130). / In this dissertation we review the Sharpe Index Model and an innovation on this model introduced by Hossain, Troskie and Guo (2005b). These models are extended to the multi index framework. We then empirically investigate the impact of the models on portfolio creation over an extensive data set. Next we extend these models by modelling the regression residuals as ARMA and GARCH(l, 1) processes and investigate the effect on the resulting portfolios. We then introduce the topic of bounded influence regression and apply it to financial data by down weighting extreme returns prior to regression. A new weighting function is introduced in this dissertation and the effects on the efficient frontiers and resulting market portfolios for the chosen set of shares are investigated.

Asian options, also known as average value options, are exotic options whose payoffs are dependent on the average prices of the underlying assets over the life of the options. The Asian options are very popular among the market participants when dealing with thinly traded commodities because the average property of the Asian options makes it very difficult to manipulate the payoffs of the options. Another reason for the popularity of Asian options is that they are cheaper than the corresponding portfolio of standard options to hedge the same exposure. The pricing of Asian options has been the subject of continuous studies. In previous studies, Asian options have been priced based on the assumption that the underlying asset follows a geometric Brownian motion. This dissertation, however, assumes that the underlying asset follows a geometric Ornstein-Uhlenbeck process and provides an explicit formula for the geometric Asian options. The geometric Ornstein-Uhlenbeck process is more economically appropriate than the geometric Brownian motion for modelling commodity prices, exchange rates and interest rates due to its mean-reverting property.

Includes bibliographical references (leaves [51] - 55). / The purpose of this study is to price options under jump diffusions using Fourier Transforms and obtain the implied volatility surface from these option prices.

Includes bibliographical references. / Rather than following a normal distribution, share returns and market proxies have been shown to follow skewed distributions, with long tails in some cases. In this dissertation various robust estimation techniques are investigated in an attempt to minimise the influence that outliers may have on the estimation and to better estimate the input parameters for the Markowitz and Sharpe portfolio models. The main goal is to ascertain whether or not the input parameters determined, using the robust procedures, yield better results than the Ordinary Least Squares (OLS) procedure.

Includes bibliographical references (leaves 86-87). / This study investigates the seasonality in agricultural commodity futures prices. Futures prices are modelled using the model developed by Sørensen (2002). The model defines the commodity spot price as the sum of a nonstationary state variable, a stationary state variable and a deterministic seasonal component. Standard no-arbitrage arguments are applied in order to derive futures and option prices. Model parameters are estimated using Kalman filter methodology and maximum likelihood estimation. Model parameters are estimated for white maize, yellow maize and wheat futures traded on the South African Futures Exchange (SAFEX). Furthermore, this research considers other models for commodity derivatives as well as pricing futures contracts in the presence of price limits.

Includes bibliographical references. / Quantitative investment practitioners typically model the performance of a stock relative to its benchmark and the stock's fundamental factors in a classical linear framework. However, these models have empirically been found to be unsuitable for capturing higher-order relationships between a stock's return relative to a benchmark and its fundamental factors. This dissertation studies the use of Classification and Regression Tree (CART) models for stock selection within the South African context, with the focus being on the period from when the Global Financial Crisis began in early 2007 until December 2012. By utilising four types of portfolios, a CART model is directly compared against two traditional linear models. It is seen that during the period focused upon, the portfolios based on the CART model deliver the best excess return and risk-adjusted return, albeit in most cases modestly above the returns delivered by the portfolios based upon the linear models. This is observed in the hedge-fund style and long-only portfolios constructed. Moreover, it is observed that the CART-based portfolios' returns are not correlated with those from the linear-model-based portfolios. This observation suggests that CART models offer an attractive option to diversify model risk within the South African context.

Includes bibliographical references (leaves 70-71). / The celebrated Black-Scholes option pricing model is unable to produce closed-form solutions for arithmetic basket options. This problem stems from the lack of an analitical form for the distribution of a sum of lognormal random variables. lVlarket participants commonly price basket options by assuming the basket follows lognormal dynamics, although it is known that this approximation performs poorly in some cicumstances. The problem of finding an analytical approximation to the sum of lognormally distributed random variables has been widely studied. In this dissertation we seek to draw these studies together and apply them in an option pricing setting. We propose some new option pricing formulae based on these approximations. In order to examine the utility of these new formulae and compare them to commonly used market approximations we present rigorous analytical bounds for the price of arithmetic basket options using the theory of comonotonicity. In this we follow the ideas in Deelstra et al. [7]. Additionally we provide an interval of hedge parameters (the Greeks). We carry out a numerical sensitivity analysis and identify circumstances under which the market approximation misprices basket options.