This thesis contributes to the econometric literature in two ways. Firstly, it introduces a new multivariate count model that presents advances in several aspects. Our multivariate time series count model can deal with issues of discreteness, overdispersion (variance greater than the mean) and both cross- and serial correlation, all at the same time. We follow a fully parametric approach and specify a marginal distribution for the counts where, conditionally on past observations the means follow a vector autoregressive process (VAR). This enables to attain improved inference on coefficients of exogenous regressors relative to the static Poisson regression, while modelling the serial correlation in a flexible way. The method is also innovative in the use of copulas, which builds the dependence structure between variables with given marginal distributions. This makes it possible to model the contemporaneous correlation between individual series in a very flexible way. Secondly, this thesis introduces a new approach to estimate the multivariate reduced rank regressions when the normality assumption is not satisfied. We propose to use the copula tool to generate multivariate distributions and, we show that this method can be applied in multivariate settings. In terms of financial literature, this thesis provides two contributions. Firstly, with our multivariate count model we analyze diverse market microstructure issues about the submission of different types of orders by traders on stock markets. With this model, we can fully take into account the interactions between submissions of the various types of orders, which represent an advantage with respect to univariate models such as the autoregressive conditional duration model. Secondly, it contributes to portfolio research proposing a new dynamic optimal portfolio allocation model in a Value-at-Risk setup. This model allows for time varying skewness and kurtosis of portfolio distributions and the model parameters are estimated by weighted maximum likelihood in an increasing window setup. This last property allows us to have more accurate portfolio recommendations in terms of the amount to invest in the risk-free interest rate and in the risky portfolio.

This paper considers dynamic optimal portfolio strategies of utility maximizing investors in the presence of risk constraints. In particular, we investigate the optimization problem with an additional constraint modeling bounded shortfall risk measured by Value at Risk or Expected Loss. Using the Black-Scholes model of a complete financial market and applying martingale methods we give analytic expressions for the optimal terminal wealth and the optimal portfolio strategies and present some numerical results.

The paper investigates dynamic optimal portfolio strategies of utility maximizing portfolio managers in the presence of risk constraints. Especially we consider the risk, that the terminal wealth of the portfolio falls short of a certain benchmark level which is proportional to the stock price. This risk is measured by the Expected Utility Loss. We generalize the findings our previous papers to this case. Using the Black-Scholes model of a complete financial market and applying martingale methods, analytic expressions for the optimal terminal wealth and the optimal portfolio strategies are given. Numerical examples illustrate the analytic results.

Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / In the Benchmark Approach to Finance, it has been shown that by taking the Growth Optimal Portfolio as numéraire, a candidate for a pricing derivatives formula under the real world probability can be given. This result allows us to price in an incomplete financial market model. The result comes from two different approaches. In the first approach we use the supermartingale property of portfolios in units of the benchmark portfolio which leads to the fact that an equivalent measure is not needed. In the second approach the numéraire property of the Growth Optimal Portfolio is used. The numéraire portfolio defines an equivalent martingale measure and by change of measure using the Radon-Nikodým derivative, a real world pricing formula is derived which is the same as the one given by the first approach stated above.

A presente dissertação expõe o ambiente em que o Problema de Merton é construído e, baseando-se na bibliografia apresentada, constrói exemplos em softwares cujas especificidades podem colaborar na clareza da resolução. O software Matlab engloba as soluções numéricas, enquanto o software Maple é responsável pela solução de equações diferenciais ordinárias e parciais de forma simbólica. Apresenta-se modificações do Problema de Merton original como exercícios para melhor esclarecer os diferentes parâmetros abordados. Na seção final é apresentada a solução de viscosidade, uma alternativa quando a função valor não apresenta características desejáveis para a análise apresentada. / This dissertation explicit the environment which Merton’s problem is built, according to the presented bibliography, exemples are built in softwares whose specificity might help to clarify the solution. The Matlab software embraces numeric solutions, while Maple software is appropriate to solve ordinary and parcial differential equations in symbolic form. Some modifications are presented to Merton’s Problem as exercise to improve understanding on the variations adopted. On final section, viscosity solutions are presented as an alternative solution for when the value function does not possess the desirables properties that allow the analysis on focus.

A presente dissertação expõe o ambiente em que o Problema de Merton é construído e, baseando-se na bibliografia apresentada, constrói exemplos em softwares cujas especificidades podem colaborar na clareza da resolução. O software Matlab engloba as soluções numéricas, enquanto o software Maple é responsável pela solução de equações diferenciais ordinárias e parciais de forma simbólica. Apresenta-se modificações do Problema de Merton original como exercícios para melhor esclarecer os diferentes parâmetros abordados. Na seção final é apresentada a solução de viscosidade, uma alternativa quando a função valor não apresenta características desejáveis para a análise apresentada. / This dissertation explicit the environment which Merton’s problem is built, according to the presented bibliography, exemples are built in softwares whose specificity might help to clarify the solution. The Matlab software embraces numeric solutions, while Maple software is appropriate to solve ordinary and parcial differential equations in symbolic form. Some modifications are presented to Merton’s Problem as exercise to improve understanding on the variations adopted. On final section, viscosity solutions are presented as an alternative solution for when the value function does not possess the desirables properties that allow the analysis on focus.

A presente dissertação expõe o ambiente em que o Problema de Merton é construído e, baseando-se na bibliografia apresentada, constrói exemplos em softwares cujas especificidades podem colaborar na clareza da resolução. O software Matlab engloba as soluções numéricas, enquanto o software Maple é responsável pela solução de equações diferenciais ordinárias e parciais de forma simbólica. Apresenta-se modificações do Problema de Merton original como exercícios para melhor esclarecer os diferentes parâmetros abordados. Na seção final é apresentada a solução de viscosidade, uma alternativa quando a função valor não apresenta características desejáveis para a análise apresentada. / This dissertation explicit the environment which Merton’s problem is built, according to the presented bibliography, exemples are built in softwares whose specificity might help to clarify the solution. The Matlab software embraces numeric solutions, while Maple software is appropriate to solve ordinary and parcial differential equations in symbolic form. Some modifications are presented to Merton’s Problem as exercise to improve understanding on the variations adopted. On final section, viscosity solutions are presented as an alternative solution for when the value function does not possess the desirables properties that allow the analysis on focus.

In this thesis a mathematical description and analysis of the Cumulative Prospect Theory is presented. Conditions that ensure well-posedness of the problem are provided, as well as existence results concerning optimal policies for discrete-time incomplete market models and for a family of diffusion market models. A brief outline of how this work is organised follows. In Chapter 2 important results on weak convergence and discrete time finance models are described, these facts form the main background to introduce in Chapter 3 the problem of optimal investment under the CPT theorem in a discrete time setting. We describe our model, present some assumptions and main results are derived. The second part of this work comprises the description of the martingale problem formulation of diffusion processes in Chapter 4. A key result on the limits and topological properties of the set of laws of a class of Itô processes is described in Chapter 5. Finally, we introduce a factor model that includes a class of stochastic volatility models, possibly with path-depending coefficients. Under this model, the problem of optimal investment with a behavioural investor is analysed and our main results on well-posedness and existence of optimal strategies are described under the framework of weak solutions. Further research and challenges when applying the techniques developed in this work are described.

This master’s thesis deals with problematics of portfolio theory, which helps to create optimal portfolios for the selected investment company. Portfolios consist of shares, which are traded on New York Stock Exchange and which include a historical value at least for two years. There are two ways of creating portfolios. The first way is the portfolio with minimal risk and no required return and the second way is the portfolio with minimal risk and required return. In this thesis are used mathematical methods, which include a linear algebra, an optimization and a statistics.

The main goal of the thesis "Technical analysis" is to analyze the issue of technical analysis itself, more specifically the CAMP model, which deals with optimizing the portfolio, then design a program to calculate it (generate the required results). This analysis corresponds to the structure of the thesis, while in the beginning of the work are mentioned an important theoretical basis. There is described the financial system, financial and capital markets in connection to the Burza cenných papírů Praha, a.s. and other major capital markets. In addition, I discuss the methodology of technical analysis, CAMP model and the source code and the description of Visual Basic. Then I use this methodology in practice and focus on the goal of the work. Finally I analyze the options, benefits as well as weaknesses of the CAMP model and my program in the Czech capital market conditions.