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.

In this thesis, we have built an optimal portfolio using five assets from the Japanese market. We have investigated the use of GARCH(1,1) when forecasting the volatility of our optimal portfolio. Different time periods have been considered for optimizing our results. An equally-weighted portfolio has been used as a benchmark. Our results show that the optimal portfolio we constructed is more efficient than the equally-weighted portfolio in all chosen situations.

In this article, we consider the portfolio selection problem as a Bayesian decision problem. We compare the traditional mean–variance and mean–variance–skewness efficient portfolios. We develop bi-level programming problem to investigate the market’s preference for risk by using observed (market) weights. Numerical experiments are conducted on a portfolio formed by the 30 stocks in the Dow Jones Industrial Average. Numerical results show that the market’s preferences are better explained when skewness is included.

Our aim is to examine the problem of optimal asset allocation for investors exhibiting a behaviour in the face of uncertainty which is not consistent with the usual axioms of Expected Utility Theory. This thesis is divided into two main parts. In the first one, comprising Chapter II, we consider an arbitrage-free discrete-time financial model and an investor whose risk preferences are represented by a possibly nonconcave utility function (defined on the non-negative half-line only). Under straightforward conditions, we establish the existence of an optimal portfolio. As for Chapter III, it consists of the study of the optimal investment problem within a continuous-time and (essentially) complete market framework, where asset prices are modelled by semi-martingales. We deal with an investor who behaves in accordance with Kahneman and Tversky's Cumulative Prospect Theory, and we begin by analysing the well-posedness of the optimisation problem. In the case where the investor's utility function is not bounded above, we derive necessary conditions for well-posedness, which are related only to the behaviour of the distortion functions near the origin and to that of the utility function as wealth becomes arbitrarily large (both positive and negative). Next, we focus on an investor whose utility is bounded above. The problem's wellposedness is trivial, and a necessary condition for the existence of an optimal trading strategy is obtained. This condition requires that the investor's probability distortion function on losses does not tend to zero faster than a given rate, which is determined by the utility function. Provided that certain additional assumptions are satisfied, we show that this condition is indeed the borderline for attainability, in the sense that, for slower convergence of the distortion function, there does exist an optimal portfolio. Finally, we turn to the case of an investor with a piecewise power-like utility function and with power-like distortion functions. Easily verifiable necessary conditions for wellposedness are found to be sufficient as well, and the existence of an optimal strategy is demonstrated.

Problem When choosing topic for this study the economy was on the brink of a recession. Many experts made varying statements regarding this fact, and further readings in this area led us to question: can an in- clusion of gold enhance the performance in an index portfolio dur- ing recessions? And if so, how much should be allocated to gold? Purpose The purpose of this thesis is to look back at the historical price de- velopment of gold and DJIA during recessions in order to find out whether an inclusion of gold can improve a DJIA index portfolio held in today’s recession. In addition, by analyzing the risks and pos- sibilities with gold, the optimal allocation of gold in a DJIA portfolio will be investigated in.   Method The methodological approach will be of a quantitative data analysis approach. By using historical data, new empirical findings will be found by using the deductive approach. This method has been cho- sen due to the nature of the purpose and in order to best give a gen- eral answer to our research questions. Conclusion The gold price is strongly influenced by uncertainty, and even though an optimal allocation of gold in each recession could be found, no general optimal allocation applicable in today’s recession could be found. Gold has higher risk (higher variance) than DJIA, but is compensated with higher return as well.

This thesis investigates a technique to estimate the risk of the mean-variance (MV) portfolio optimization problem. We call this technique the Scaling technique. It provides a better estimator of the risk of the MV optimal portfolio. We obtain this result for a general estimator of the covariance matrix of the returns which includes the correlated sampling case as well as the independent sampling case and the exponentially weighted moving average case. This gave rise to the paper, [CMcS]. Our result concerning the Scaling technique relies on the moments of the inverse of compound Wishart matrices. This is an open problem in the theory of random matrices. We actually tackle a much more general setup, where we consider any random matrix provided that its distribution has an appropriate invariance property (orthogonal or unitary) under an appropriate action (by conjugation, or by a left-right action). Our approach is based on Weingarten calculus. As an interesting byproduct of our study - and as a preliminary to the solution of our problem of computing the moments of the inverse of a compound Wishart random matrix, we obtain explicit moment formulas for the pseudo-inverse of Ginibre random matrices. These results are also given in the paper, [CMS]. Using the moments of the inverse of compound Wishart matrices, we obtain asymptotically unbiased estimators of the risk and the weights of the MV portfolio. Finally, we have some numerical results which are part of our future work.

The paper deals with the problem of finding an optimal one-time rebalancing strategy for the Bachelier model, and makes some remarks for the similar problem within Black-Scholes model. The problem is studied on finite time interval under mean-square criterion of optimality. The methods of the paper are based on the results for optimal stopping problem and standard mean-square criterion. The solution of the problem, considered in the paper, let us interpret how and - that is more important for us -when investor should rebalance the portfolio, if he wants to hedge it in the best way.