This thesis primarily looks at estimation error problems and other related issues arising in connection with portfolio optimization. With some available assets, a portfolio program or optimizer seeks to distribute a fixed amount of capital among these available assets to optimize some cost function. In this regard, Markowitz portfolio selection basis defines the variance of the portfolio return to being that of the portfolio risk and tries to find an allocation that reduces or minimizes the risk subject to a target mean or expected return. Should the mean return vector and the covariance matrix of returns for the underlying assets be known, the Markowitz problem is said to have a closed-form solution. In practice, however, an estimation is made from historical data for unknown expected returns and the covariance matrix of the returns, and this brings into the domain several problems such as estimation problems and renders the Markowitz theory impracticable in real-life portfolio applications. Estimators necessary to remedy these problems would be made bare to show how possible it is to tackle such issues. In the concept demonstration sections, the analysis starts with the price data of 40 stocks and the S\&P index. The efficient frontier is introduced and used to show how the estimators take effect. Finally, implementation is made possible using the R Programming Language to demonstrate the necessary concepts with the conclusion presented at the end.

The main objective of this thesis is to implement stochastic correlation into the existing structural credit risk models. There are two stochastic models suggested for the covariance matrix of the assets' prices. In our first model, to induce the stochasticity into the structure of the correlation, we assume that the eigenvectors of the covariance matrix are constant but the eigenvalues are driven by independent Cox-Ingersoll-Ross processes. To price equity options on this framework we first transform the calculations from the pricing domain to the frequency domain. Then we derive a closed formula for the Fourier transform of the Green's function of the pricing PDE. Finally we use the method of images to find the price of the equity options. The same method is used to find closed formulas for marginal probabilities of defaults and CDS prices. In our second model, the covariance of the assets follows a Wishart process, which is an extension of the CIR model to dimensions greater than one. The popularity of the Heston model, which uses the CIR process to model the stochastic volatility, could be a promising point for using Wishart process to model stochastic correlation. We give closed form solutions for equity options, marginal probabilities of defaults, and some other major financial derivatives. For the calculation of our pricing formulas we make a bridge between two recent trends in pricing theory; from one side, pricing of barrier options by Lipton (2001) and Sepp (2006) and from other side the development of Wishart processes by Bru (1991), Gourieroux (2005) and Fonseca et al. (2006, 2007a, 2007b). After obtaining the mathematical results above, we then estimate the parameters of the two models we have developed by an evolutionary algorithm. We prove a theorem which guarantees the convergence of the evolutionary algorithm to the set of optimizing parameters. After estimating the parameters of the two stochastic correlation models, we conduct a comparative analysis of our stochastic correlation models. We give an approximation formula for the joint and marginal probabilities of default for General Motors and Ford. For the marginal probabilities of default, a closed formula is given and for the joint probabilities of default an approximation formula is suggested. To show the convergence properties of this approximation method, we perform the Monte Carlo simulation in two forms: a full and a partial Monte Carlo simulation. At the end, we compare the marginal and joint probabilities with full and partial Monte Carlo simulations.

The main objective of this thesis is to implement stochastic correlation into the existing structural credit risk models. There are two stochastic models suggested for the covariance matrix of the assets' prices. In our first model, to induce the stochasticity into the structure of the correlation, we assume that the eigenvectors of the covariance matrix are constant but the eigenvalues are driven by independent Cox-Ingersoll-Ross processes. To price equity options on this framework we first transform the calculations from the pricing domain to the frequency domain. Then we derive a closed formula for the Fourier transform of the Green's function of the pricing PDE. Finally we use the method of images to find the price of the equity options. The same method is used to find closed formulas for marginal probabilities of defaults and CDS prices. In our second model, the covariance of the assets follows a Wishart process, which is an extension of the CIR model to dimensions greater than one. The popularity of the Heston model, which uses the CIR process to model the stochastic volatility, could be a promising point for using Wishart process to model stochastic correlation. We give closed form solutions for equity options, marginal probabilities of defaults, and some other major financial derivatives. For the calculation of our pricing formulas we make a bridge between two recent trends in pricing theory; from one side, pricing of barrier options by Lipton (2001) and Sepp (2006) and from other side the development of Wishart processes by Bru (1991), Gourieroux (2005) and Fonseca et al. (2006, 2007a, 2007b). After obtaining the mathematical results above, we then estimate the parameters of the two models we have developed by an evolutionary algorithm. We prove a theorem which guarantees the convergence of the evolutionary algorithm to the set of optimizing parameters. After estimating the parameters of the two stochastic correlation models, we conduct a comparative analysis of our stochastic correlation models. We give an approximation formula for the joint and marginal probabilities of default for General Motors and Ford. For the marginal probabilities of default, a closed formula is given and for the joint probabilities of default an approximation formula is suggested. To show the convergence properties of this approximation method, we perform the Monte Carlo simulation in two forms: a full and a partial Monte Carlo simulation. At the end, we compare the marginal and joint probabilities with full and partial Monte Carlo simulations.

M.Phil. / The provision of adequate infrastructure and the economic growth of a country's highly interrelated population growth and rapid urbanization has placed enormous pressure on existing infrastructure. The provision of new and maintenance of existing infrastructure presents a challenge to the government. In South Africa infrastructure expenditure were generally funded directly from the country's fiscal budgets. Macroeconomic instability and growing investment requirements have shown that public financing is too volatile and rarely meets crucial infrastructure expenditure requirements in a timely and adequate manner. On the other hand, private sector organisations have a larger pool of sources from which to seek funding, equity investors, capital markets, banks etc., this can be from local to international markets. Therefore, private sector involvement in infrastructure provision has been widely used as a preferred method of financing these projects. The South African government can no longer carry the financial burden in it's fiscal policy to finance all the infrastructure projects needed in this country, and it is also up to the private sector to seek funding for projects that will be economical strong enough to service its own debt. Research objectivesobjectives of this study are to give a background of project financing by addressing the risks involved, finance structures, funding alternatives and strategies.

The statistical distribution of financial returns plays a key role in evaluating Value-at-Risk using parametric methods. Traditionally, when evaluating parametric Value-at-Risk, the statistical distribution of the financial returns is assumed to be normally distributed. However, though simple to implement, the Normal distribution underestimates the kurtosis and skewness of the observed financial returns. This dissertation focuses on the evaluation of the South African equity markets in a Value-at-Risk framework. Value-at- Risk is estimated on five equity stocks listed on the Johannesburg Stock Exchange, including the FTSE/JSE TOP40 index and the S&P 500 index. The statistical distribution of the financial returns is modelled using the Normal Inverse Gaussian and is compared to the financial returns modelled using the Normal, Skew t-distribution and Student t-distribution. We then estimate Value-at-Risk under the assumption that financial returns follow the Normal Inverse Gaussian, Normal, Skew t-distribution, Student t-distribution and Extreme Value Theory and backtesting was performed under each distribution assumption. The results of these distributions are compared and discussed. / Dissertation (MSc)--University of Pretoria, 2015. / Mathematics and Applied Mathematics / MSc / Unrestricted

Large data sets are required to store all of the information contained in option chains. The data set we work with includes all U.S. exchange traded put and call options. This data set is part of a larger data set commonly referred to as the National Best Bid Offer (NBBO) data set. The national bid best offer is a Securities and Exchange Commission (SEC) term for the best available ask price and bid price. Brokers must guarantee investors these prices on their trades. We have acquired data for the 5 year period from 2005 to 2009 for all U.S. traded options. Each year of data is approximately 6 gigabytes. The company, (DeltaNeutral - Options Data And End Of Day Downloads, 2010), from which we acquired the data, also has a software package, OptimalTrader, to process the data. For this data to be used in research projects, the data must be accessible by specific underlying security for selected date ranges. This type of data is more useful to the financial mathematics student than the output given by the software provided by DeltaNeutral. The software used in this data manipulation is Matlab. Each individual file of original data was parsed, and new files were written with some reformatting in which the original data was largely reorganized. The new organization will make searching for information from one stock or any specific group of stocks easier to achieve. We have created 3 m-files in Matlab which deal with reformatting the data, error handling, and searching through the original or reformatted data. The result is that new datasets can be created for further studying and manipulation. Future students working with this data should find this method, toolset, and the newly constructed datasets to be useful tools in working with options data and examining option chains.

Portfolio design is one of the central topics in finance. The original attempt dates back to the mean-variance model developed for a single period portfolio selection. To have a more realistic approach, multi-period selections were developed in order to manage uncertainties associated with the financial markets. This thesis presents a multi-period financial model proposed on the basis of the network flow structure with many planning advantages. This approach comprises two main steps, dynamic portfolio selection, and dynamic portfolio monitoring and rebalancing throughout the investment horizon. To build a realistic yet practical model that can capture the real characteristics of a portfolio a set of proper constraints is designed including restrictions on the size of the portfolio as well as the number of transactions, and consequently the management costs. The model is solved for two-stage financial planning problems to demonstrate the main advantages as well as characteristics of the presented approach.

Portfolio design is one of the central topics in finance. The original attempt dates back to the mean-variance model developed for a single period portfolio selection. To have a more realistic approach, multi-period selections were developed in order to manage uncertainties associated with the financial markets. This thesis presents a multi-period financial model proposed on the basis of the network flow structure with many planning advantages. This approach comprises two main steps, dynamic portfolio selection, and dynamic portfolio monitoring and rebalancing throughout the investment horizon. To build a realistic yet practical model that can capture the real characteristics of a portfolio a set of proper constraints is designed including restrictions on the size of the portfolio as well as the number of transactions, and consequently the management costs. The model is solved for two-stage financial planning problems to demonstrate the main advantages as well as characteristics of the presented approach.

Stochastic Portfolio Theory (SPT) as a methodology aims to move away from the e cient market hypothesis which was developed mainly as a way of explaining the relationship between risk and returns. SPT attempts to explain stock market behaviour using only the assumption of a logarithmic model of stocks, which is widely used in derivative pricing and hedging. This provides a potential tool for portfolio management and an alternative to the commonly used mean-variance approach of Markowitz. The aim of this dissertation is to provide an overview of the foundations of Stochastic Portfolio Theory, the consequences for portfolio construction and behaviour and apply these concepts to the South African Equity Market. / Dissertation (MSc)--University of Pretoria, 2015. / Mathematics and Applied Mathematics / MSc / Unrestricted

In this thesis we study dynamic strategies for index tracking and algorithmic trading. Tracking problems have become ever more important in Financial Engineering as investors seek to precisely control their portfolio risks and exposures over different time horizons. This thesis analyzes various tracking problems and elucidates the tracking errors and strategies one can employ to minimize those errors and maximize profit. In Chapters 2 and 3, we study the empirical tracking properties of exchange traded funds (ETFs), leveraged ETFs (LETFs), and futures products related to spot gold and the Chicago Board Option Exchange (CBOE) Volatility Index (VIX), respectively. These two markets provide interesting and differing examples for understanding index tracking. We find that static strategies work well in the nonleveraged case for gold, but fail to track well in the corresponding leveraged case. For VIX, tracking via neither ETFs, nor futures portfolios succeeds, even in the nonleveraged case. This motivates the need for dynamic strategies, some of which we construct in these two chapters and further expand on in Chapter 4. There, we analyze a framework for index tracking and risk exposure control through financial derivatives. We derive a tracking condition that restricts our exposure choices and also define a slippage process that characterizes the deviations from the index over longer horizons. The framework is applied to a number of models, for example, Black-Scholes model and Heston model for equity index tracking, as well as the Square Root (SQR) model and the Concatenated Square Root (CSQR) model for VIX tracking. By specifying how each of these models fall into our framework, we are able to understand the tracking errors in each of these models. Finally, Chapter 5 analyzes a tracking problem of a different kind that arises in algorithmic trading: schedule following for optimal execution. We formulate and solve a stochastic control problem to obtain the optimal trading rates using both market and limit orders. There is a quadratic terminal penalty to ensure complete liquidation as well as a trade speed limiter and trader director to provide better control on the trading rates. The latter two penalties allow the trader to tailor the magnitude and sign (respectively) of the optimal trading rates. We demonstrate the applicability of the model to following a benchmark schedule. In addition, we identify conditions on the model parameters to ensure optimality of the controls and finiteness of the associated value functions. Throughout the chapter, numerical simulations are provided to demonstrate the properties of the optimal trading rates.