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<p>In Black-Scholes model, the parameters -a volatility and an interest rate were assumed as constants. In this thesis we concentrate on behaviour of the volatility as</p><p>a function and we find more realistic models for the volatility, which elimate a risk</p><p>connected with behaviour of the volatility of an underlying asset. That is</p><p>the reason why we will study the Uncertain Volatility Model. In Chapter</p><p>1 we will make some theoretical introduction to the Uncertain Volatility Model</p><p>introduced by Avellaneda, Levy and Paras and study how it behaves in the different scenarios. In</p><p>Chapter 2 we choose one of the scenarios. We also introduce the BSB equation</p><p>and try to make some modification to narrow the uncertainty bands using</p><p>the idea of a static hedging. In Chapter 3 we try to construct the proper</p><p>portfolio for the static hedging and compare the theoretical results with the real</p><p>market data from the Stockholm Stock Exchange.</p>

In Black-Scholes model, the parameters -a volatility and an interest rate were assumed as constants. In this thesis we concentrate on behaviour of the volatility as a function and we find more realistic models for the volatility, which elimate a risk connected with behaviour of the volatility of an underlying asset. That is the reason why we will study the Uncertain Volatility Model. In Chapter 1 we will make some theoretical introduction to the Uncertain Volatility Model introduced by Avellaneda, Levy and Paras and study how it behaves in the different scenarios. In Chapter 2 we choose one of the scenarios. We also introduce the BSB equation and try to make some modification to narrow the uncertainty bands using the idea of a static hedging. In Chapter 3 we try to construct the proper portfolio for the static hedging and compare the theoretical results with the real market data from the Stockholm Stock Exchange.

The Guaranteed Minimum Withdrawal Benefits (GMWBs) are optional riders provided by insurance companies in variable annuities. They guarantee the policyholders' ability to get the initial investment back by making periodic withdrawals regardless of the impact of poor market performance. With GMWBs attached, variable annuities become more attractive. This type of guarantee can be challenging to price and hedge. We employ two approaches to price GMWBs. Under the constant static withdrawal assumption, the first approach is to decompose the GMWB and the variable annuity into an arithmetic average strike Asian call option and an annuity certain. The second approach is to treat the GMWB alone as a put option whose maturity and payoff are random. Hedging helps insurers specify and manage the risks of writing GMWBs, as well as find their fair prices. We propose semi-static hedging strategies that offer several advantages over dynamic hedging. The idea is to construct a portfolio of European options that replicate the conditional expected GMWB liability in a short time period, and update the portfolio after the options expire. This strategy requires fewer portfolio adjustments, and outperforms the dynamic strategy when there are random jumps in the underlying price. We also extend the semi-static hedging strategies to the Heston stochastic volatility model.

The Guaranteed Minimum Withdrawal Benefits (GMWBs) are optional riders provided by insurance companies in variable annuities. They guarantee the policyholders' ability to get the initial investment back by making periodic withdrawals regardless of the impact of poor market performance. With GMWBs attached, variable annuities become more attractive. This type of guarantee can be challenging to price and hedge. We employ two approaches to price GMWBs. Under the constant static withdrawal assumption, the first approach is to decompose the GMWB and the variable annuity into an arithmetic average strike Asian call option and an annuity certain. The second approach is to treat the GMWB alone as a put option whose maturity and payoff are random. Hedging helps insurers specify and manage the risks of writing GMWBs, as well as find their fair prices. We propose semi-static hedging strategies that offer several advantages over dynamic hedging. The idea is to construct a portfolio of European options that replicate the conditional expected GMWB liability in a short time period, and update the portfolio after the options expire. This strategy requires fewer portfolio adjustments, and outperforms the dynamic strategy when there are random jumps in the underlying price. We also extend the semi-static hedging strategies to the Heston stochastic volatility model.

With the rapid increase in the usage of barrier options on the OTC markets, pricing and especially hedging of these exotic instruments became an important field of research. This paper aims to explain, apply and compare current methods used for pricing and hedging barrier options with a simulation approach. An overview of most popular methods for pricing and hedging is presented in the first part, followed by application of these pricing methods and comparing the performances of different dynamic and static hedging techniques in Black-Scholes environment by simulation in the second part. In the third part different models such as ARCH type and Stochastic Volatility are used with different jump terms to relax the assumptions of the Black-Scholes and examine the effects of these incomplete models on both pricing and performance of different hedging techniques. In the fourth part diffusion models such as Constant Variance Elasticity, Heston Stochastic Volatility and Merton Jump Diffusion are used to complete the picture.

A guaranteed minimum income benefit (GMIB) is a long-dated option that can be embedded in a deferred variable annuity. The GMIB is attractive because, for policyholders who plan to annuitize, it offers protection against poor market performance during the accumulation phase, and adverse interest rate experience at annuitization. The GMIB also provides an upside equity guarantee that resembles the benefit provided by a lookback option. We price the GMIB, and determine the fair fee rate that should be charged. Due to the long dated nature of the option, conventional hedging methods, such as delta hedging, will only be partially successful. Therefore, we are motivated to find alternative hedging methods which are practicable for long-dated options. First, we measure the effectiveness of static hedging strategies for the GMIB. Static hedging portfolios are constructed based on minimizing the Conditional Tail Expectation of the hedging loss distribution, or minimizing the mean squared hedging loss. Next, we measure the performance of semi-static hedging strategies for the GMIB. We present a practical method for testing semi-static strategies applied to long term options, which employs nested Monte Carlo simulations and standard optimization methods. The semi-static strategies involve periodically rebalancing the hedging portfolio at certain time intervals during the accumulation phase, such that, at the option maturity date, the hedging portfolio payoff is equal to or exceeds the option value, subject to an acceptable level of risk. While we focus on the GMIB as a case study, the methods we utilize are extendable to other types of long-dated options with similar features.

A guaranteed minimum income benefit (GMIB) is a long-dated option that can be embedded in a deferred variable annuity. The GMIB is attractive because, for policyholders who plan to annuitize, it offers protection against poor market performance during the accumulation phase, and adverse interest rate experience at annuitization. The GMIB also provides an upside equity guarantee that resembles the benefit provided by a lookback option. We price the GMIB, and determine the fair fee rate that should be charged. Due to the long dated nature of the option, conventional hedging methods, such as delta hedging, will only be partially successful. Therefore, we are motivated to find alternative hedging methods which are practicable for long-dated options. First, we measure the effectiveness of static hedging strategies for the GMIB. Static hedging portfolios are constructed based on minimizing the Conditional Tail Expectation of the hedging loss distribution, or minimizing the mean squared hedging loss. Next, we measure the performance of semi-static hedging strategies for the GMIB. We present a practical method for testing semi-static strategies applied to long term options, which employs nested Monte Carlo simulations and standard optimization methods. The semi-static strategies involve periodically rebalancing the hedging portfolio at certain time intervals during the accumulation phase, such that, at the option maturity date, the hedging portfolio payoff is equal to or exceeds the option value, subject to an acceptable level of risk. While we focus on the GMIB as a case study, the methods we utilize are extendable to other types of long-dated options with similar features.

Dentre as diversas ferramentas disponíveis para gestão de risco no mercado financeiro, este trabalho analisa estratégias de hedging para commodities agrícolas, utilizando o mercado futuro. Isto posto, efetua-se uma revisão das diferentes estratégias apresentadas pela literatura e analisa-se sua aplicação para o mercado brasileiro. Ao construir uma estratégia de hedging no mercado futuro, busca-se determinar o número de contratos a ser adquirido ou vendido, de forma a reduzir o risco financeiro, resultante de oscilações adversas no preço dos ativos. Ou seja, considerando-se um portfólio composto por dois ativos, um no mercado à vista e outro no futuro, as diferentes medidas de desempenho caracterizadas pelas diversas estratégias - conduzem a diferentes portfólios ótimos. Dessa forma, pretende-se analisar qual a melhor estratégia, determinando, implicitamente, qual a composição de portfolio mais adequada a um agente específico no mercado de commodities. São analisados o mercado do café, da soja, do açúcar e do álcool. Ativos financeiros, como o câmbio e o Ibovespa, também são considerados, a fim de averiguar eventuais diferenças de comportamento das estratégias, resultantes de peculiaridades do mercado de commodities. As estratégias estudadas foram: de mínima variância; de mínima variância condicionada ao período de carregamento, de maximização do índice de Sharpe; de maximização da utilidade esperada; de minimização do coeficiente de Gini estendido; de regressão linear; de regressão linear condicionada ao conjunto de informações e regressão linear condicinada ao conjunto de informações e ao período de carregamento. Apesar de o trabalho considerar somente estratégias estáticas, que se caracterizam por, uma vez determinado a quantidade de contratos a se posicionar no mercado futuro, não mais se alterar até o vencimento dos mesmos, adotou-se uma abordagem dinâmica para análise, presumindo que o portfólio pudesse ser reestruturado ao longo do tempo, de acordo com o comportamento do mercado, permitindo empregar uma abordagem mais próxima da realidade. Os resultados indicaram que as estratégias possuem diferenças, derivadas de sua estrutura, mas não variaram significativamente em função do tipo de commodity analisada. Não foi possível também identificar uma estratégia que fosse superior às demais, ou mais adequada, para uma commodity específica, do ponto de vista de resultado financeiro. Os resultados sugerem entretanto, que a seleção de uma estratégia por parte do investidor, deverá considerar as tendências de mercado, abrindo espaço para a incorporação desta informação nos modelos empregados. / Among the various tools available for managing risk in the financial market, this research analyzes hedging strategies for agricultural commodities, using the future market. It given makes up a review of different strategies presented by the literature and looks to its application to the Brazilian market. By constructing a strategy of hedging in the future market, seeks to determine the number of contracts to be purchased or sold, in order to reduce the financial risk, resulting from adverse fluctuations in the price of assets. In other words, considering a portfolio consisting of two assets, one in the spot market and one in the future, the various measures of performance - characterized by different strategies - leading to different portfolios optimum. Thus, it is intended to examine the best strategy, determining, implicitly, what the composition of portfolio best suited to a specific agent on the market of commodities. The markets analyzed were the coffee market, soybean market, sugar and alcohol market. Financial assets, such as exchange and the Ibovespa Index, are also considered in order to ascertain any differences in behavior of strategies, from peculiarities of the commodities market. The strategies studied were: the minimum- variance, the minimum-variance on the time lifted, the maximum Sharpe index; the maximum expected utility, the minimum extended Gini coefficient; the regression method; the regression method conditional on the set of information, and regression method conditional on the set of information and the time lifted. Although this research considered only static strategies which have since determined the amount of contracts to position itself in the future market, no more changes until the expiration of them, took up a dynamic approach for analysis, assuming that the portfolio could be restructured over time, according to the behavior of the market and will allow an approach closer to reality. The results indicated that strategies have differences, derived from its structure, but did not vary significantly depending on the type of commodity examined. Unable also identify a strategy with superiority than other, or more appropriately, for a specific commodity, from the viewpoint of financial results. The results suggest, however that the selection of a strategy by the investor should consider the trends of the market, opening up space to incorporate this information into the model employed.

Dentre as diversas ferramentas disponíveis para gestão de risco no mercado financeiro, este trabalho analisa estratégias de hedging para commodities agrícolas, utilizando o mercado futuro. Isto posto, efetua-se uma revisão das diferentes estratégias apresentadas pela literatura e analisa-se sua aplicação para o mercado brasileiro. Ao construir uma estratégia de hedging no mercado futuro, busca-se determinar o número de contratos a ser adquirido ou vendido, de forma a reduzir o risco financeiro, resultante de oscilações adversas no preço dos ativos. Ou seja, considerando-se um portfólio composto por dois ativos, um no mercado à vista e outro no futuro, as diferentes medidas de desempenho caracterizadas pelas diversas estratégias - conduzem a diferentes portfólios ótimos. Dessa forma, pretende-se analisar qual a melhor estratégia, determinando, implicitamente, qual a composição de portfolio mais adequada a um agente específico no mercado de commodities. São analisados o mercado do café, da soja, do açúcar e do álcool. Ativos financeiros, como o câmbio e o Ibovespa, também são considerados, a fim de averiguar eventuais diferenças de comportamento das estratégias, resultantes de peculiaridades do mercado de commodities. As estratégias estudadas foram: de mínima variância; de mínima variância condicionada ao período de carregamento, de maximização do índice de Sharpe; de maximização da utilidade esperada; de minimização do coeficiente de Gini estendido; de regressão linear; de regressão linear condicionada ao conjunto de informações e regressão linear condicinada ao conjunto de informações e ao período de carregamento. Apesar de o trabalho considerar somente estratégias estáticas, que se caracterizam por, uma vez determinado a quantidade de contratos a se posicionar no mercado futuro, não mais se alterar até o vencimento dos mesmos, adotou-se uma abordagem dinâmica para análise, presumindo que o portfólio pudesse ser reestruturado ao longo do tempo, de acordo com o comportamento do mercado, permitindo empregar uma abordagem mais próxima da realidade. Os resultados indicaram que as estratégias possuem diferenças, derivadas de sua estrutura, mas não variaram significativamente em função do tipo de commodity analisada. Não foi possível também identificar uma estratégia que fosse superior às demais, ou mais adequada, para uma commodity específica, do ponto de vista de resultado financeiro. Os resultados sugerem entretanto, que a seleção de uma estratégia por parte do investidor, deverá considerar as tendências de mercado, abrindo espaço para a incorporação desta informação nos modelos empregados. / Among the various tools available for managing risk in the financial market, this research analyzes hedging strategies for agricultural commodities, using the future market. It given makes up a review of different strategies presented by the literature and looks to its application to the Brazilian market. By constructing a strategy of hedging in the future market, seeks to determine the number of contracts to be purchased or sold, in order to reduce the financial risk, resulting from adverse fluctuations in the price of assets. In other words, considering a portfolio consisting of two assets, one in the spot market and one in the future, the various measures of performance - characterized by different strategies - leading to different portfolios optimum. Thus, it is intended to examine the best strategy, determining, implicitly, what the composition of portfolio best suited to a specific agent on the market of commodities. The markets analyzed were the coffee market, soybean market, sugar and alcohol market. Financial assets, such as exchange and the Ibovespa Index, are also considered in order to ascertain any differences in behavior of strategies, from peculiarities of the commodities market. The strategies studied were: the minimum- variance, the minimum-variance on the time lifted, the maximum Sharpe index; the maximum expected utility, the minimum extended Gini coefficient; the regression method; the regression method conditional on the set of information, and regression method conditional on the set of information and the time lifted. Although this research considered only static strategies which have since determined the amount of contracts to position itself in the future market, no more changes until the expiration of them, took up a dynamic approach for analysis, assuming that the portfolio could be restructured over time, according to the behavior of the market and will allow an approach closer to reality. The results indicated that strategies have differences, derived from its structure, but did not vary significantly depending on the type of commodity examined. Unable also identify a strategy with superiority than other, or more appropriately, for a specific commodity, from the viewpoint of financial results. The results suggest, however that the selection of a strategy by the investor should consider the trends of the market, opening up space to incorporate this information into the model employed.