<p>The present mini-thesis seeks to explore and investigate the mathematical theory and concepts that underpins the valuation of derivative securities, particularly European plainvanilla options. The main argument that we emphasise is that novel models of option pricing, as is suggested by Hull and White (1987) [1] and others, must account for the discrepancy observed on the implied volatility &ldquo / smile&rdquo / curve. To achieve this we also propose that market volatility be modeled as random or stochastic as opposed to certain standard option pricing models such as Black-Scholes, in which volatility is assumed to be constant.</p>

We discuss the Normal inverse Gaussian (NIG) distribution in modeling volatility in the financial markets. Refining the work of Barndorff-Nielsen (1997) and Andersson (2001), we introduce a new parameterization of the NIG distribution to build the GARCH(p,q)-NIG model. This new parameterization allows the model to be a strong GARCH in the sense of Drost and Nijman (1993). It also allows us to standardized the observed returns to be i.i.d., so that we can use standard inference methods when we evaluate the fit of the model. We use the realized volatility (RV), calculated from intraday data, to standardize the returns of the ECU/USD foreign exchange rate. We show that normality cannot be rejected for the RV-standardized returns, i.e., the Mixture-of-Distributions Hypothesis (MDH) of Clark (1973) holds. {We build a link between the conditional RV and the conditional variance. This link allows us to use the conditional RV as a proxy for the conditional variance. We give an empirical justification of the GARCH-NIG model using this approximation. In addition, we introduce a new General GARCH(p,q)-NIG model. This model has as special cases the Threshold-GARCH(p,q)-NIG model to model the leverage effect, the Absolute Value GARCH(p,q)-NIG model, to model conditional standard deviation, and the Threshold Absolute Value GARCH(p,q)-NIG model to model asymmetry in the conditional standard deviation. The properties of the maximum likelihood estimates of the parameters of the models are investigated in a simulation study.

In this study, we show that realized bipower variation which is an extension of realized power variation is an alternative method that estimates integrated variance like realized variance. It is seen that realized bipower variation is robust to rare jumps. Robustness means that if we add rare jumps to a stochastic volatility process, realized bipower variation process continues to estimate integrated variance although realized variance estimates integrated variance plus the quadratic variation of the jump component. This robustness is crucial since it separates the discontinuous component of quadratic variation which comes from the jump part of the logarithmic price process. Thus, we demonstrate that if the logarithmic price process is in the class of stochastic volatility plus rare jumps processes then the difference between realized variance and realized bipower variation process estimates the discontinuous component of the quadratic variation. So, quadratic variation of the jump component can be estimated and jump detection can be achieved.

<p>The present mini-thesis seeks to explore and investigate the mathematical theory and concepts that underpins the valuation of derivative securities, particularly European plainvanilla options. The main argument that we emphasise is that novel models of option pricing, as is suggested by Hull and White (1987) [1] and others, must account for the discrepancy observed on the implied volatility &ldquo / smile&rdquo / curve. To achieve this we also propose that market volatility be modeled as random or stochastic as opposed to certain standard option pricing models such as Black-Scholes, in which volatility is assumed to be constant.</p>

It is widely believed among South African agricultural market participants that the United States' corn price, as represented by the Chicago Board of Trade-listed corn contract, is causal to the price of white and yellow maize traded on the South African Futures Exchange. Although a strong correlation exists between these markets, the corn contract is far from causal to the South African maize price, as indicated by Auret and Schmitt (2008). Similarly, South African market participants believe that volatility generated in the United States corn market spills over to the South African market. Given the perceived volatility spill-over from the corn market to the maize market, market participants might inadvertently include a higher volatility component in an option price in the South African maize market than is necessary. This study sought to quantify the amount of volatility spill-over to the South African white and yellow maize market from the United States corn contract. This task was accomplished by applying an Exponential Generalised Auto Regressive Conditional Heteroscedasticity model, within an aggregate shock framework, to the data. The findings indicated that the volatility spill-over from the United States corn market to the South African maize market is not statistically significant. This result suggests that volatility in the South African market is locally driven; hence, it should not be necessary for a South African listed option contract to carry an international volatility component in its price. It was also found that the returns data of the South African maize market is asymmetrically skewed, indicating that bad news will have a greater effect on the price of maize compared with good news. / Thesis (MCom (Risk Management))--North-West University, Potchefstroom Campus, 2013.

It is widely believed among South African agricultural market participants that the United States' corn price, as represented by the Chicago Board of Trade-listed corn contract, is causal to the price of white and yellow maize traded on the South African Futures Exchange. Although a strong correlation exists between these markets, the corn contract is far from causal to the South African maize price, as indicated by Auret and Schmitt (2008). Similarly, South African market participants believe that volatility generated in the United States corn market spills over to the South African market. Given the perceived volatility spill-over from the corn market to the maize market, market participants might inadvertently include a higher volatility component in an option price in the South African maize market than is necessary. This study sought to quantify the amount of volatility spill-over to the South African white and yellow maize market from the United States corn contract. This task was accomplished by applying an Exponential Generalised Auto Regressive Conditional Heteroscedasticity model, within an aggregate shock framework, to the data. The findings indicated that the volatility spill-over from the United States corn market to the South African maize market is not statistically significant. This result suggests that volatility in the South African market is locally driven; hence, it should not be necessary for a South African listed option contract to carry an international volatility component in its price. It was also found that the returns data of the South African maize market is asymmetrically skewed, indicating that bad news will have a greater effect on the price of maize compared with good news. / Thesis (MCom (Risk Management))--North-West University, Potchefstroom Campus, 2013.

由選擇權價格反推求算出的隱含波動率，可表示市場對未來波動的預期，亦間接反映出該選擇權的價值高低，成為投資者在制定交易策略時重要的依據。經由實證研究發現，CBOE VXO及VIX都可反應投資人的恐慌心理，因此能作為標的走勢的逆向指標，所以又稱為「投資人恐慌指標」。而台指市場並沒有波動率的指標可供投資人參考，所以本研究的目的，是依照臺灣指數選擇權之市場特性，修改多種隱含波動率的估計方法。依照下列比較基準，找出適合台指市場的波動率指數。 1. 報酬反向指標： 分析波動率指數變動與市場報酬之間的關係，觀察「反向非對稱變動行為」，以Vega指數的表現最明顯。 2. 週期行為： 所有波動率指數，在日內行為的偏離幅度都很有限，且週內行為並沒有異常的週期性。分析到期日效果，只有ATM指數在到期日前二日及交易當日顯著下降，顯示台指報酬在到期日前並沒有大幅的異常波動。 3. 預測能力： 比較各波動指數的預測能力優劣。使用避免假性迴歸的模型、每分鐘報價來計算實際波動率，以VIX指數的解釋能力最佳。 綜觀以上分析結果，發現無法找出單一最佳的台指波動率指標。所以若需要最佳的「投資人恐慌指標」，必須使用Vega指數；若想做預測分析，則必須使用VIX指數。

Submitted by Ricardo Consonni (ricardo.consonni@gmail.com) on 2011-09-09T12:42:48Z No. of bitstreams: 1 +Dissetacao RC 2011 - v1g - Revisado.pdf: 2814391 bytes, checksum: f5888f2effddf979aa33bd15b87760d2 (MD5) / Approved for entry into archive by Gisele Isaura Hannickel (gisele.hannickel@fgv.br) on 2011-09-09T13:14:56Z (GMT) No. of bitstreams: 1 +Dissetacao RC 2011 - v1g - Revisado.pdf: 2814391 bytes, checksum: f5888f2effddf979aa33bd15b87760d2 (MD5) / Approved for entry into archive by Gisele Isaura Hannickel (gisele.hannickel@fgv.br) on 2011-09-09T13:15:04Z (GMT) No. of bitstreams: 1 +Dissetacao RC 2011 - v1g - Revisado.pdf: 2814391 bytes, checksum: f5888f2effddf979aa33bd15b87760d2 (MD5) / Made available in DSpace on 2011-09-09T13:17:32Z (GMT). No. of bitstreams: 1 +Dissetacao RC 2011 - v1g - Revisado.pdf: 2814391 bytes, checksum: f5888f2effddf979aa33bd15b87760d2 (MD5) Previous issue date: 2011-08-12 / Este trabalho apresenta um modelo para determinação da superfície de volatilidades de um par de moedas cujas opções têm baixa liquidez, utilizando superfícies de volatilidade com maior liquidez, de pares de moedas em que as moedas estudadas sejam uma de suas componentes. Esse objetivo é atingido através da utilização de um modelo de volatilidade estocástica. A calibração de seus parâmetros é feita a partir dos valores de mercado de Butterfly Spreads e Risk Reversals dos pares de moedas líquidos. O trabalho contribui em relação à literatura no sentido de ampliar a cobertura de strikes e vencimentos considerados, permitindo que, tanto opções pouco líquidas e fora do dinheiro, como notas estruturadas com opções embutidas possam ser mais adequadamente apreçadas. / This work presents a model for determining the volatility surface of a currency pair whose options have low liquidity, using higher liquidity volatility surfaces of other currency pairs, in which the desired currencies are one of their components. This goal is achieved through the use of a stochastic volatility model. The calibration of its parameters is done from market values of the Butterfly Spreads and Risk Reversals of the liquid-currency pairs. This work contributes to the literature in an effort to broaden the scope of strikes and maturities considered, allowing for both illiquid and out of-the-money options, as well as structured notes with embedded options, to be more appropriately priced.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-07-18T14:26:58Z No. of bitstreams: 1 daniellealdepaulaestevesdosreis.pdf: 1239258 bytes, checksum: 75cc07cdf6eba15d62c43b78ac783fbc (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-07-22T15:03:54Z (GMT) No. of bitstreams: 1 daniellealdepaulaestevesdosreis.pdf: 1239258 bytes, checksum: 75cc07cdf6eba15d62c43b78ac783fbc (MD5) / Made available in DSpace on 2016-07-22T15:03:54Z (GMT). No. of bitstreams: 1 daniellealdepaulaestevesdosreis.pdf: 1239258 bytes, checksum: 75cc07cdf6eba15d62c43b78ac783fbc (MD5) Previous issue date: 2011-12-16 / FAPEMIG - Fundação de Amparo à Pesquisa do Estado de Minas Gerais / Medidas de volatilidade se constituem numa preocupação por parte de estudiosos e profissionais do mercado financeiro. Modelos da família ARCH/GARCH a partir dos retornos diários produzem um indicador de volatilidade, mas, não conferem ao pesquisador uma medida observável do grau de variabilidade dos retornos em torno de seu valor esperado. A recente disponibilidade de dados de frequência inferior a um dia de negociação permitiu a elaboração de indicadores de volatilidade observáveis por meio de uma medida conhecida como volatilidade realizada. A partir de então, é possível elaborar um indicador observável de volatilidade diária com base em dados de natureza intradiária, de modo a representar uma medida mais apropriada do grau de risco de um ativo ou carteira de ativos, e, a partir de então, estimar a volatilidade por meio de processo da família ARIMA. De posse dos dados de alta-frequência de um papel preferencial da Petrobrás S.A., o presente trabalho se propõe, portanto, em construir a medida de volatilidade realizada por meio da soma dos quadrados dos retornos obtidos em intervalos regulares (5, 15 e 30 minutos) durante cada dia de negociação do papel PETR4 durante o período de 02/01/2007 à 29/10/2010. Posteriormente à criação do indicador de volatilidade realizada que se supõe como mais apropriado para se mensurar o grau de risco, pretende-se comparar a qualidade do ajustamento e a capacidade preditiva de cada um dos métodos de modelagem da volatilidade. A comparação dos modelos baseados em dados diários e intradiários dar-se-á por meio do cômputo do erro quadrático médio (EQM) e dos testes de Diebold e Mariano e de Harvey para avaliação da acurácia preditiva dos modelos. Os resultados mostraram que, em geral, os modelos da família ARIMA são mais apropriados para a avaliação do grau de ajustamento, e produz previsões mais satisfatórias que os modelos da família ARCH/GARCH. / Volatility measures constitute a concern among scholars and professionals of the financial market. Models of the ARCH/GARCH class from the daily returns produce an indicator of volatility, but do not give the researcher an observable measure of the degree of variability of returns around their expected value. The recent availability of data at frequencies below a trading day allowed the development of indicators of volatility observable through a measurement known as realized volatility. Since then, they can build an observable indicator of daily volatility based on intraday data, so as to represent a more appropriate measure of the riskiness of an asset, and from then estimate volatility through a process of ARIMA family. Provided with the data of a high frequency preferential role of Petrobrás S. A., the present paper therefore proposes to construct a measure of realized volatility by the sum of the squares of the returns obtained at regular intervals (5, 15 and 30 minutes ) during each trading day for the paper PETR4 during 02/01/2007 to 29/10/2010. After the creation of the realized volatility indicator that is supposed to be more appropriate to measure the degree of risk, the intent is to compare the goodness of fit and predictive ability of each of the methods of volatility’s models. The comparison of models based on daily data and intraday give will be through the calculation of the mean square error (MSE) and tests of Diebold and Mariano and Harvey to evaluate the predictive accuracy of models. The results in general showed that the models of the ARIMA class are more suitable for assessing the degree of adjustment and produces predictions more satisfactory than the models of the ARCH/GARCH class.

A presente pesquisa objetivou verificar se medidas contábeis calculadas por meio do lucro abrangente e de seus componentes são relevantes na explicação do risco das empresas. Para isso, analisou-se amostra composta por 105 companhias brasileiras de capital aberto com ações negociadas na BM&FBovespa. Foram selecionadas empresas não financeiras com dados disponíveis no período compreendido entre o primeiro trimestre de 2011 e o primeiro de trimestre de 2015, sendo excluídas empresas com ações de baixa liquidez. Inicialmente, avaliou-se por meio de análise de estatísticas descritivas e do Teste de Wilcoxon se o lucro abrangente é mais volátil que o lucro líquido. Esta primeira hipótese de pesquisa não foi refutada e se constatou que, para as empresas que compõem a amostra estudada, a volatilidade do lucro abrangente foi 30,84% superior à volatilidade do lucro líquido, evidenciando que a análise focada única e exclusivamente no lucro líquido induziria o usuário da informação contábil a ignorar possíveis fontes de risco da empresa. Para a análise da relevância do lucro abrangente e de seus componentes na explicação do risco, foi feita análise de regressão com dados em painel por meio de oito modelos estimados, cada qual com diferentes variáveis explicativas e abordagens de risco, estimado por meio da volatilidade dos retornos das ações (risco total) e do beta de mercado (risco sistemático). Os resultados apresentados evidenciaram que a relação entre a volatilidade do lucro abrangente e o risco da empresa é superior àquela observada entre a volatilidade do lucro líquido e o risco, porém tal relação não se mostrou estatisticamente significante. Todavia, verificou-se que a volatilidade do valor referente a outros resultados abrangentes possui relação negativa e estatisticamente significante com seu risco sistemático. Ganhos e perdas com hedges de fluxo de caixa e com ativos financeiros classificados como disponíveis para venda apresentaram relação negativa e estatisticamente significante com o risco da empresa, o que, de acordo com estudos anteriores, deve-se ao fato de que resultados não realizados estariam além do controle dos gestores. As evidências apresentadas pelo presente estudo corroboram a importância do assunto, fornecendo insumos para discussões sobre políticas contábeis relacionadas ao lucro e para o desenvolvimento de métricas contábeis para a avaliação do risco das empresas. / This research aimed to verify if the comprehensive income and its components are relevant in explanation of the firm risk. Thus, it analyzed sample of 105 Brazilian public companies listed on the BM&FBovespa. Non-financial companies with available data from the first quarter of 2011 to first quarter of 2015 were selected, being excluded companies with low stock liquidity. Initially, it was evaluated by analysis of descriptive statistics and the Wilcoxon test if the comprehensive income is more volatile than net income. This first research hypothesis was not refuted and found that for the companies in the sample, the volatility of comprehensive income was 30.84% higher than the volatility of net income, showing that an analysis focused exclusively on net income induce the accounting information user to ignore possible sources of firm risk. For the analysis of the relevance of comprehensive income and its components in risk explanation, regression analysis was done with panel data through eight estimated models, each with different explanatory variables and risk approaches, estimated by the volatility of stock returns (total risk) and market beta (systematic risk). The results showed that the relationship between the volatility of comprehensive income and the company\'s risk is greater than that observed between the volatility of net income and risk, but this relationship was not statistically significant. However, it was found that the volatility of the value related to other comprehensive income has negative and statistically significant relationship with its systematic risk. Gains and losses from cash flow hedges and financial assets classified as available for sale showed a negative and statistically significant relationship to the risk of the firm, which, according to previous studies, is due to the fact that unrealized results would be beyond the control of managers. The evidence presented in this study confirm the importance of the subject, providing inputs for discussions on accounting policies related to income and to the development of accounting metrics for risk assessment of companies.