Barrier options are becoming more popular, mainly due to the reduced cost to hold a barrier option when compared to holding a standard call/put options, but exotic options are difficult to price since the payoff functions depend on the whole path of the underlying process, rather than on its value at a specific time instant. It is a path dependent option, which implies that the payoff depends on the path followed by the price of the underlying asset, meaning that barrier options prices are especially sensitive to volatility. For basic exchange traded options, analytical prices, based on the Black-Scholes formula, can be computed. These prices are influenced by supply and demand. There is not always an analytical solution for an exotic option. Hence it is advantageous to have methods that efficiently provide accurate numerical solutions. This study gives a literature overview and compares implementation of some available numerical methods applied to barrier options. The three numerical methods that will be adapted and compared for the pricing of barrier options are: • Binomial Tree Methods • Monte-Carlo Methods • Finite Difference Methods / Thesis (MSc (Applied Mathematics))--North-West University, Potchefstroom Campus, 2013

Barrier options are becoming more popular, mainly due to the reduced cost to hold a barrier option when compared to holding a standard call/put options, but exotic options are difficult to price since the payoff functions depend on the whole path of the underlying process, rather than on its value at a specific time instant. It is a path dependent option, which implies that the payoff depends on the path followed by the price of the underlying asset, meaning that barrier options prices are especially sensitive to volatility. For basic exchange traded options, analytical prices, based on the Black-Scholes formula, can be computed. These prices are influenced by supply and demand. There is not always an analytical solution for an exotic option. Hence it is advantageous to have methods that efficiently provide accurate numerical solutions. This study gives a literature overview and compares implementation of some available numerical methods applied to barrier options. The three numerical methods that will be adapted and compared for the pricing of barrier options are: • Binomial Tree Methods • Monte-Carlo Methods • Finite Difference Methods / Thesis (MSc (Applied Mathematics))--North-West University, Potchefstroom Campus, 2013

Over the last decades the market for financial derivatives has grown dramatically to values of global importance. With the digital automation of the markets, programs able to efficiently value financial derivatives has become key to market competitiveness and thus garnered considerable interest. This report explores the potential efficiency gains of employing modern technology in GPU computing to price financial options, using the binomial option pricing model. The model is implemented using both CPU and GPU hardware and results compared in terms of computational efficiency. According to this thesis, GPU computing can considerably improve option pricing runtimes. / Under de senaste decennierna har marknaden för finansiella derivatinstrument vuxit till värden av global betydelse. Med ökande digitalisering av marknaden har program som effektivt kan värdera derivatinstrument blivit avgörande för konkurrenskraft och därför givits avsevärt intresse. Denna rapport utforskar vilka möjliga ökningar i effektivitet som kan nås genom att använda modern teknik för GPU-beräkningar för att värdera finansiella optioner genom den binomiala optionsvärderingsmodellen. Modellen implementeras både med CPU-, och GPU-hårdvara och resultaten jämförs i termer av beräkningseffektivitet. Enligt denna studie kan GPU-beräkingar avsevärt förbättra körtider för optionsvärderingar.

[pt] Com o objetivo de reduzir o efeito estufa e atender a alta demanda de energia elétrica, muitos países desenvolvidos e emergentes têm optado pelo uso de energias essencialmente de origem renovável, incluindo energia eólica, biomassa e solar fotovoltaica. Para determinar a viabilidade econômica dos projetos de energias renováveis é necessário capturar as variáveis de maior interesse. As opções reais ao considerar a dinâmica e incertezas, típicas dos mercados de eletricidade, surgem como uma alternativa para a tomada de decisões neste tipo de projetos. Este trabalho aplica a técnica tradicional de avaliação de projetos através do fluxo de caixa descontado (FCD) e a teoria de opções reais (TOR) para avaliação de um parque eólico. A flexibilidade foi incluída ao considerar a opção de fazer um investimento em retrofit para atualizar a tecnologia das turbinas eólicas e estender o contrato de operação. A opção real analisada é análoga a uma opção de compra europeia, seu valor é calculado através de uma árvore binomial recombinante com barreira no preço da energia eólica, principal fonte de incerteza do modelo. Realizaram-se análises de sensibilidade para avaliar o comportamento do preço da opção frente às mudanças em alguns parâmetros do modelo. Conclui-se a viabilidade econômica de estender a operação do parque eólico através de um investimento em retrofit, utilizando a TOR, uma vez que o seu valor é muito superior frente ao calculado pelo FCD, apesar do valor da opção ter uma forte correlação com o limite da barreira e a volatilidade do preço. / [en] In order to reduce the greenhouse effect and meet the high demand of electricity, many developed and emerging countries have opted for the use of renewable energies, for example, wind energy, biomass and solar photovoltaic. To determine the economic viability of renewable energy projects it is necessary to capture the most relevant variables that suggest greater interest. Real options allows to consider the dynamic and uncertainties present in the power markets, therefore it is an attractive tool for decision-making in this kind of markets. In this work we apply the traditional technique of project evaluation, the discounted cash flow (DCF), as well as the real real options theory (ROT) for the financial evaluation of a wind farm. The flexibility is the option of making a retrofit investment that allows upgrading the wind turbines technology in order to extend the operating contract. The real option analyzed is analogous to a european call option and its value was calculated through a recombinant binomial tree with barrier in the wind energy price, considered the main source of uncertainty in the model. Sensitivity analyzes were performed to assess the option price behavior when some parameters of the model change. We finally conclude for the economic viability of extending the operation of the wind farm through an investment in retrofit, using the ROT, since its value is much higher than the calculated by the DCF, even though the option value is strongly dependent on the barrier and the volatility of electricity prices.

[pt] O presente trabalho tem como objetivo final a verificação da aplicabilidade da Teoria das Opções Reais (TOR) em investimentos de papel e celulose, considerando o Movimento de Reversão à Média (MRM) nos fatores de risco, dado um modelo de gerenciamento de curto prazo, no âmbito de um estudo de caso da Fibria Celulose S.A. para o setor de papel e celulose no Brasil. Nesta dissertação, testa-se a aderência da série histórica de preços da celulose de fibra curta da Fibria, no período entre 2003 e 2013, a um modelo estocástico de reversão à média, sendo este modelo validado para o presente estudo. Uma vez o modelo validado, determinam-se os parâmetros para realização de cálculos e análises fundamentais para se chegar aos objetivos intermediários, etapa preliminar aos resultados do objetivo final. Dentre os cálculos e análises citados, ressaltam-se: determinação dos VPLs dinâmicos e os valores das Opções Reais europeias sequenciais para a Simulação de Monte Carlo com Processo Neutro ao Risco; construção e análise da Árvore Binomial com Processo Neutro ao Risco; construção e análise das Regiões de Gatilho para preços e lucros marginais em um Processo Real; comparação das Regiões de Gatilho com as determinadas pelas Árvores Binomiais. Assim, com tais análises, confirma-se, nesta dissertação, a aplicabilidade da Teoria das Opções Reais na Análise de Investimento no setor celulósico-papeleiro. / [en] The present work has the ultimate purpose of verifying the applicability of the Real Options Theory in the pulp and paper investment, considering the Mean Reversion Movement in the risk factors, given a short-term management model, within a study of Fibria Celulose S.A. for the pulp and paper industry in Brazil. This dissertation tests the adherence of Fibria s short fiber pulp historical price series, between 2003 and 2013 to a stochastic mean reversion model, being this model validated in the present study. Once the model is validated, the parameters for calculations and fundamental analyzes are determined to reach intermediate goals, preliminary step to the results of the final goal. Among the cited calculations and analyzes, it is emphasized: determination of dynamic NPVs and values of the sequential European Real Options for the Monte Carlo Simulation with Risk Neutral Process; construction and analysis of the Binomial Tree with Risk Neutral Process; construction and analysis of the Trigger Regions for prices and marginal profits in a Real Case; Comparison between Trigger Regions and those determined by the Binomial Trees. So with such analyzes, it is confirmed, in this work, the applicability of the Real Option Theory on Investment Analysis in pulp and paper industry.