In this thesis we have created a computer program in Java language which calculates European call- and put options with four different models based on the article The Pricing of Options on Assets with Stochastic Volatilities by John Hull and Alan White. Two of the models use stochastic volatility as an input. The paper describes the foundations of stochastic volatility option pricing and compares the output of the models. The model which better estimates the real option price is dependent on further research of the model parameters involved.

[pt] Esta Dissertação apresenta um modelo para extrair fatores capazes de prever a volatilidade do índice de ações IBOVESPA, representativo do mercado de ações brasileiro. Esta metodologia é diferenciada por utilizar fatores que não incluem ativos da classe de ações. São utilizados fatores extraídos de classes de ativos de crédito, taxas de juros, moedas e commodities para precificar a volatilidade de um índice de ações. Além disso, os fatores são extraídos de painéis de volatilidades filtradas por modelos do tipo GARCH. / [en] It will be presented a model that is able to extract factors capable of predicting the volatility of IBOVESPA market index, which is representative of Brazilian equity market. This methodology is different from others because it won t use any inputs from equity asset classes. It will be used factors extracted from credit risk, interest rates, exchange rates and commodities data for pricing the volatility of an equity index. Besides that, those factors will be extracted from panels of volatility filtered by GARCH models.

This diploma thesis focuses on modeling volatility in financial time series. The main approach to modelling volatility is using GARCH models which can capture the variability of conditional volatility of time series. For modelling a conditional mean value in time series are used ARMA models. In the series there are usually not fulfilled the assumption of earnings normality, therefore, are the earnings in most cased characterized by the leptokurtic shape of distribution. The thesis introduces some more distribution types, which can be more easily used for the earnings distribution - above all the Students t distribution. The aim of the thesis in the first part is to present the topic of financial time series and description of the GARCH models including their further modification. There are used e.g. IGARCH or other models capturing asymmetric impact of shocks such as GJR-GARCH. The second part deals with generated data, where are more in detail explored the volatility models and their behavior in corresponding financial time series. The third part focuses on the volatility estimation and forecasting for the financial time series. Firstly this concerns development of stock index MICEX secondly currency pair Russian Ruble to Czech Crown and eventually price development of the Brent crude oil. The goal of the third part is to present the impacts on volatility of chosen time series applied on the example of economic sanctions against Russia after annexation of the Crimea peninsula which happened in the first quarter 2014.

This thesis focuses on comparing an online parameter estimator to an offline estimator, both based on the PaRIS-algorithm, when estimating parameter values for a stochastic volatility model. By modeling the stochastic volatility model as a hidden Markov model, estimators based on particle filters can be implemented in order to estimate the unknown parameters of the model. The results from this thesis implies that the proposed online estimator could be considered as a superior method to the offline counterpart. The results are however somewhat inconclusive, and further research regarding the subject is recommended. / Detta examensarbetefokuserar på att jämföra en online och offline parameter-skattare i stokastiskavolatilitets modeller. De två parameter-skattarna som jämförs är båda baseradepå PaRIS-algoritmen. Genom att modellera en stokastisk volatilitets-model somen dold Markov kedja, kunde partikelbaserade parameter-skattare användas föratt uppskatta de okända parametrarna i modellen. Resultaten presenterade idetta examensarbete tyder på att online-implementationen av PaRIS-algorimen kanses som det bästa alternativet, jämfört med offline-implementationen.Resultaten är dock inte helt övertygande, och ytterligare forskning inomområdet

Este trabalho apresenta um estudo através da abordagem Bayesiana em modelos de volatilidade estocástica, para modelagem de séries temporais financeiras, com o uso do método de Monte Carlo Hamiltoniano (HMC). Propomos o uso de outras distribuições para os erros da equação de observações do modelos de volatilidade estocástica, além da distribuição Gaussiana, para tratar problemas como caudas pesadas e assimetria nos retornos. Além disso, utilizamos critérios de informações, recentemente desenvolvidos, WAIC e LOO que aproximam a metodologia de validação cruzada, para realizar a seleção de modelos. No decorrer do trabalho, estudamos a qualidade do método HMC através de exemplos, estudo de simulação e aplicação a conjunto de dados. Adicionalmente, avaliamos a performance dos modelos e métodos propostos através do cálculo de estimativas para o Valor em Risco (VaR) para múltiplos horizontes de tempo. / This paper presents a study using Bayesian approach in stochastic volatility models for modeling financial time series, using Hamiltonian Monte Carlo methods (HMC). We propose the use of other distributions for the errors of the equation at stochastic volatiliy model, besides the Gaussian distribution, to treat the problem as heavy tails and asymmetry in the returns. Moreover, we use recently developed information criteria WAIC and LOO that approximate the crossvalidation methodology, to perform the selection of models. Throughout this work, we study the quality of the HMC methods through examples, simulation study and application to dataset. In addition, we evaluated the performance of the proposed models and methods by calculating estimates for Value at Risk (VaR) for multiple time horizons.

Conselho Nacional de Desenvolvimento Científico e Tecnológico / The analysis of macro-economic variables through time-series models is widely used in the literature supporting economic theory, showing the actual behaviour of these variables. One of these macroeconomic variables have two variables that interfere with eou has relationships with other variables justifying the relevance in studying their behaviors. The first is the interest rate, which is very important in driving the economy, influencing the intention to spend and save of all economic agents, whether personal, commercial or industrial level (State or private). The second is the exchange rate, where its buoyancy determines the level of imports and exports affecting the trade balance. In this context the present research aims to describe the behavior of SELIC interest rates and Brazilian Exchange from January 1974 to June 2012 and January 1980 to may 2012, respectively. To this end, at first was used the Box-Jenkins model where the models showed through the analysis of residues which both had conditional heteroscedasticity in the waste of the models. Then joint modeling was used to the level of the process and the process variance (ARCH family models). The results showed that, for the SELIC interest rate series, the model selected was an ARIMA (1,1,1)-EGARCH (3,1) and, to the exchange rate, an ARIMA (0,1,1)-EGARCH (1,1). It is evidenced through these models that there is asymmetry of information, yet there was the leverage effect. In a second moment was chosen a model representing each one of the models of family ARCH (ARCH, GARCH, TARCH, EGARCH) and later held the combination of prediction by methods: ACP, middle and MMQO. The results obtained show that, in General, the performance measures MAPE, MSE and U-THEIL are superior to the combinations of prediction. In addition, the combination of forecast for different weights with ACP to check which of the types of weights provide better results. Therefore, it is concluded that the different weights allow the researcher to achieve greater accuracy in the choice of models combined, allowing aid managers in prior decision of the behavior of these variables that affect so scathing the health of the Brazilian economy. / A análise de variáveis macroeconômicas por meio de modelos de séries temporais é amplamente utilizada na literatura dando suporte à teoria econômica, mostrando o real comportamento dessas variáveis. Dentre essas variáveis macroeconômicas tem-se duas variáveis que interferem e/ou tem relações com outras variáveis justificando-se assim a relevância em estudar seus comportamentos. A primeira é a taxa de juros, que é muito importante na condução da economia, influenciando a intenção de gastar e poupar de todos os agentes econômicos, seja no nível pessoal, comercial ou industrial (privado e/ou estatal). A segunda é a taxa de câmbio, onde sua flutuação determina o nível das importações e exportações afetando assim a balança comercial. Nesse contexto a presente pesquisa tem como objetivo descrever o comportamento das taxas de juros SELIC e câmbio brasileiras no período de janeiro de 1974 a junho de 2012 e de janeiro de 1980 a maio de 2012, respectivamente. Para tanto, num primeiro momento foi utilizada a modelagem Box-Jenkins onde os modelos evidenciaram por meio da análise de resíduos que ambos possuíam heterocedasticidade condicional nos resíduos dos modelos. Em seguida, utilizou-se a modelagem conjunta para o nível do processo e para a variância do processo (modelos da família ARCH). Os resultados obtidos mostraram que, para a série da taxa de juros SELIC, o modelo elegido foi um ARIMA (1,1,1)- EGARCH (3,1) e, para a taxa de câmbio, um ARIMA (0,1,1)- EGARCH (1,1). Evidencia-se por meio destes modelos que há assimetria das informações, contudo não se verificou o efeito de alavancagem. Num segundo momento foi escolhido um modelo representando cada um dos modelos da família ARCH (ARCH, GARCH, EGARCH, TARCH) e posteriormente realizada a combinação de previsão pelos métodos: ACP, Média e MMQO. Os resultados obtidos evidenciam que, no geral, as medidas de desempenho MAPE, MSE e U-THEIL são superiores para as combinações de previsão. Além disso, foi realizada a combinação de previsão por ACP com ponderações diferentes para verificar qual dos tipos de ponderações propiciam resultados melhores. Logo, conclui-se que as diferentes ponderações permitem ao pesquisador conseguir maior acurácia na escolha dos modelos combinados, permitindo auxiliar gestores na decisão prévia do comportamento dessas variáveis que afetam de maneira contundente a saúde da economia brasileira.

Submitted by Pedro Nielsen Rotta (pedro.rotta@gvmail.br) on 2013-01-17T01:23:43Z No. of bitstreams: 1 Dissertação_MPE_Pedro_Nielsen_Rotta.pdf: 1289831 bytes, checksum: e326f8c88be4c40b8581a8dd53dc83a3 (MD5) / Approved for entry into archive by Eliene Soares da Silva (eliene.silva@fgv.br) on 2013-01-17T15:46:21Z (GMT) No. of bitstreams: 1 Dissertação_MPE_Pedro_Nielsen_Rotta.pdf: 1289831 bytes, checksum: e326f8c88be4c40b8581a8dd53dc83a3 (MD5) / Made available in DSpace on 2013-01-17T16:11:12Z (GMT). No. of bitstreams: 1 Dissertação_MPE_Pedro_Nielsen_Rotta.pdf: 1289831 bytes, checksum: e326f8c88be4c40b8581a8dd53dc83a3 (MD5) Previous issue date: 2012-12-18 / Nas últimas décadas, a análise dos padrões de propagação internacional de eventos financeiros se tornou o tema de grande parte dos estudos acadêmicos focados em modelos de volatilidade multivariados. Diante deste contexto, objetivo central do presente estudo é avaliar o fenômeno de contágio financeiro entre retornos de índices de Bolsas de Valores de diferentes países a partir de uma abordagem econométrica, apresentada originalmente em Pelletier (2006), sobre a denominação de Regime Switching Dynamic Correlation (RSDC). Tal metodologia envolve a combinação do Modelo de Correlação Condicional Constante (CCC) proposto por Bollerslev (1990) com o Modelo de Mudança de Regime de Markov sugerido por Hamilton e Susmel (1994). Foi feita uma modificação no modelo original RSDC, a introdução do modelo GJR-GARCH formulado em Glosten, Jagannathan e Runkle (1993), na equação das variâncias condicionais individuais das séries para permitir capturar os efeitos assimétricos na volatilidade. A base de dados foi construída com as séries diárias de fechamento dos índices das Bolsas de Valores dos Estados Unidos (SP500), Reino Unido (FTSE100), Brasil (IBOVESPA) e Coréia do Sul (KOSPI) para o período de 02/01/2003 até 20/09/2012. Ao longo do trabalho a metodologia utilizada foi confrontada com outras mais difundidos na literatura, e o modelo RSDC com dois regimes foi definido como o mais apropriado para a amostra selecionada. O conjunto de resultados encontrados fornecem evidências a favor da existência de contágio financeiro entre os mercados dos quatro países considerando a definição de contágio financeiro do Banco Mundial denominada de 'muito restritiva'. Tal conclusão deve ser avaliada com cautela considerando a extensa diversidade de definições de contágio existentes na literatura. / Over the last decades, the analysis of the transmissions of international financial events has become the subject of many academic studies focused on multivariate volatility models volatility. The goal of this study is to evaluate the financial contagion between stock market returns. The econometric approach employed was originally presented by Pelletier (2006), named Regime Switching Dynamic Correlation (RSDC). This methodology involves the combination of Constant Conditional Correlation Model (CCC) proposed by Bollerslev (1990) with Markov Regime Switching Model suggested by Hamilton and Susmel (1994). A modification was made in the original model RSDC, the introduction of the GJR-GARCH Glosten model formulated in Glosten, Jagannathan e Runkle (1993), on the equation of the conditional univariate variances to allow asymmetric effects in volatility be captured. The database was built with the series of daily closing stock market indices in the United States (SP500), United Kingdom (FTSE100), Brazil (IBOVESPA) and South Korea (KOSPI) for the period from 02/01/2003 to 20/09/2012. Throughout the work the methodology was compared with others most widespread in the literature, and the model RSDC with two regimes was defined as the most appropriate for the selected sample. The set of results provide evidence for the existence of financial contagion between markets of the four countries considering the definition of financial contagion from the World Bank called 'very restrictive'. Such a conclusion should be evaluated carefully considering the wide diversity of definitions of contagion in the literature.

Doctor Scientiae - DSc / Robust Spectral Methods for Solving Option Pricing Problems by Edson Pindza PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of Natural Sciences, University of the Western Cape Ever since the invention of the classical Black-Scholes formula to price the financial derivatives, a number of mathematical models have been proposed by numerous researchers in this direction. Many of these models are in general very complex, thus closed form analytical solutions are rarely obtainable. In view of this, we present a class of efficient spectral methods to numerically solve several mathematical models of pricing options. We begin with solving European options. Then we move to solve their American counterparts which involve a free boundary and therefore normally difficult to price by other conventional numerical methods. We obtain very promising results for the above two types of options and therefore we extend this approach to solve some more difficult problems for pricing options, viz., jump-diffusion models and local volatility models. The numerical methods involve solving partial differential equations, partial integro-differential equations and associated complementary problems which are used to model the financial derivatives. In order to retain their exponential accuracy, we discuss the necessary modification of the spectral methods. Finally, we present several comparative numerical results showing the superiority of our spectral methods.

La première partie de cette thèse est consacrée aux méthodes numériques pour la simulation de processus aléatoires définis par des équations différentielles stochastiques (EDS). Nous commençons par l’étude de l’algorithme de Beskos et al. [13] qui permet de simuler exactement les trajectoires d’un processus solution d’une EDS en dimension 1. Nous en proposons une extension à des fins de calcul exact d’espérances et nous étudions l’application de ces idées à l’évaluation du prix d’options asiatiques dans le modèle de Black & Scholes. Nous nous intéressons ensuite aux schémas numériques. Dans le deuxième chapitre, nous proposons deux schémas de discrétisation pour une famille de modèles à volatilité stochastique et nous en étudions les propriétés de convergence. Le premier schéma est adapté à l’évaluation du prix d’options path-dependent et le deuxième aux options vanilles. Nous étudions également le cas particulier où le processus qui dirige la volatilité est un processus d’Ornstein-Uhlenbeck et nous exhibons un schéma de discrétisation qui possède de meilleures propriétés de convergence. Enfin, dans le troisième chapitre, il est question de la convergence faible trajectorielle du schéma d’Euler. Nous apportons un début de réponse en contrôlant la distance de Wasserstein entre les marginales du processus solution et du schéma d’Euler, uniformément en temps. La deuxième partie de la thèse porte sur la modélisation de la dépendance en finance et ce à travers deux problématiques distinctes : la modélisation jointe entre un indice boursier et les actions qui le composent et la gestion du risque de défaut dans les portefeuilles de crédit. Dans le quatrième chapitre, nous proposons un cadre de modélisation original dans lequel les volatilités de l’indice et de ses composantes sont reliées. Nous obtenons un modèle simplifié quand la taille de l’indice est grande, dans lequel l’indice suit un modèle à volatilité locale et les actions individuelles suivent un modèle à volatilité stochastique composé d’une partie intrinsèque et d’une partie commune dirigée par l’indice. Nous étudions la calibration de ces modèles et montrons qu’il est possible de se caler sur les prix d’options observés sur le marché, à la fois pour l’indice et pour les actions, ce qui constitue un avantage considérable. Enfin, dans le dernier chapitre de la thèse, nous développons un modèle à intensités permettant de modéliser simultanément, et de manière consistante, toutes les transitions de ratings qui surviennent dans un grand portefeuille de crédit. Afin de générer des niveaux de dépendance plus élevés, nous introduisons le modèle dynamic frailty dans lequel une variable dynamique inobservable agit de manière multiplicative sur les intensités de transitions. Notre approche est purement historique et nous étudions l’estimation par maximum de vraisemblance des paramètres de nos modèles sur la base de données de transitions de ratings passées / The first part of this thesis deals with probabilistic numerical methods for simulating the solution of a stochastic differential equation (SDE). We start with the algorithm of Beskos et al. [13] which allows exact simulation of the solution of a one dimensional SDE. We present an extension for the exact computation of expectations and we study the application of these techniques for the pricing of Asian options in the Black & Scholes model. Then, in the second chapter, we propose and study the convergence of two discretization schemes for a family of stochastic volatility models. The first one is well adapted for the pricing of vanilla options and the second one is efficient for the pricing of path-dependent options. We also study the particular case of an Orstein-Uhlenbeck process driving the volatility and we exhibit a third discretization scheme which has better convergence properties. Finally, in the third chapter, we tackle the trajectorial weak convergence of the Euler scheme by providing a simple proof for the estimation of the Wasserstein distance between the solution and its Euler scheme, uniformly in time. The second part of the thesis is dedicated to the modelling of dependence in finance through two examples : the joint modelling of an index together with its composing stocks and intensity-based credit portfolio models. In the forth chapter, we propose a new modelling framework in which the volatility of an index and the volatilities of its composing stocks are connected. When the number of stocks is large, we obtain a simplified model consisting of a local volatility model for the index and a stochastic volatility model for the stocks composed of an intrinsic part and a systemic part driven by the index. We study the calibration of these models and show that it is possible to fit the market prices of both the index and the stocks. Finally, in the last chapter of the thesis, we define an intensity-based credit portfolio model. In order to obtain stronger dependence levels between rating transitions, we extend it by introducing an unobservable random process (frailty) which acts multiplicatively on the intensities of the firms of the portfolio. Our approach is fully historical and we estimate the parameters of our model to past rating transitions using maximum likelihood techniques

[pt] Este trabalho tem como objetivo investigar o efeito das intervenções do Banco Central na volatilidade da taxa de câmbio no Brasil no período de 2000 a 2003 e entender se a autoridade monetária intervém com o intuito de suavizar a volatilidade do câmbio. Para abordarmos o primeiro ponto, utilizamos o modelo EGARCH de Nelson (1991) que nos permitiu estimar o impacto das intervenções na volatilidade da taxa de câmbio levando em conta a possibilidade de que choques positivos e negativos no retorno do câmbio tenham efeitos distintos na volatilidade. Como principal resultado, encontrou-se que as intervenções do Banco Central estariam adicionando volatilidade na taxa de câmbio. Entretanto, devido à possibilidade de simultaneidade, utilizou-se a metodologia desenvolvida por Vella (1993) que nos permite estimar o efeito das intervenções na volatilidade de forma consistente e testar a endogeneidade das intervenções. Concluímos que as estimativas anteriores eram inconsistentes uma vez que encontramos que as intervenções contribuíram para uma redução de volatilidade e o teste de endogeneidade confirmou que as intervenções são endógenas ao modelo. Podemos também depreender deste resultado que possivelmente o Banco Central tem suavizado movimentos na taxa de câmbio. / [en] This work investigates the effect of Central Bank interventions on the exchange rate volatility from 2000 to 2003 and tries to understand whether or not the monetary authority smoothes exchange rate volatility. Referring to the first issue, we estimated an EGARCH model developed by Nelson (1991) that allows us to estimate the effect of interventions on the volatility regarding the possibility that positive and negative shocks have different impacts on volatility. The results found indicate that Central Bank interventions are adding volatility to the exchange rate. However, because of the possibility of simultaneity, we implemented the methodology developed by Vella (1993) that allows us to test consistently the effect of interventions on the volatility and test the endogeneity of interventions. We conclude that previous estimates were inconsistent as new results reveal that interventions contribute to reduce volatility and the endogeneity test confirms that interventions are an endogenous variable of the model. This result also indicates that possibly Central Bank smoothes exchange rate movements.