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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

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

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Nutrient pollution has been identified as a significant threat to U.S. coastal and estuarine water quality. Though coastal and estuarine waters need nutrients to maintain a healthy, productive ecosystem, excess nutrients can lead to eutrophication. There are significant potential negative consequences associated with eutrophication, including loss of habitat, loss of economic activity, and direct threats to human health. Hillsborough Bay experienced eutrophication in the 1960s and 1970s due to a rapidly growing population and associated increases in nutrient pollution. These eutrophic conditions led to more frequent phytoplankton and macroalgae blooms and declines in seagrasses. To address these problems, a series of actions were taken including legislation limiting nutrient concentrations from domestic wastewater treatment plants, development of water quality and nutrient loading targets, and establishment of seagrass restoration and protection goals. Since the 1970s, water quality improvements and increasing seagrass acreages have been documented throughout Tampa Bay. In the current study, a series of analyses and tools are developed to obtain a more in depth understanding of water quality in Hillsborough Bay. The first tool is a linked hydrodynamic and water quality model (Environmental Fluid Dynamics Code) of Hillsborough Bay which can be employed to predict water quality responses to proposed management actions. In the second part of the study, a series of water quality indices were evaluated. The most appropriate index for determining overall water quality in Hillsborough Bay was identified. Chlorophyll a is one of the constituents in the water quality index and is currently used to evaluate annual water quality conditions in Hillsborough Bay. Therefore, the statistical distribution that describes chlorophyll a concentrations in Hillsborough Bay was identified and robust confidence intervals were developed to better understand the uncertainty associated with chlorophyll a measurements. Previous work linked chlorophyll a concentrations in Hillsborough Bay to explanatory variables based on monthly estimates. These relationships were used to develop water quality targets for the system. In this study, the previously developed relationship was revisited, resulting in an improved statistical model that is more robust. This improved model can also be used to evaluate the previously proposed targets and to better predict future changes due to climate change, sea level rise, and management actions. Lastly, a new method was developed to estimate atmospheric temperature in the contiguous United States.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

No / The probability density and exceedance probability functions of the induced currents in a screened cable connecting two enclosures, resulting from the close. presence of single and multiple mobile phones working at 900 MHz, are investigated. The analysis of the problem is undertaken using the Method of Moments, but due to weak coupling, the impedance matrix was modified to reduce the memory and time requirements for the problem, to enable it to be executed multiple times. The empirical probability distribution functions (PDFs) and exceedance probabilities for the induced currents are presented. The form of the PDFs is seen to be quite well approximated by a log-normal distribution for a single source and by a Weibull distribution for multiple sources

Like coral reefs worldwide, the Florida Reef Tract has dramatically declined within the past two decades. Monitoring of 40 sites throughout the Florida Keys National Marine Sanctuary has undertaken a multiple-parameter approach to assess spatial and temporal changes in the status of the ecosystem. The objectives of the present study consist of the following: In chapter one, we review past coral reef studies; emphasis is placed on recent studies on the stony corals of reefs in the lower Florida Keys. We also review the economic impact of coral reefs on the state of Florida. In chapter two, we identify the underlying probability distribution function of the stony coral cover proportions and we obtain better estimates of the statistical properties of stony coral cover proportions. Furthermore, we improve present procedures in constructing confidence intervals of the true median and mean for the underlying probability distribution. In chapter three, we investigate the applicability of the normal probability distribution assumption made on the pseudovalues obtained from the jackknife procedure for the Shannon-Wiener diversity index used in previous studies. We investigate a new and more effective approach to estimating the Shannon-Wiener and Simpson's diversity index. In chapter four, we develop the best possible estimate of the probability distribution function of the jackknifing pseudovalues, obtained from the jackknife procedure for the Shannon-Wiener diversity index used in previous studies, using the xi nonparametric kernel density estimate method. This nonparametric procedure gives very effective estimates of the statistical measures for the jackknifing pseudovalues. Lastly, the present study develops a predictive statistical model for stony coral cover. In addition to identifying the attributable variables that influence the stony coral cover data of the lower Florida Keys, we investigate the possible interactions present. The final form of the developed statistical model gives good estimates of the stony coral cover given some information of the attributable variables. Our nonparametric and parametric approach to analyzing coral reef data provides a sound basis for developing efficient ecosystem models that estimate future trends in coral reef diversity. This will give the scientists and managers another tool to help monitor and maintain a healthy ecosystem.

Le but de ce mémoire de maîtrise est de décrire les propriétés de la loi double Pareto-lognormale, de montrer comment on peut introduire des variables explicatives dans le modèle et de présenter son large potentiel d'applications dans le domaine de la science actuarielle et de la finance. Tout d'abord, nous donnons la définition de la loi double Pareto-lognormale et présentons certaines de ses propriétés basées sur les travaux de Reed et Jorgensen (2004). Les paramètres peuvent être estimés en utilisant la méthode des moments ou le maximum de vraisemblance. Ensuite, nous ajoutons une variable explicative à notre modèle. La procédure d'estimation des paramètres de ce mo-\\dèle est également discutée. Troisièmement, des applications numériques de notre modèle sont illustrées et quelques tests statistiques utiles sont effectués. / The purpose of this Master's thesis is to describe the double Pareto-lognormal distribution, show how the model can be extended by introducing explanatory variables in the model and present its large potential of applications in actuarial science and finance. First, we give the definition of the double Pareto-lognormal distribution and present some of its properties based on the work of Reed and Jorgensen (2004). The parameters could be estimated by using the method of moments or maximum likelihood. Next, we add an explanatory variable to our model. The procedure of estimation for this model is also discussed. Finally, some numerical applications of our model are illustrated and some useful statistical tests are conducted.

Príons são patógenos infecciosos que causam um grupo de doenças neurodegenerativas fatais. A proteína normal, PrP celular, denominada PrPC, é convertida em PrPSc, isoforma anormal e patogênica de PrP, através de um processo no qual uma porção de -hélice da estrutura é reenovelada em folhas . A conversão de PrPC em PrPSc ocorre por um mecanismo auto-catalítico. Para um melhor entendimento do mecanismo de propagação dos príons, têm sido propostos vários modelos matemáticos. Nesse trabalho, estudamos o tempo de incubação de algumas doenças causadas por príons: Encefalopatia Espongiforme Bovina (BSE), ou mal da vaca louca; doença variante de Creutzfeldt-Jakob (vCJD), que afeta humanos, através da exposição ao agente de BSE; e Scrapie murina, uma infecção priônica experimental em camundongos. A distribuição de probabilidades da duração do período de incubação foi suposta ser lognormal, modelo este extensamente aceito em doenças infecciosas. Os objetivos desse trabalho foram esclarecer aspectos obscuros sobre a cinética de replicação priônica e o mecanismo de toxicidade das doenças priônicas, através de comparação dos resultados de simulações computacionais com os perfis de distribuição de tempos de incubação de BSE, vCJD e Scrapie murina. Foram realizadas simulações computacionais, utilizando o Método Monte Carlo Dinâmico (MCD) e o modelo Difusão Limitada à Agregação. Primeiramente, estudamos o modelo de Eigen (1996), através de simulações computacionais usando o MCD, para verificar quais termos são importantes para a cinética priônica. De posse desse resultado, partimos então para o estudo sobre a toxicidade das doenças priônicas, usando o modelo DLA e o método MCD: considerando que PrPC se converte em PrPSc quando existe contato (auto-catálise); e PrPCs são livres e podem se movimentar por uma rede, enquanto PrPScs, ou agregados de PrPScs são fixos. Confirmamos a suspeita de Eigen de que o termo mais importante nas equações de cinética priônica é o termo de Michaelis-Menten, ou termo auto-catalítico. Os resultados obtidos através das simulações MCD e modelo DLA foram comparados com os perfis de distribuições de tempos dessas doenças (BSE, vCJD e Scrapie murina). Conseguimos o ajuste de diferentes perfis de distribuição de tempos de incubação para algumas doenças priônicas, lognormal para BSE e vCJD, e lognormal com segundo pico para Scrapie murina. A auto-catálise é o mecanismo mais importante na cinética priônica, a conversão espontânea de PrPC em PrPSc pode ser negligenciada. A partir do modelo DLA, fica reforçada a hipótese de que para BSE e vCJD, doenças priônicas de ocorrência natural, a toxicidade é causada, principalmente, pela formação das placas amilóides. Para Scrapie murina, uma infecção experimentalmente induzida, a toxicidade é, possivelmente, causada por dois mecanismos: formação das placas amilóides e depleção de PrPC. Apenas com a mudança dos parâmetros iniciais e finais, conseguimos ajustar as distribuições de tempos de incubação das três doenças priônicas estudadas, apesar de o modelo ser bastante simples. A lognormalidade, de acordo com o modelo, é resultado do processo difusivo. As concentrações de PrPC devem ser baixas, menores que 1% e o número de PrPScs deve ser menor que 10 para que a lognormalidade ocorra sem a depleção de PrPC. / Prions are infectious agents responsible for a group of fatal neurodegenerative disorders. A pathogenic isoform of the prion protein (PrPSc) generated by a posttranslational process involving the conversion of alpha helices into beta sheets of the normal cellular prion protein (PrPC) is believed to be the main component of these infectious agents. The conversion of a normal PrPC into an abnormal isoform PrPSc, kinetically follows through an autocatalytic process. For better understanding of this kind of abnormal protein propagation, many analytical models have been proposed. Thus, we studied, using the Monte Carlo method, the distribution of the incubation periods in some of these neurodegenerative disorders, such as: bovine spongiform encephalopathy well known as mad cow disease (BSE), Variant Creutzfeldt Jakob disease (vCJD) and murine scrapie, an experimental murine prionic disease. The distribution of the incubation times of these diseases were considered lognormal. The aim of this study was to investigate some aspects of toxicity and replication of the prionic diseases, by comparing the results of computational simulations with the incubation times of BSE, vCJD and murine scrapie, previously established. Computational simulations, using a Dynamical Monte Carlo method (DMC) and the diffusion limited aggregation model (DLA), were worked out. At first, we evaluate the Eigen model through computational simulations using the DMC to verify the essential parameters in the kinetic of the prionic diseases. Following the results, we studied the toxicity of the prionic diseases using the DMC and the DLA model; by considering that PrPC converting in PrPSc just when exists contact (autocatalysis) and free PrPCs are allowed to diffuse randomly to their nearest neighbour sites in a square lattice, while isolated PrPScs or aggregate of PrPScs are fixed. Confirming the Eigen suspicion, the most important parameter in the equation of the prionic kinetic is the Michaelis Menten term (or the autocatalytic term). The results obtained through simulations using DMC and DLA model were compared with the time distribution profiles of the prionic diseases already established (BSE, vCJD and murine Scrapie). We get the fitting in different profiles of the distribution of the incubation periods (lognormal to BSE and vCJD and lognormal with a second peak to murine scrapie). It is concluded that autocatalysis is an essential mechanism for the prionic kinetics and the spontaneous conversion of PrPC in PrPSc can be neglected. Starting from the DLA model, is reinforced that the hypothesis for BSE and vCJD, prionic diseases of natural occurrence, the toxicity is caused, mainly, by the formation of amyloid plaques. For Scrapie murina, an experimentally induced infection, the toxicity is, possibly, caused by two mechanisms: formation of amyloid plaques and depletion of PrPC. Just with the change of the initial and final parameters, we fitted all studied prionic diseases, in spite of the model to be quite simple. The lognormality from the model, is resulting of a diffusive process. Concentrations of PrPC should be low, smaller than 1% and the number of PrPScs should be smaller than 10 for the lognormality take place without the depletion of PrPC.