Includes bibliographical references. / Liquidation strategies consider the problem of minimising transaction costs occurring in a portfolio liquidation. Transaction costs are the difference between current market value and the realised value after the liquidation. A strategy to follow to perform a liquidation is especially important to institutional investors due the large size of their trades. Large trades can have a significant effect on the price of a security which can impact the realised returns of the liquidation. These models solve for trading trajectories that maximise this. The models investigated do this in a mean-variance framework where the expected return of the strategy is constrained by its variance and the investors risk preference. Parameters used in liquidity functions are estimated for securities on the South African JSE Securities Exchange. The effects of security liquidity, volatility, stock correlation and length of liquidation horizon on the optimal strategy are investigated. There is little or no existing literature that attempts to model these functions in the South African market. Due to the smaller size of the South African market as well as the number of thinly traded shares compared to most markets studied in the literature, many securities are highly illiquid. We investigate relationships between firm size and daily traded value and these liquidity parameters. General rules are presented to help traders improve a liquidation strategy without the need to estimate all parameters needed to calculate an optimal strategy using one of these models.

Includes bibliographical references (leaves 140-149). / Identifying outliers and/or influential observations is a fundamental step in any statistical analysis, since their presence is likely to lead to erroneous results. Numerous measures have been proposed for detecting outliers and assessing the influence of observations on least squares regression results. Since outliers can arise in different ways, the above mentioned measures are based on motivational arguments and they are designed to measure the influence of observations on different aspects of various regression results. In what follows, we investigate how one can combine different test statistics based on residuals and diagnostic plots to identify outliers and influential observations (both in the single and multiple case) in general linear regression models.

Column-space conditions are shown to be at the heart of a number of identities linking generalized inverses of rectangular matrices. These identities give some new insights into reparametrizations of the general linear model, and into the imposition of constraints, when the variance-covariance structure is σ².I. Hypothesis-test statistics for non-estimable functions are shown to give no further information than underlying estimable functions. For an arbitrary variance-covariance structure the "sweep-out" method is generalized. The John and Draper model for outliers is extended, and distributional results established. Some diagnostic statistics for outlying or influential observations are considered. A Bayesian formulation of outliers in the general linear model is attempted.

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Bibliography: leaves 225-233. / After critically reviewing developments in line transect estimation theory to date, general likelihood functions are derived for the case in which detection probabilities are modelled as functions of any number of explanatory variables and detection of animals on the trackline (i.e. directly in the observer's path) is not certain. Existing models are shown to correspond to special cases of the general models. Maximum likelihood estimators are derived for some special cases of the general model and some existing line transect estimators are shown to correspond to maximum likelihood estimators for other special cases. The likelihoods are shown to be extensions of existing mark-recapture likelihoods as well as being generalizations of existing line transect likelihoods. Two new abundance estimators are developed. The first is a Horvitz-Thompson-like estimator which utilizes the fact that for point estimation of abundance the density of perpendicular distances in the population can be treated as known in appropriately designed line transect surveys. The second is based on modelling the probability density function of detection probabilities in the population. Existing line transect estimators are shown to correspond to special cases of the new Horvitz-Thompson-like estimator, so that this estimator, together with the general likelihoods, provides a unifying framework for estimating abundance from line transect surveys.

Includes bibliography.

In this thesis an algorithm for the noncentral linear density and cumulative distribution function of Wilks' likelihood ratio criterion in MANOVA is derived and it is shown how this algorithm, with modifications, can be used to find the distributions of a number of test criteria for different hypotheses. At the same time previous results regarding percentiles and powers of these criteria are examined and discussed.

Using a geometric approach to best linear unbiased estimation in the general linear model, the additional sum of squares principle, used to generate decompositions, can be generalized allowing for an efficient treatment of augmented linear models. The notion of the admissibility of a new variable is useful in augmenting models. Best linear unbiased estimation and tests of hypotheses can be performed through transformations and reparametrizations of the general linear model. The theory of outliers and influential observations can be generalized so as to be applicable for the general univariate linear model, where three types of outlier and influence may be distinguished. The adjusted models, adjusted parameter estimates, and test statistics corresponding to each type of outlier are obtained, and data adjustments can be effected. Relationships to missing data problems are exhibited. A unified approach to outliers in the general linear model is developed. The concept of recursive residuals admits generalization. The typification of outliers and influential observations in the general linear model can be extended to normal multivariate models. When the outliers in a multivariate regression model follow a nested pattern, maximum likelihood estimation of the parameters in the model adjusted for the different types of outlier can be performed in closed form, and the corresponding likelihood ratio test statistic is obtained in closed form. For an arbitrary outlier pattern, and for the problem of outliers in the generalized multivariate regression model, three versions of the EM-algorithm corresponding to three types of outlier are used to obtain maximum likelihood estimates iteratively. A fundamental principle is the comparison of observations with a choice of distribution appropriate to the presumed type of outlier present. Applications are not necessarily restricted to multivariate normality.

This dissertation explores a key challenge of the financial industry — the efficient computation of sensitivities of financial instruments. The adjoint approach to solving affine recursion problems (ARPs) is presented as a solution to this challenge. A Monte Carlo setting is adopted and it is illustrated how computational efficiency in sensitivity calculation may be significantly improved via the pathwise derivatives method through adapting an adjoint approach. This is achieved through the reversal of the order of differentiation in the pathwise derivatives algorithm in comparison to the standard, intuitive ‘forward’ approach. The Libor market model (LMM) framework is selected for examples to demonstrate these computational savings, with varying degrees of complexity of the LMM explored, from a one-factor model with constant volatility to a full factor model with time homogeneous volatilities.

This dissertation focuses on the application of neural networks to financial model calibration. It provides an introduction to the mathematics of basic neural networks and training algorithms. Two simplified experiments based on the Black-Scholes and constant elasticity of variance models are used to demonstrate the potential usefulness of neural networks in calibration. In addition, the main experiment features the calibration of the Heston model using model-generated data. In the experiment, we show that the calibrated model parameters reprice a set of options to a mean relative implied volatility error of less than one per cent. The limitations and shortcomings of neural networks in model calibration are also investigated and discussed.