The thesis describes commonly used measures of risk, such as volatility, Value at Risk (VaR) and Expected Shortfall (ES), and is tasked with creating models for measuring market risk. It is concerned with the risk over daily and over monthly horizons and shows the shortcomings of a square-root-of-time approach for converting VaR and ES between horizons. Parametric models, geometric Brownian motion (GBM) and GARCH process, and non-parametric models, historical simulation (HS) and some its possible improvements, are presented. The application of these mentioned models is demonstrated using real data. The accuracy of VaR models is proved through backtesting and the results are discussed. Part of this thesis is also a simulation study, which reveals the precision of VaR and ES estimates.

[pt] O objetivo deste trabalho é determinar o melhor timing para a abertura de capital de empresas fechadas. Empresas fechadas detêm a opção de abertura de capital por tempo indeterminado deixando a decisão nas mãos do empreendedor ou do corpo administrativo. A aplicação da teoria de opções reais neste contexto tem como objetivo valorar a flexibilidade da administração em um ambiente dinâmico, no qual métodos tradicionais falham. O momento ideal para lançamento de ações no mercado primário torna-se viável a partir de certo estágio de vida da firma e, a partir deste momento, busca-se o momento mais oportuno para a captação. Sabendo que os preços das ações são variáveis estocásticas e seguem um Movimento Geométrico Browniano, a Simulação de Monte Carlo é capaz de gerar os preços de ações futuros para a tomada de decisão e valoração da opção de espera no tempo corrente. Neste cenário, o gestor tenderá a esperar choques positivos nos preços antes de lançar ações e, como consequência, o período ótimo de exercício da opção de abertura de capital deve ocorrer após aumentos anormais nos preços das ações. Seja o motivo principal do lançamento dar liquidez às ações ou financiar investimentos, o benefício da abertura deverá ser maior do que o valor da opção de espera. / [en] The purpose of this study is to identify the best timing for private firms to go public. Private firms possess an option to go public at the time their managers believe it would be a valuable strategy. In the same way, exchange traded companies also have the option of issuing more equities or even taking them over. In this sense, the theory of real options can be applied in order to value management flexibility in an uncertain and dynamic ambience in which traditional methods such as net present value fail to evaluate investments. For the management team, the more uncertain the market is, the more value it adds to the option to wait and postpone the issuance. Given that stock prices fluctuate randomly over time and follow the Geometric Brownian Motion, the Monte Carlo Simulation provides the forecasted stock prices to help the decision making process and also to value the timing option. Therefore, the entrepreneur is willing to wait for positive price shocks before taking the firm public. As a consequence of optimally exercising the timing option IPOs should occur after abnormal price increases. Any firm contemplating an IPO should factor the timing option into the decision. The main reason for the IPO can be either capital raising or the desire to increase the shares’ liquidity but, in all cases, the benefit of going public should be greater than the value of the timing option and issuing shares from private equity.

[pt] As vantagens de incluir a flexibilidade gerencial e a analogia às opções financeiras nos critérios de avaliação de projetos têm sido alvo de discussões teóricas no ramo das finanças. Diversos autores criticam os métodos de análise de investimentos utilizados correntemente, que têm como principal representante o fluxo de caixa descontado (FCD), apoiando-se na noção de que os gerentes ao tomarem decisões devem utilizar técnicas que reflitam as flexibilidades disponíveis. Nesse sentido, a presente dissertação tem por finalidade validar as vantagens sugeridas na utilização da teoria de Opções Reais através de um back-testing, que tem como objeto campos de petróleo com características representativas da indústria petrolífera brasileira. Estes testes serão realizados para o período de 1970 a 1990, sendo contemplada a incerteza econômica e excluindo-se as incertezas técnicas. O modelo desenvolvido em Excel e VBA (Visual Basic for Applications) para decisões de investimento considera as opções de espera de até dois anos e de escolha entre três intensidades de produção. O Movimento Geométrico Browniano foi assumido como o processo estocástico para representar a evolução dos preços reais do petróleo em dólares americanos ao longo do tempo, e sua volatilidade foi variada a título de análise de sensibilidade. Por fim, cabe ressaltar que os resultados obtidos não devem ser aceitos como definitivos, e sim como base de futuros trabalhos na linha de estudos empíricos para verificar e validar as vantagens teóricas das Opções Reais em relação aos demais critérios utilizados na prática. / [en] Financial researchers have discussed a lot about the theoretical advantages of including the managerial flexibility and the financial options analogy in projects valuation criteria. Plenty of authors criticize the currently used investment analysis methods, mainly represented by the discounted cash flow, supported by the notion that the managers should use techniques that better reflect the available flexibility to take their decisions. In this sense, the present dissertation has the objective of validating the suggested advantages of using the Real Options theory through a back-testing focused on oil fields with Brazilian oil industry representative characteristics. Those tests will be carried out for the 1970-1990 period, considering the economic uncertainty and excluding the technical uncertainties. The investment decisions model developed in Excel and VBA (Visual Basic for Applications) contemplates the options of waiting till two years and of choosing among three exploitation intensities. The Geometric Brownian Motion was assumed as the stochastic process to represent the real oil prices time evolution, and its volatility was varied to generate a sensibility analysis. Finally it is worthy to state that the results shall not be accepted as definitive, and just as a foundation to future studies on the empirical research line of verifying and validating the theoretical advantages of the Real Options with regard to others currently used criteria.

<p>This study examines the behaviour of the South African financial markets with regards to the Geometric Brownian motion process. It uses the daily, weekly, and monthly stock returns time series of some major securities trading in the South African financial market, more specifically the US dollar/Euro, JSE ALSI Total Returns Index, South African All Bond Index, Anglo American Corporation, Standard Bank, Sasol, US dollar Gold Price , Brent spot oil price, and South African white maize near future. The assumptions underlying the&nbsp / Geometric Brownian motion in finance, namely the stationarity, the normality and the independence of stock returns, are tested using both graphical (histograms and normal plots)&nbsp / and statistical test (Kolmogorov-Simirnov test, Box-Ljung statistic and Augmented Dickey-Fuller test) methods to check whether or not the Brownian motion as a model for South&nbsp / African financial markets holds. The Hurst exponent or independence index is also applied to support the results from the previous test. Theoretically, the independent or Geometric&nbsp / Brownian motion time series should be characterised by the Hurst exponent of &frac12 / . A value of a Hurst exponent different from that would indicate the presence of long memory or&nbsp / fractional Brownian motion in a time series. The study shows that at least one assumption is violated when the Geometric Brownian motion process is examined assumption by&nbsp / assumption. It also reveals the presence of both long memory and random walk or Geometric Brownian motion in the South African financial markets returns when the Hurst index analysis is used and finds that the Currency market is the most efficient of the South African financial markets. The study concludes that although some assumptions underlying the&nbsp / rocess are violated, the Brownian motion as a model in South African financial markets can not be rejected. It can be accepted in some instances if some parameters such as the Hurst exponent are added.</p>

基於SEC之要求，越來越多美國金融機構於其財務報表附註中揭露金融交易資產之風險值；然而計算風險值涉及到許多假設，於是導致過去部分文獻對於風險值資訊揭露之可靠性產生質疑。本研究以風險值之揭露對於報表使用者之資訊價值作為研究課題；為求與附註揭露之風險值資訊比較，本研究以公司帳列之金融交易資產(Trading Assets)、金融交易收入(Trading Revenue)為基礎，利用蒙地卡羅模擬法模擬帳列金融交易資產於次期可能產生之最大潛在損失，並且透過OLS regression及panel data model探討： (1)風險值及金融交易資產潛在可能損失是否可以預測次期金融交易收入波動 (2)風險值與金融交易資產潛在可能損失資訊之提供是否影響次期股票交易量 (3)風險值與金融交易資產潛在可能損失資訊是否可以有效預測次期股價報酬 率變異。 實證結果顯示，風險值之揭露與金融交易資產潛在可能損失之資訊對於預計次期金融交易收入之波動與股價報酬率變異均呈現顯著正相關；易言之，上述兩種資訊之揭露均提供增額之資訊價值。惟另方面，風險值之揭露與金融交易資產潛在可能損失之資訊卻與次期股票交易量呈正相關，也就是說上述兩種資訊的揭露反而會造成投資人降低長期投資持有的意願。 / Financial institutions in the United States are required by the Securities and Exchange Commission to disclose their Value at Risk (VaR) in the footnotes of the financial statements. Over the years, VaR has been used by institutional investors, industry consultants, and regulators as one of the key measures of market risk. However, there are a number of approaches to calculating VaR and some of them may involve various statistical models and assumptions. Due to the fact that different models and assumptions may be used, the VaR numbers produced by different institutions are difficult to compare with one another. The usefulness of these numbers is therefore decreased. This thesis examines the usefulness of disclosures of VaR information. In order to compare with VaR disclosures, the implied potential maximum losses of trading assets are simulated by using Monte Carlo simulation. We then employ the OLS regression and the panel data models to investigate the following research questions: (1)whether VaR and implied potential maximum losses of trading assets disclosed by financial institutions are instrumental in predicting the variability of trading revenue for the next quarter; (2)whether VaR and implied potential maximum losses of trading assets disclosed by financial institutions affect investors' investing decision; (3)how useful are VaR and implied potential maximum losses of trading assets in predicting the volatility of daily stock return next quarter. The empirical results indicate that VaR and implied potential maximum losses of trading assets are significantly related to the variability of trading revenue and the volatility of daily stock returns next quarter. This evidence suggests that both types of disclosures provide incremental information on predicting the variability of trading revenue and investment risk in the future. Nevertheless, we also find that both VaR disclosures and implied potential maximum losses of trading assets are positively associated with future average stock trading volume, implying that investors tend to trade stock more frequently when the market risk information is disclosed.

<p>This study examines the behaviour of the South African financial markets with regards to the Geometric Brownian motion process. It uses the daily, weekly, and monthly stock returns time series of some major securities trading in the South African financial market, more specifically the US dollar/Euro, JSE ALSI Total Returns Index, South African All Bond Index, Anglo American Corporation, Standard Bank, Sasol, US dollar Gold Price , Brent spot oil price, and South African white maize near future. The assumptions underlying the&nbsp / Geometric Brownian motion in finance, namely the stationarity, the normality and the independence of stock returns, are tested using both graphical (histograms and normal plots)&nbsp / and statistical test (Kolmogorov-Simirnov test, Box-Ljung statistic and Augmented Dickey-Fuller test) methods to check whether or not the Brownian motion as a model for South&nbsp / African financial markets holds. The Hurst exponent or independence index is also applied to support the results from the previous test. Theoretically, the independent or Geometric&nbsp / Brownian motion time series should be characterised by the Hurst exponent of &frac12 / . A value of a Hurst exponent different from that would indicate the presence of long memory or&nbsp / fractional Brownian motion in a time series. The study shows that at least one assumption is violated when the Geometric Brownian motion process is examined assumption by&nbsp / assumption. It also reveals the presence of both long memory and random walk or Geometric Brownian motion in the South African financial markets returns when the Hurst index analysis is used and finds that the Currency market is the most efficient of the South African financial markets. The study concludes that although some assumptions underlying the&nbsp / rocess are violated, the Brownian motion as a model in South African financial markets can not be rejected. It can be accepted in some instances if some parameters such as the Hurst exponent are added.</p>

[pt] No seu planejamento plurianual de investimentos, as organizações do setor de Exploração e Produção (EeP) estruturam alternativas de projetos de produção de petróleo e gás natural, sujeitas a diversas restrições e a incertezas técnicas e econômicas. Como não há como assegurar que os resultados dos projetos ocorram conforme o previsto, é possível que seu retorno seja inferior ao esperado, o que, dependendo da relevância, pode provocar um efeito adverso no resultado operacional e nas condições financeiras da companhia. Nesse mérito, a dissertação apresenta e aplica um modelo de programação estocástica linear inteira mista para seleção de portfólio de projetos que permita a maximização dos resultados, com restrição de risco. A aplicação considerou dados realistas do segmento de upstream de uma empresa do setor. Para representar os cenários econômicos, optou-se pela utilização da simulação de Monte Carlo do modelo Movimento Geométrico Browniano. Com o Valor Presente Líquido como retorno e Conditional Value-at-Risk representando a medida de risco, foi possível estabelecer a fronteira eficiente do risco-retorno, com a qual o decisor pode definir uma solução de portfólio, conforme sua aversão ao risco. / [en] In their multi-annual investment planning, oil and gas companies consider alternatives of production projects, subject to a variety of constraints, and technical and economic uncertainties. Considering that it is not possible to guarantee that these projects will perform as predicted, the return can be less than expected and can lead to a significant adverse effect to the operational results and to financial conditions of a given organization. Therefore, this dissertation proposes a mixed integer linear stochastic programming model for project portfolio selection that maximizes the return with risk constraint. The application considered realistic data from the upstream segment of an oil and gas company. Monte Carlo simulation of the Geometric Brownian Motion model was considered to represent the economic scenarios. Using the Net Present Value as the function and Conditional Value-at-Risk as a risk measure, it was possible to establish the efficient frontier of risk-return, which can assist the decision-maker to define the project portfolio according to their risk aversion.

[pt] O objetivo da presente dissertação é capturar o valor da opção de parada temporária que uma planta GTL oferece em cenários econômicos desfavoráveis para mantê-la operando. Desta forma, o autor considera que a metodologia das opções reais é a mais indicada para avaliar tal flexibilidade, sendo assim, o objetivo principal deste estudo é a análise da opção de parar temporariamente através da utilização do processo estocástico (Movimento geométrico Browniano)e a correlação de Cholesky entre cada input e os outputs da planta. O autor acredita que os resultados desta dissertação podem auxiliar os gestores que enxergam flexibilidades em seus projetos a antecipar prejuízos prolongados em cenários desfavoráveis para manter uma planta funcionando. / [en] The objective of this dissertation is to capture the value of the option to temporarily stop that a GTL plant offers in unfavorable economic scenarios to keep it operating. Thus the author believes that the methodology of real options is the best placed to assess such flexibility, and thus the main objective of this study is the analysis of the option to temporarily stop using the stochastic process (geometric Brownian Motion) and the correlation Cholesky between each input and outputs of the plant. The author believes that the results of this dissertion can assist the managers who see flexibilities in its projects to anticipate damages drawn out in favorable scenes to keep a plant functioning.

Magister Scientiae - MSc / This study examines the behaviour of the South African financial markets with regards to the Geometric Brownian motion process. It uses the daily, weekly, and monthly stock returns time series of some major securities trading in the South African financial market, more specifically the US dollar/Euro, JSE ALSI Total Returns Index, South African All Bond Index, Anglo American Corporation, Standard Bank, Sasol, US dollar Gold Price , Brent spot oil price, and South African white maize near future. The assumptions underlying the Geometric Brownian motion in finance, namely the stationarity, the normality and the independence of stock returns, are tested using both graphical (histograms and normal plots) and statistical test (Kolmogorov-Simirnov test, Box-Ljung statistic and Augmented Dickey-Fuller test) methods to check whether or not the Brownian motion as a model for South African financial markets holds. The Hurst exponent or independence index is also applied to support the results from the previous test. Theoretically, the independent or Geometric Brownian motion time series should be characterised by the Hurst exponent of ½. A value of a Hurst exponent different from that would indicate the presence of long memory or fractional Brownian motion in a time series. The study shows that at least one assumption is violated when the Geometric Brownian motion process is examined assumption by assumption. It also reveals the presence of both long memory and random walk or Geometric Brownian motion in the South African financial markets returns when the Hurst index analysis is used and finds that the Currency market is the most efficient of the South African financial markets. The study concludes that although some assumptions underlying the rocess are violated, the Brownian motion as a model in South African financial markets can not be rejected. It can be accepted in some instances if some parameters such as the Hurst exponent are added. / South Africa

<p>In this thesis we investigate portfolio optimization under Value at Risk, Average Value at Risk and Limited expected loss constraints in a framework, where stocks follow a geometric Brownian motion. We solve the problem of minimizing Value at Risk and Average Value at Risk, and the problem of finding maximal expected wealth with Value at Risk, Average Value at Risk, Limited expected loss and Variance constraints. Furthermore, in a model where the stocks follow an exponential Ornstein-Uhlenbeck process, we examine portfolio selection under Value at Risk and Average Value at Risk constraints. In both geometric Brownian motion (GBM) and exponential Ornstein-Uhlenbeck (O.U) models, the risk-reward criterion is employed and the optimal strategy is found. Secondly, the Value at Risk, Average Value at Risk and Variance is minimized subject to an expected return constraint. By running numerical experiments we illustrate the effect of Value at Risk, Average Value at Risk, Limited expected loss and Variance on the optimal portfolios. Furthermore, in the exponential O.U model we study the effect of mean-reversion on the optimal strategies. Lastly we compare the leverage in a portfolio where the stocks follow a GBM model to that of a portfolio where the stocks follow the exponential O.U model.</p> / Master of Science (MSc)