In computational finance, finite differences are a widely used tool in the valuation of standard derivative contracts. In a lower-dimensional setting, high accuracy and speed often characterize such methods, which gives them a competitive advantage against Monte Carlo methods. For option contracts with discontinuous payoff functions, however, finite differences encounter problems to maintain the order of convergence of the employed finite difference scheme. Therefore the timesteps are often computed in a conservative manner, which might increase the total execution time of the solver more than necessary.     It can be shown that for American call options written on dividend paying stocks, it may be optimal to exercise the option right before a dividend is paid out. The result is that yet another discontinuity is introduced in the solution and the timestep is often reduced to preserve the intrinsic convergence order. However, it is thought that at least in theory the optimal length of the timestep is an increasing function of the time elapsed since the last discontinuity occured. The objective thus becomes that of finding an explicit method for adjusting the timestep both at the dividend instants and between dividend instants. Keeping the discretization in space constant leads to a time non-equidistant finite difference problem.     The aim of this thesis is to propose a time non-equidistant numerical finite difference algorithm for valuation of American call options on stocks with dividends known in advance. In particular, an explicit formula is proposed for computing timesteps at the dividend instants and between dividend payments given a user-specified error tolerance. A portion of the report is also devoted to numerical stabilization techniques that are applied to maintain the convergence order, including Rannacher time-marching and mollification.

The objective of this study is to implement a Java applet for calculating Bermudan/American-Asian call option prices and to obtain their respective optimal exercise strategies. Additionally, the study presents a computational time analysis and the effect of the variables on the option price.

Over 22 billion derivative contracts were traded on different stock exchanges globally during the year 2010 of which almost 50% were futures while the remaining 50% were options. An overall 25% increase in such contracts was registered as compared to those traded in the year 2009 (International Options Market Association (IOMA) Report, 2011).Investors often use a wide array of trading tools, market indicators and market trading strategies to get the best possible returns for the money that was invested. The main objective of this paper is to focus on the use of market sentiment indicators, specifically the Put-Call Ratio (PCR) as a predictor of returns for an investor.The Put-Call Ratio is defined as a ratio of the trading volume of put options to call options. It is called a sentiment indicator because it measures the “feelings” of option traders. Additionally, it has longed been viewed as an indicator of investors’ sentiment in the market (Put-Call Ratio, 2012) and is possibly the most favoured description of market psychology (James, 2011). / Dissertation (MBA)--University of Pretoria, 2012. / Gordon Institute of Business Science (GIBS) / unrestricted

[pt] O preço da energia no mercado de curto-prazo é caracterizado pela sua alta volatilidade e dificuldade de previsão, representando um alto risco para agentes produtores de energia, especialmente para geradores por fontes renováveis. A abordagem típica empregada por tais empresas para obter a estratégia de contratação ótima de médio e longo prazos é simular um conjunto de caminhos para os fatores de incerteza a fim de caracterizar a distribuição de probabilidade da receita futura e, então, otimizar o portfólio da empresa, maximizando o seu equivalente certo. Contudo, na prática, a modelagem e simulação do preço de curto prazo da energia é um grande desafio para os agentes do setor elétrico devido a sua alta dependência a parâmetros que são difíceis de prever no médio e longo, como o crescimento do PIB, variação da demanda, entrada de novos agentes no mercado, alterações regulatórias, entre outras. Neste sentido, nesta dissertação, utilizamos otimização robusta para tratar a incerteza presente na distribuição do preço de curto-prazo da energia, enquanto a produção de energia renovável é tratada com cenários simulados exógenos, como é comum em programação estocástica. Mostramos, também, que esta abordagem pode ser interpretada a partir de dois pontos de vista: teste de estresse e aversão à ambiguidade. Com relação ao último, apresentamos um link entre otimização robusta e teoria de ambiguidade. Além disso, incluímos no modelo de formação de portfólio ótimo a possibilidade de considerar um contrato de opção térmica de compra para o hedge do portfólio do agente contra a irregularidade do preço de curto-prazo. Por fim, é apresentado um estudo de caso com dados realistas do sistema elétrico brasileiro para ilustrar a aplicabilidade da metodologia proposta. / [en] Energy spot price is characterized by its high volatility and difficult prediction, representing a major risk for energy companies, especially those that rely on renewable generation. The typical approach employed by such companies to address their mid- and long-term optimal contracting strategy is to simulate a large set of paths for the uncertainty factors to characterize the probability distribution of the future income and, then, optimize the company s portfolio to maximize its certainty equivalent. In practice, however, spot price modeling and simulation is a big challenge for agents due to its high dependence on parameters that are difficult to predict, e.g., GDP growth, demand variation, entrance of new market players, regulatory changes, just to name a few. In this sense, in this dissertation, we make use of robust optimization to treat the uncertainty on spot price distribution while renewable production remains accounted for by exogenously simulated scenarios, as is customary in stochastic programming. We show that this approach can be interpreted from two different point of views: stress test and aversion to ambiguity. Regarding the latter, we provide a link between robust optimization and ambiguity theory, which was an open gap in decision theory. Moreover, we include into the optimal portfolio model, the possibility to consider an energy call option contract to hedge the agent s portfolio against price spikes. A case study with realistic data from the Brazilian system is shown to illustrate the applicability of the proposed methodology.