In this thesis we consider the calibration of models based on Lévy processes to option prices observed in some market. This means that we choose the parameters of the option pricing models such that the prices calculated using the models correspond as closely as possible to these option prices. We demonstrate the ability of relatively simple Lévy option pricing models to nearly perfectly replicate option prices observed in nancial markets. We speci cally consider calibrating option pricing models to barrier option prices and we demonstrate that the option prices obtained under one model can be very accurately replicated using another. Various types of calibration are considered in the thesis. We calibrate a wide range of Lévy option pricing models to option price data. We con- sider exponential Lévy models under which the log-return process of the stock is assumed to follow a Lévy process. We also consider linear Lévy models; under these models the stock price itself follows a Lévy process. Further, we consider time changed models. Under these models time does not pass at a constant rate, but follows some non-decreasing Lévy process. We model the passage of time using the lognormal, Pareto and gamma processes. In the context of time changed models we consider linear as well as exponential models. The normal inverse Gaussian (N IG) model plays an important role in the thesis. The numerical problems associated with the N IG distribution are explored and we propose ways of circumventing these problems. Parameter estimation for this distribution is discussed in detail. Changes of measure play a central role in option pricing. We discuss two well-known changes of measure; the Esscher transform and the mean correcting martingale measure. We also propose a generalisation of the latter and we consider the use of the resulting measure in the calculation of arbitrage free option prices under exponential Lévy models. / PhD (Risk Analysis), North-West University, Potchefstroom Campus, 2015

The generalized inverse Gaussian distribution has become quite popular in financial engineering. The most popular random variate generator is due to Dagpunar (1989). It is an acceptance-rejection algorithm method based on the Ratio-of-uniforms method. However, it is not uniformly fast as it has a prohibitive large rejection constant when the distribution is close to the gamma distribution. Recently some papers have discussed universal methods that are suitable for this distribution. However, these methods require an expensive setup and are therefore not suitable for the varying parameter case which occurs in, e.g., Gibbs sampling. In this paper we analyze the performance of Dagpunar's algorithm and combine it with a new rejection method which ensures a uniformly fast generator. As its setup is rather short it is in particular suitable for the varying parameter case. (authors' abstract) / Series: Research Report Series / Department of Statistics and Mathematics

Submitted by Aelson Maciera (aelsoncm@terra.com.br) on 2017-08-29T19:13:47Z No. of bitstreams: 1 TeseLHMM.pdf: 2605456 bytes, checksum: b07c268a8fc9a1af8f14ac26deeec97e (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-09-25T18:22:48Z (GMT) No. of bitstreams: 1 TeseLHMM.pdf: 2605456 bytes, checksum: b07c268a8fc9a1af8f14ac26deeec97e (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-09-25T18:22:55Z (GMT) No. of bitstreams: 1 TeseLHMM.pdf: 2605456 bytes, checksum: b07c268a8fc9a1af8f14ac26deeec97e (MD5) / Made available in DSpace on 2017-09-25T18:27:54Z (GMT). No. of bitstreams: 1 TeseLHMM.pdf: 2605456 bytes, checksum: b07c268a8fc9a1af8f14ac26deeec97e (MD5) Previous issue date: 2017-04-07 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Conventional reliability analysis techniques are focused on the occurrence of failures over time. However, in certain situations where the occurrence of failures is tiny or almost null, the estimation of the quantities that describe the failure process is compromised. In this context the degradation models were developed, which have as experimental data not the failure, but some quality characteristic attached to it. Degradation analysis can provide information about the components lifetime distribution without actually observing failures. In this thesis we proposed different methodologies for degradation data based on the inverse Gaussian distribution. Initially, we introduced the inverse Gaussian deterioration rate model for degradation data and a study of its asymptotic properties with simulated data. We then proposed an inverse Gaussian process model with frailty as a feasible tool to explore the influence of unobserved covariates, and a comparative study with the traditional inverse Gaussian process based on simulated data was made. We also presented a mixture inverse Gaussian process model in burn-in tests, whose main interest is to determine the burn-in time and the optimal cutoff point that screen out the weak units from the normal ones in a production row, and a misspecification study was carried out with the Wiener and gamma processes. Finally, we considered a more flexible model with a set of cutoff points, wherein the misclassification probabilities are obtained by the exact method with the bivariate inverse Gaussian distribution or an approximate method based on copula theory. The application of the methodology was based on three real datasets in the literature: the degradation of LASER components, locomotive wheels and cracks in metals. / As técnicas convencionais de análise de confiabilidade são voltadas para a ocorrência de falhas ao longo do tempo. Contudo, em determinadas situações nas quais a ocorrência de falhas é pequena ou quase nula, a estimação das quantidades que descrevem os tempos de falha fica comprometida. Neste contexto foram desenvolvidos os modelos de degradação, que possuem como dado experimental não a falha, mas sim alguma característica mensurável a ela atrelada. A análise de degradação pode fornecer informações sobre a distribuição de vida dos componentes sem realmente observar falhas. Assim, nesta tese nós propusemos diferentes metodologias para dados de degradação baseados na distribuição gaussiana inversa. Inicialmente, nós introduzimos o modelo de taxa de deterioração gaussiana inversa para dados de degradação e um estudo de suas propriedades assintóticas com dados simulados. Em seguida, nós apresentamos um modelo de processo gaussiano inverso com fragilidade considerando que a fragilidade é uma boa ferramenta para explorar a influência de covariáveis não observadas, e um estudo comparativo com o processo gaussiano inverso usual baseado em dados simulados foi realizado. Também mostramos um modelo de mistura de processos gaussianos inversos em testes de burn-in, onde o principal interesse é determinar o tempo de burn-in e o ponto de corte ótimo para separar os itens bons dos itens ruins em uma linha de produção, e foi realizado um estudo de má especificação com os processos de Wiener e gamma. Por fim, nós consideramos um modelo mais flexível com um conjunto de pontos de corte, em que as probabilidades de má classificação são estimadas através do método exato com distribuição gaussiana inversa bivariada ou em um método aproximado baseado na teoria de cópulas. A aplicação da metodologia foi realizada com três conjuntos de dados reais de degradação de componentes de LASER, rodas de locomotivas e trincas em metais.

Submitted by Bruna Rodrigues (bruna92rodrigues@yahoo.com.br) on 2016-10-03T11:30:55Z No. of bitstreams: 1 TeseRFR.pdf: 5229141 bytes, checksum: 6f0e842f89ed4a41892f27532248ba4a (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-10T17:37:43Z (GMT) No. of bitstreams: 1 TeseRFR.pdf: 5229141 bytes, checksum: 6f0e842f89ed4a41892f27532248ba4a (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-10T17:37:50Z (GMT) No. of bitstreams: 1 TeseRFR.pdf: 5229141 bytes, checksum: 6f0e842f89ed4a41892f27532248ba4a (MD5) / Made available in DSpace on 2016-10-10T17:37:59Z (GMT). No. of bitstreams: 1 TeseRFR.pdf: 5229141 bytes, checksum: 6f0e842f89ed4a41892f27532248ba4a (MD5) Previous issue date: 2016-04-01 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Modeling of a cure fraction, also known as long-term survivors, is a part of survival analysis. It studies cases where supposedly there are observations not susceptible to the event of interest. Such cases require special theoretical treatment, in a way that the modeling assumes the existence of such observations. We need to use some strategy to make the survival function converge to a value p 2 (0; 1), representing the cure rate. A way to model cure rates is to use defective distributions. These distributions are characterized by having probability density functions which integrate to values less than one when the domain of some of their parameters is di erent from that usually de ned. There is not so much literature about these distributions. There are at least two distributions in the literature that can be used for defective modeling: the Gompertz and inverse Gaussian distribution. The defective models have the advantage of not need the assumption of the presence of immune individuals in the data set. In order to use the defective distributions theory in a competitive way, we need a larger variety of these distributions. Therefore, the main objective of this work is to increase the number of defective distributions that can be used in the cure rate modeling. We investigate how to extend baseline models using some family of distributions. In addition, we derive a property of the Marshall-Olkin family of distributions that allows one to generate new defective models. / A modelagem da fração de cura e uma parte importante da an álise de sobrevivência. Essa área estuda os casos em que, supostamente, existem observa ções não suscetíveis ao evento de interesse. Tais casos requerem um tratamento teórico especial, de forma que a modelagem pressuponha a existência de tais observações. E necessário usar alguma estratégia para tornar a função de sobrevivência convergente para um valor p 2 (0; 1), que represente a taxa de cura. Uma forma de modelar tais frações e por meio de distribui ções defeituosas. Essas distribuições são caracterizadas por possuirem funções de densidade de probabilidade que integram em valores inferiores a um quando o domínio de alguns dos seus parâmetros e diferente daquele em que e usualmente definido. Existem, pelo menos, duas distribuições defeituosas na literatura: a Gompertz e a inversa Gaussiana. Os modelos defeituosos têm a vantagem de não precisar pressupor a presença de indivíduos imunes no conjunto de dados. Para utilizar a teoria de d istribuições defeituosas de forma competitiva e necessário uma maior variedade dessas distribuições. Portanto, o principal objetivo deste trabalho e aumentar o n úmero de distribuições defeituosas que podem ser utilizadas na modelagem de frações de curas. Nós investigamos como estender os modelos defeituosos básicos utilizando certas famílias de distribuições. Além disso, derivamos uma propriedade da famí lia Marshall-Olkin de distribuições que permite gerar uma nova classe de modelos defeituosos.

Orientador: Filidor Edilfonso Vilca Labra / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-27T16:25:27Z (GMT). No. of bitstreams: 1 RamosQuispe_LuzMarina_M.pdf: 6411257 bytes, checksum: 6e1e798cf8f6d7586fe5d9a057492a77 (MD5) Previous issue date: 2015 / Resumo: Vários trabalhos têm sido feitos sobre a distribuição Birnbaum-Saunders (BS) univariada e suas extensões. A distribuição bivariada Birnbaum-Saunders (BS) foi apresentada apenas recentemente por Kundu et al. (2010) e algumas extensões já foram discutidas por Vilca et al. (2014) e Kundu et al. (2013). Eles propuseram uma distribuição BS bivariada com estrutura de dependência e estabeleceram várias propriedades atraentes. Este trabalho fornece extensões, univariada e bivariada, da distribuição BS. Estas extensões são baseadas na distribuição Gaussiana Inversa (IG) que é usada como uma distribuição de mistura no contexto de misturas de escala normal. As distribuições resultantes são distribuições absolutamente contínuas e muitas propriedades da distribuição BS são preservadas. Sob caso bivariado, as marginais e condicionais são do tipo Birnbaum-Saunders univariada. Para a obtenção da estimativa de máxima verossimilhança (EMV) é desenvolvido um algoritmo EM. Ilustramos os resultados obtidos com dados reais e simulados / Abstract: Several works have been done on the univariate Birnbaum-Saunders (BS) distribution and its extensions. The bivariate Birnbaum-Saunders (BS) distribution was presented only recently by Kundu et al. (2010) and some extensions have already been discussed by Vilca et al. (2014) and Kundu et al. (2013). They proposed a bivariate BS distribution with dependence structure and established several attractive properties. This work provides extensions, univariate and bivariate, of the BS distribution. These extensions are based on the Inverse Gaussian (IG) distribution that is used as a mixing distribution in the context of scale mixtures of normal. The resulting distributions are absolutely continuous distributions and many properties of the BS distribution are preserved. Under bivariate case, the marginals and conditionals are of type univariate Birnbaum-Saunders. For obtaining the maximum likelihood estimates (MLE) of the model parameters is developed an algorithm EM. We illustrate the obtained results with real and simulated dataset / Mestrado / Estatistica / Mestra em Estatística

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住宅抵押貸款保險(Mortgage Insurance)為管理違約風險的重要工具，在2008年次級房貸風暴後更加受到金融機構的關注。為了能更準確且更有效率的預測房價及合理評價住宅抵押貸款保險，本文延續Christoffersen, Heston and Jacobs (2006)對股票報酬率的研究，提出新的GARCH模型，利用Inverse Gaussian分配取代常態分配來捕捉房價序列中存在的自我相關以及典型現象(stylized facts)，並且同時考慮房價市場中所隱含的價格跳躍現象。本文將新模型命名為IG-GARJI模型，以便和傳統GARCH模型作區分。由於傳統的GARCH模型在計算保險價格時，通常不存在封閉解，必須藉由模擬的方法來計算價格，會增加預測的誤差，本文提供IG-GARJI模型半封閉解以增進預測效率與準確度，並利用Bühlmann et al. (1996)提出的Esscher transform方法找出其風險中立機率測度，而後運用Heston and Nandi (2000)提出之遞迴方法，找出適合的住宅抵押貸款保險評價模型。實證結果顯示，在新建房屋市場中，使用Inverse Gaussian分配會比常態分配的表現要好；對於非新建房屋，不同模型間沒有顯著的差異。另外，本文亦引用Bardhan, Karapandža, and Urošević (2006)的觀點，利用不同評價模型來比較若房屋所有權無法及時轉換時，對住宅抵押貸款保險價格帶來的影響，為住宅抵押貸款保險提供更準確的評價方法。 / Mortgage insurance products represent an attractive alternative for managing default risk. After the subprime crisis in 2008, more and more financial institutions have paid highly attention on the credit risk and default risk in mortgage market. For the purpose of giving a more accurate and more efficient model in forecasting the house price and evaluate mortgage insurance contracts properly, we follow Christoffersen, Heston and Jacobs (2006) approach to propose a new GARCH model with Inverse Gaussian innovation instead of normal distribution which is capable of capturing the auto-correlated characteristic as well as the stylized facts revealed in house price series. In addition, we consider the jump risk within the model, which is widely discussed in the house market. In order to separate our new model from traditional GARCH model, we named our model IG-GARJI model. Generally, traditional GARCH model do not exist an analytical solution, it may increase the prediction error with respect to the simulation procedure for evaluating mortgage insurance. We propose a semi-analytical solution of our model to enhance the efficiency and accuracy. Furthermore, our approach is implemented the Esscher transform introduced by Bühlmann et al. (1996) to identify a martingale measure. Then use the recursive procedure proposed by Heston and Nandi (2000) to evaluate the mortgage insurance contract. The empirical results indicate that the model with Inverse Gaussian distribution gives better performance than the model with normal distribution in newly-built house market and we could not find any significant difference between each model in previously occupied house market. Moreover, we follow Bardhan, Karapandža, and Urošević (2006) approach to investigate the impact on the mortgage insurance premium due to the legal efficiency. Our model gives another alternative to value the mortgage contracts.

Characterization of distributions has been an important topic in statistical theory for decades. Although there have been many well known results already developed, it is still of great interest to find new characterizations of commonly used distributions in application, such as normal or gamma distribution. In practice, sometimes we make guesses on the distribution to be fitted to the data observed, sometimes we use the characteristic properties of those distributions to do so. In this paper we will restrict our attention to the characterizations of gamma distribution as well as some related studies on the corresponding parameter estimation based on the characterization properties. Some simulation studies are also given.

Made available in DSpace on 2016-06-02T20:06:04Z (GMT). No. of bitstreams: 1 3603.pdf: 602959 bytes, checksum: 3078931e73ff3d01b4122cbac2c7f0a0 (MD5) Previous issue date: 2011-03-24 / Financiadora de Estudos e Projetos / In this work, the aim was to analyze in the classic and bayesian context, the GARMA model with three different continuous distributions: Gaussian, Inverse Gaussian and Gamma. We analyzed the performance and the goodness of fit of the three models, as well as the performance of the coverage percentile. In the classic analyze we consider the maximum likelihood estimator and by simulation study, we verified the consistency, the bias and de mean square error of the models. To the bayesian approach we proposed a non-informative prior distribution for the parameters of the model, resulting in a posterior distribution, which we found the bayesian estimatives for the parameters. This study still was not found in the literature. So, we can observe that the bayesian inference showed a good quality in the analysis of the serie, which can be comprove with the last section of this work. This, consist in the analyze of a real data set corresponding in the rate of tuberculosis cases in metropolitan area of Sao Paulo. The results show that, either the classical and bayesian approach, are good alternatives to describe the behavior of the real time serie. / Neste trabalho, o objetivo foi analisar no contexto clássico e bayesiano, o modelo GARMA com três distribuições contínuas: Gaussiana (Normal), Inversa Gaussiana e Gama, e também o desempenho e a qualidade do ajuste dos modelos de interesse, bem como o desempenho dos percentis de cobertura para eles. Para o estudo clássico foi considerado os estimadores de máxima verossimilhança e por meio de simulação verificou-se a consistência, o viés e o erro quadrático médio dos mesmos. Para a abordagem bayesiana é proposta uma distribuição a priori não informativa para os parâmetros dos modelos resultando em uma distribuição a posteriori, o qual a partir daí pode-se encontrar as estimativas bayesianas para os parâmetros, sendo que este estudo ainda não foi encontrado na literatura. Com isso pode-se observar que a inferência bayesiana mostrou boa eficiência no processo de análise da série, o que pode ser comprovado também com a última etapa do trabalho. Esta, consiste na análise de um conjunto de dados reais correspondente a taxa de casos de tuberculose na região metropolitana de São Paulo. Os resultados mostram que, tanto o estudo clássico quanto o bayesiano, são capazes de descrever bem o comportamento da série.

本篇論文主要考慮成群資料的存活分析，其特點為群內個體間具有相關性，並假定群內個體具有相同但無法觀測到的事故傾向。首先，探討事故傾向服從任一連續分配時混合Weibull迴歸模式的特性，接著，推導出事故傾向服從血Inverse Gaussian吧時之混合Weibull模式，並介紹參數的估計問題。然後，推導出群內個體是否獨立之分數檢定統計量，以分別就兩種最常見的存活資料型態一完整型態與右設限型態：檢定模式中事故傾向的效應是否存在。最後，並以實例說明分數檢定之程序。 / In this paper, we study survival analysis for grouped data, where the within group correlations are considered. It is also assumed that individuals within the same group share a common but unobservable random frailty. First, we discuss the properties of the Weibull regression model mixed by any continuous distribution. Next, we derive an Inverse Gaussan mixture of Weibull regression model, and discuss the estimation problem. Then, we derive the score test for testing independence between components within the same group, where the two most common cases are discussed the complete data case and the right censoring case. Finally, the testing procedures are illustrated by two examples.