Made available in DSpace on 2015-04-16T14:49:06Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1909548 bytes, checksum: f063691d27b5e27e8ff2e452860bc508 (MD5) Previous issue date: 2013-01-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The beef sector is one of the main highlights of the Brazilian agribusiness in the global scenario, the recent stabilization of the economy, the advantages of production costs based on abundant natural resources and few environmental restrictions have ensured the growth and competitiveness of the sector. With decreasing government intervention, the policies private of risk management began to become a concern among those involved in agribusiness and also between agents of the beef sector that started to seek ways of managing risk, among them the futures markets and options, to ensure profitability by reducing exposure to the risk of price fluctuations. Given the importance of the beef sector, the developments presented by derivative instruments and use yet inexpressive these contracts for risk management, sought to apply the pricing model for options on futures contracts, developed by Black, to the reality of the beef cattle future market. The method consisted of applying different types of volatility (historical, implied and deterministic) to pricing model of Black and then held the performance analysis of the models by calculating the errors. The results show that historical volatility for the different windows mobile subpricing values prizes traded in the market, whereas the models calculated with EWMA and THARCH volatility superprecificam the option premiums. Generally the pricing model with historical volatility by moving window showed the best performance in analysis. The results were also evaluated according to the maturity periods and degrees of moneyness, seeking to provide information that most agents have access to these instruments familiarizing themselves with the existing pricing methods and thus improve market liquidity. / O setor de carne bovina é um dos principais destaques do agronegócio brasileiro no cenário mundial, o recente processo de estabilização da economia, as vantagens de custos de produção com base em recursos naturais abundantes e poucas restrições ambientais têm garantido o crescimento e a competitividade do setor. Com a diminuição da intervenção governamental as políticas privadas de gestão de risco começaram a tornar-se uma preocupação entre os agentes envolvidos no agronegócio e também entre os agentes do setor de carne bovina que passaram a buscar formas de gerenciamento de risco, dentre elas os mercados futuros e de opções, a fim de garantir antecipadamente a lucratividade, reduzindo à exposição ao risco de oscilações dos preços. Em face à importância do setor de carne bovina, e a evolução apresentada pelos instrumentos derivativos, e o uso, ainda pouco expressivo, desses contratos para a administração do risco, buscou-se aplicar o modelo de precificação de opções sobre contratos futuros, desenvolvido por Black, à realidade do mercado futuro de boi gordo. O método consistiu em aplicar diferentes tipos de volatilidade (histórica, implícita e determinística) ao modelo de precificação de Black e em seguida realizou-se a análise de desempenho dos modelos por meio do cálculo dos erros. Foi encontrado que volatilidade histórica para as diferentes janelas móveis, subprecifica os valores dos prêmios negociados no mercado; enquanto que os modelos calculados com volatilidade EWMA e TARCH superprecificam os prêmios das opções. De modo geral o modelo de precificação com volatilidade histórica por janela móvel foi o que apresentou melhor desempenho nas análises realizadas. Os resultados foram avaliados ainda de acordo com os períodos de maturidade e os graus de moneyness, buscando proporcionar informações para que mais agentes tenham acesso a esses instrumentos familiarizando-se com os métodos de precificação existentes e assim melhorando a liquidez desse mercado.

Title: Black-Scholes Models of Option Pricing Author: Martin Cekal Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc., Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics. Abstract: In the present master thesis we study a generalization of Black-Scholes model using fractional Brownian motion and jump processes. The main goal is a derivation of the price of call option in a fractional jump market model. The first chapter introduces long memory and its modelling by discrete and continuous time models. In the second chapter fractional Brownian motion is defined, appropriate stochastic analysis is developed and we generalize the notion of Lévy and jump processes. The third chapter introduces fractional Black-Scholes model. In the fourth chapter, tools developed in the second chapter are used for the construction of jump fractional Black-Scholes model and derivation of explicit formula for the price of european call option. In the fifth chapter, we analyze long memory contained in simulated and empirical time series. Keywords: Black-Scholes model, fractional Brownian motion, fractional jump process, long- memory, options pricing.

Este trabalho tem como objetivo utilizar medidas de causalidade entre séries temporais de grandezas financeiras para determinar a dependência entre os ativos do mercado e utilizar as medidas obtidas para fazer inferências sobre a dinâmica desses ativos. Essa metodologia define um previsor para os valores das séries que, juntamente com a determinação das distribuições de probabilidades empíricas dos erros desse previsor por meio do método de Kernel, permite a amostragem aleatória de cenários multivariados, com diversas aplicações. Os ativos considerados para os testes de causalidade são o índice Ibovespa, o valor da paridade da moeda dólar-real USDBRL (utilizando suas séries de preços e retornos de preços), além da taxa de juros negociada diariamente (CDI). O uso do Método de Monte Carlo (MMC) é abordado para a precificação de opções de compra europeias (calls) de USDBRL e Ibovespa, e a comparação dos resultados gerados por essa metodologia com valores calculados pela fórmula de Black-Scholes (método mais utilizado no mercado financeiro, atualmente), evidenciando suas vantagens e desvantagens. Conclui-se, com este estudo, que, por meio da metodologia proposta, é possível replicar alguns comportamentos intrínsecos do mercado (como a observação de tendências nas séries de preços devido a dependências implícitas, e a presença de caudas pesadas nas distribuições dos retornos) que são desprezados pela maioria dos modelos paramétricos utilizados hoje, bem como o efeito do uso dessas informações no preço de derivativos. / This paper proposes the use of causality measures applied over time-series of financial values to determine the dependency relations between market assets and a way to use the obtained measures to make inferences about the dynamics of these assets. This methodology defines a predictor for values of the time-series that, by determining the empirical probability distributions of the errors generated by this predictor based on the Kernel method, allows a random sampling of multivariated scenarios with many applications. The assets considered for the causality tests are the Ibovespa index, the dollar-real parity value USDBRL (using their price and price-return series), in addition to the daily traded interest rate (CDI). The use of the Monte Carlo Method (MMC) for the pricing of European call options (USDBRL) and Ibovespa was discussed, in addition to a comparison of the results generated by this methodology with values calculated by the Black-Scholes formula (currently the most used method by finance institutions), showing its advantages and disadvantages. The conclusion is that, based on the proposed methodology, it is possible to replicate some intrinsic market behaviors (such as the existence of trends in price series, due to implicit dependencies, and the presence of fat tails in the distributions of price-returns) that are neglected by most of parametric models, currently, as well as the effect of using this information for pricing derivatives.

Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.

Exploration and production (E&P) involves the upstream activities from looking for promising reservoirs to extracting oil and selling it to downstream companies. E&P is the most profitable business in the oil industry. However, it is also the most capital-intensive and risky. Hence, the proper assessment of E&P projects with effective management of uncertainties is crucial to the success of any upstream business. This dissertation is concentrated on developing portfolio optimization models to manage E&P projects. The idea is not new, but it has been mostly restricted to the conceptual level due to the inherent complications to capture interactions among projects. We disentangle the complications by modeling the project portfolio optimization problem as multistage stochastic programs with mixed integer programming (MIP) techniques. Due to the disparate nature of uncertainties, we separately consider explored and unexplored oil fields. We model portfolios of real options and portfolios of decision trees for the two cases, respectively. The resulting project portfolio models provide rigorous and consistent treatments to optimally balance the total rewards and the overall risk. For explored oil fields, oil price fluctuations dominate the geologic risk. The field development process hence can be modeled and assessed as sequentially compounded options with our optimization based option pricing models. We can further model the portfolio of real options to solve the dynamic capital budgeting problem for oil projects. For unexplored oil fields, the geologic risk plays the dominating role to determine how a field is optimally explored and developed. We can model the E&P process as a decision tree in the form of an optimization model with MIP techniques. By applying the inventory-style budget constraints, we can pool multiple project-specific decision trees to get the multistage E&P project portfolio optimization (MEPPO) model. The resulting large scale MILP is efficiently solved by a decomposition-based primal heuristic algorithm. The MEPPO model requires a scenario tree to approximate the stochastic process of the geologic parameters. We apply statistical learning, Monte Carlo simulation, and scenario reduction methods to generate the scenario tree, in which prior beliefs can be progressively refined with new information. / text

Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.

Thesis (PhD)--Macquarie University, Division of Economic and Financial Studies, Dept. of Statistics, 2008. / Bibliography: p. 73-77. / Introduction -- Lévy processes used in option pricing -- Option pricing for Lévy processes -- Option pricing based on empirical characteristic functions -- Performance of the five models on historical data -- Conclusions -- References -- Appendix A. Proofs -- Appendix B. Supplements -- Appendix C. Matlab programs. / Pricing problems of financial derivatives are among the most important ones in Quantitative Finance. Since 1973 when a Nobel prize winning model was introduced by Black, Merton and Scholes the Brownian Motion (BM) process gained huge attention of professionals professionals. It is now known, however, that stock market log-returns do not follow the very popular BM process. Derivative pricing models which are based on more general Lévy processes tend to perform better. --Carr & Madan (1999) and Lewis (2001) (CML) developed a method for vanilla options valuation based on a characteristic function of asset log-returns assuming that they follow a Lévy process. Assuming that at least part of the problem is in adequate modeling of the distribution of log-returns of the underlying price process, we use instead a nonparametric approach in the CML formula and replaced the unknown characteristic function with its empirical version, the Empirical Characteristic Functions (ECF). We consider four modifications of this model based on the ECF. The first modification requires only historical log-returns of the underlying price process. The other three modifications of the model need, in addition, a calibration based on historical option prices. We compare their performance based on the historical data of the DAX index and on ODAX options written on the index between the 1st of June 2006 and the 17th of May 2007. The resulting pricing errors show that one of our models performs, at least in the cases considered in the project, better than the Carr & Madan (1999) model based on calibration of a parametric Lévy model, called a VG model. --Our study seems to confirm a necessity of using implied parameters, apart from an adequate modeling of the probability distribution of the asset log-returns. It indicates that to precisely reproduce behaviour of the real option prices yet other factors like stochastic volatility need to be included in the option pricing model. Fortunately the discrepancies between our model and real option prices are reduced by introducing the implied parameters which seem to be easily modeled and forecasted using a mixture of regression and time series models. Such approach is computationaly less expensive than the explicit modeling of the stochastic volatility like in the Heston (1993) model and its modifications. / Mode of access: World Wide Web. / x, 111 p. ill., charts

Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / The computation of the greeks of an option is an important aspect of financial mathematics. The information gained from knowing the value of a greek of an option can help investors decide whether or not to hold on to or to sell their options to avoid losses or gain a profit. However, there are technical difficulties that arise from having to do this. Among them is the fact that the mathematical formula for the value some options is complex in nature and evaluating their greeks may be cumber- some. On the other hand the greek might have to be numerically estimated if the option does not posses an explicit evaluation formula. This could be a computationally expensive undertaking. Malliavin calculus offers us a solution to these problems. We can find formula that can be used in combination with Monte Carlo simulations to give results quickly and which are not computationally expensive to obtain and hence give us an degree of accuracy higher that non Malliavin calculus techniques. This thesis will develop the Malliavin calculus tools that will enable us to develop the tools which we will then use to compute the greeks of some known options.

Submitted by Laércio Amaia (laercio_amaia@yahoo.com.br) on 2011-09-17T17:37:24Z No. of bitstreams: 1 Dissertação_LFAMAIA.pdf: 1940979 bytes, checksum: 7e6d776999403d28525c697eee220c1d (MD5) / Rejected by Suzinei Teles Garcia Garcia (suzinei.garcia@fgv.br), reason: Prezado Laercio, Faltam as palavras-chave em português e inglês. Atenciosamente, Secretaria de Registro, Suzi on 2011-09-19T14:15:35Z (GMT) / Submitted by Laércio Amaia (laercio_amaia@yahoo.com.br) on 2011-09-19T14:52:47Z No. of bitstreams: 1 Dissertação_LFAMAIA.pdf: 1900930 bytes, checksum: d051a2aa7a9015bb784fe76affac8288 (MD5) / Approved for entry into archive by Suzinei Teles Garcia Garcia (suzinei.garcia@fgv.br) on 2011-09-19T14:57:23Z (GMT) No. of bitstreams: 1 Dissertação_LFAMAIA.pdf: 1900930 bytes, checksum: d051a2aa7a9015bb784fe76affac8288 (MD5) / Approved for entry into archive by Suzinei Teles Garcia Garcia (suzinei.garcia@fgv.br) on 2011-09-19T14:57:35Z (GMT) No. of bitstreams: 1 Dissertação_LFAMAIA.pdf: 1900930 bytes, checksum: d051a2aa7a9015bb784fe76affac8288 (MD5) / Made available in DSpace on 2011-09-19T15:00:30Z (GMT). No. of bitstreams: 1 Dissertação_LFAMAIA.pdf: 1900930 bytes, checksum: d051a2aa7a9015bb784fe76affac8288 (MD5) Previous issue date: 2011-08-19 / Muitos bancos e fundos de investimento mantêm opções de ações com pouca liquidez em suas carteiras que precisam ser apreçadas diariamente, e esta falta de liquidez gera dificuldades para o processo de apreçamento. A proposta deste trabalho para resolver este problema é utilizar um modelo derivado do CAPM para estimar a superfície de volatilidades implícitas destas opções sem liquidez através da superfície de volatilidades implícitas de opções do Índice BOVESPA, ou de outras opções com maior liquidez. O modelo testado é conhecido como modelo de um fator e é utilizado para o cálculo da variância, e conseqüentemente do risco, de uma ação, ou de uma carteira de ações. Porém, neste trabalho, testaremos a validade da aplicação deste modelo para a obtenção das volatilidades implícitas de opções ilíquidas, mas com liquidez nos ativos objeto, através dos betas obtidos pelo CAPM e de volatilidades implícitas de opções líquidas e com ativos objeto também líquidos. Para o teste, foram utilizadas séries históricas de volatilidades implícitas de opções de compra líquidas de algumas ações negociadas na BM&FBOVESPA no período de 2005 a 2010, e estas foram comparadas com as volatilidades implícitas obtidas através do modelo proposto. Com os resultados, pode-se observar que as volatilidades implícitas obtidas pelo modelo são boas estimativas para apreçarmos opções com deltas próximos de 50% e para vencimentos de até 2 meses. / Many banks and investment funds hold stock options with little liquidity in their portfolios that need to be priced daily, and this lack of liquidity creates difficulties for the pricing process. The aim of this work is to solve this problem by using a model derived from the CAPM to estimate the surface of implied volatilities of these options without liquidity obtained by the surface of implied volatilities of the BOVESPA index call options or other liquid stock options. The tested model is known as one-factor model and is used to calculate the variance, and consequently the risk of a single stock or a stock portfolio. However, in this work, we will test the validity of applying this model in order to obtain the implied volatilities of illiquid options, but whose underlying assets are liquid, through the betas obtained from the CAPM and implied volatilities of liquid options of liquid underlying assets. For the test, we used historical data of implied volatilities of call options, of some stocks traded at BM&FBOVESPA, in the period 2005 to 2010, and these were compared with the implied volatilities obtained using the proposed model. With the results, one can observe that the obtained implied volatilities are a good estimate for pricing stock options with deltas near to 50% and with maturity up to two months.

Este trabalho tem como objetivo utilizar medidas de causalidade entre séries temporais de grandezas financeiras para determinar a dependência entre os ativos do mercado e utilizar as medidas obtidas para fazer inferências sobre a dinâmica desses ativos. Essa metodologia define um previsor para os valores das séries que, juntamente com a determinação das distribuições de probabilidades empíricas dos erros desse previsor por meio do método de Kernel, permite a amostragem aleatória de cenários multivariados, com diversas aplicações. Os ativos considerados para os testes de causalidade são o índice Ibovespa, o valor da paridade da moeda dólar-real USDBRL (utilizando suas séries de preços e retornos de preços), além da taxa de juros negociada diariamente (CDI). O uso do Método de Monte Carlo (MMC) é abordado para a precificação de opções de compra europeias (calls) de USDBRL e Ibovespa, e a comparação dos resultados gerados por essa metodologia com valores calculados pela fórmula de Black-Scholes (método mais utilizado no mercado financeiro, atualmente), evidenciando suas vantagens e desvantagens. Conclui-se, com este estudo, que, por meio da metodologia proposta, é possível replicar alguns comportamentos intrínsecos do mercado (como a observação de tendências nas séries de preços devido a dependências implícitas, e a presença de caudas pesadas nas distribuições dos retornos) que são desprezados pela maioria dos modelos paramétricos utilizados hoje, bem como o efeito do uso dessas informações no preço de derivativos. / This paper proposes the use of causality measures applied over time-series of financial values to determine the dependency relations between market assets and a way to use the obtained measures to make inferences about the dynamics of these assets. This methodology defines a predictor for values of the time-series that, by determining the empirical probability distributions of the errors generated by this predictor based on the Kernel method, allows a random sampling of multivariated scenarios with many applications. The assets considered for the causality tests are the Ibovespa index, the dollar-real parity value USDBRL (using their price and price-return series), in addition to the daily traded interest rate (CDI). The use of the Monte Carlo Method (MMC) for the pricing of European call options (USDBRL) and Ibovespa was discussed, in addition to a comparison of the results generated by this methodology with values calculated by the Black-Scholes formula (currently the most used method by finance institutions), showing its advantages and disadvantages. The conclusion is that, based on the proposed methodology, it is possible to replicate some intrinsic market behaviors (such as the existence of trends in price series, due to implicit dependencies, and the presence of fat tails in the distributions of price-returns) that are neglected by most of parametric models, currently, as well as the effect of using this information for pricing derivatives.