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Made available in DSpace on 2014-06-11T19:23:32Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-12-13Bitstream added on 2014-06-13T18:09:47Z : No. of bitstreams: 1 catalao_ab_me_ift.pdf: 811288 bytes, checksum: d4e34c59801bd92233bc9f26884a19ab (MD5) / Neste trabalho estudamos a calibração de opções de câmbio no mercado brasileiro utilizando o processo estocástico proposto por Heston [Heston, 1993], como uma alternativa ao modelo de apreçamento de Black e Scholes [Black e Scholes,1973], onde as volatilidades implícitas de opções para diferentes preços de exercícios e prazos são incorporadas ad hoc. Comparamos dois métodos de apreçamento: o método de Carr e Madan [Carr e Madan, 1999], que emprega transfomada rápida de Fourier e função característica, e expansão assintótica para baixos valores de volatilidade da variância. Com a nalidade de analisar o domínio de aplicabilidade deste método, selecionamos períodos de alta volatilidade no mercado, correspondente à crise subprime de 2008, e baixa volatilidade, correspondente ao período subsequente. Adicionalmente, estudamos a incorporação de swaps de variância para melhorar a calibração do modelo / In this work we study the calibration of forex call options in the Brazilian market using the stochastic process proposed by Heston [Heston, 1993], as an alternative to the Black and Scholes [Black e Scholes,1973] pricing model, in which the implied option volatilities related to di erent strikes and maturities are incorporated in an ad hoc manner. We compare two pricing methods: one from Carr and Madan [Carr e Madan, 1999], which uses fast Fourier transform and characteristic function, and asymptotic expantion for low values of the volatility of variance. To analyze the applicability of this method, we select periods of high volatility in the market, related to the subprime crisis of 2008, and of low volatility, correspondent to the following period. In addition, we study the use of variance swaps to improve the calibration of the model

Master of Science / Department of Statistics / Abigail Jager / A discrete-time Markov chain with stationary transition probabilities is often used for the purpose of investigating treatment programs and health care protocols for chronic disease. Suppose the patients of a certain chronic disease are observed over equally spaced time intervals. If we classify the chronic disease into n distinct health states, the movement through these health states over time then represents a patient’s disease history. We can use a discrete-time Markov chain to describe such movement using the transition probabilities between the health states. The purpose of this study was to investigate the case when the observation interval coincided with the cycle length of the Markov chain as well as the case when the observational interval and the cycle length did not coincide. In particular, we are interested in how the estimated transition matrix behaves as the ratio of observation interval and cycle length changes. Our results suggest that more estimation problems arose for small sample sizes as the length of observational interval increased, and that the deviation from the known transition probability matrix got larger as the length of observational interval increased. With increasing sample size, there were fewer estimation problems and the deviation from the known transition probability matrix was reduced.

Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version of this is known to be true as well, the limit process being a stable L´evy process. In this thesis, we developed an explicit construction for the α-stable L´evy process motion with values in D([0, 1]), by considering the cases α < 1 and α > 1. The case α < 1 is the simplest since we can work with the uniform topology of the sup-norm on D([0, 1]) and the construction follows more or less by classical techniques. The case α > 1 required more work. In particular, we encountered two problems : one was related to the construction of a modification of this process (for all time), which is right-continuous and has left-limit with respect to the J1 topology. This problem was solved by using the Itob-Nisio theorem. The other problem was more difficult and we only managed to solve it by developing a criterion for tightness of probability measures on the space of cadlag fonction on [0, T] with values in D([0, 1]), equipped with a generalization of Skorohod’s J1 topology. In parallel with the construction of the infinite-dimensional process Z, we focus on the functional extension of Roueff and Soulier [29]. This part of the thesis was completed using the method of point process, which gave the convergence of the truncated sum. The case α > 1 required more work due to the presence of centering. For this case, we developed an ad-hoc result regarding the continuity of the addition for functions on [0, T] with values in D([0, 1]), which was tailored for our problem.

A contextualização de eventos próximos da realidade dos alunos aliada a utilização da informática como ferramenta auxiliar no aprendizado da probabilidade, pode ser um dos caminhos para a melhoria do ensino de Matemática. Assim, este trabalho buscou a modelagem matemática de eventos temporais do dia a dia dos alunos do ensino básico. A modelagem se baseou no conceito de Cadeias de Markov e teve o objetivo de auxiliar o professor dos ensinos fundamental e médio a introduzir o conceito de probabilidade. As aplicações das Cadeias de Markov também possibilitam apresentar aos alunos dos ensinos médio e fundamental como a Matemática pode resolver problemas do cotidiano. Para introduzir os conceitos de Cadeias de Markov foi necessário uma revisão teórica dos conceitos da teoria da probabilidade e os conceitos de Cadeias de Markov foram estudados em literatura em língua inglesa. Considerando o interesse e curiosidade demonstrado pelos alunos em experiência prévia com o material, as atividades mostraram-se muito eficientes. Espera-se que esse trabalho possa contribuir para a prática docente de outros professores. / The contextualization of events close to the reality of the students allied to the use of information technology as an auxiliary tool in the learning of probability, can be one of the ways to improve the teaching of Mathematics. Thus, this paper sought the mathematical modeling of temporal events from the daily of students of basic Education. The modeling was based on the concept of Markov Chains and aimed to help the middle and high school teachers to introduce the concept of probability. The applications of the Markov Chains also make it possible to present to the students of the middle and high school teachings how Mathematics can solve daily problems. To introduce the concepts of Markov Chains, a theoretical revision of the concepts of probability theory was necessary and the concepts of Markov Chains were studied in literature in English Language. Considering the interest and curiosity demonstrated by the students in previous experience with the material, the activities were very efficient. It is hoped that this paper may contribute to the teaching practice of other teachers.

The stochastic process algebra PEPA is a powerful modelling formalism for concurrent systems, which has enjoyed considerable success over the last decade. Such modelling can help designers by allowing aspects of a system which are not readily tested, such as protocol validity and performance, to be analysed before a system is deployed. However, model construction and analysis can be challenged by the size and complexity of large scale systems, which consist of large numbers of components and thus result in state-space explosion problems. Both structural and quantitative analysis of large scale PEPA models suffers from this problem, which has limited wider applications of the PEPA language. This thesis focuses on developing PEPA, to overcome the state-space explosion problem, and make it suitable to validate and evaluate large scale computer and communications systems, in particular a content adaption framework proposed by the Mobile VCE. In this thesis, a new representation scheme for PEPA is proposed to numerically capture the structural and timing information in a model. Through this numerical representation, we have found that there is a Place/Transition structure underlying each PEPA model. Based on this structure and the theories developed for Petri nets, some important techniques for the structural analysis of PEPA have been given. These techniques do not suffer from the state-space explosion problem. They include a new method for deriving and storing the state space and an approach to finding invariants which can be used to reason qualitatively about systems. In particular, a novel deadlock-checking algorithm has been proposed to avoid the state-space explosion problem, which can not only efficiently carry out deadlock-checking for a particular system but can tell when and how a system structure lead to deadlocks. In order to avoid the state-space explosion problem encountered in the quantitative analysis of a large scale PEPA model, a fluid approximation approach has recently been proposed, which results in a set of ordinary differential equations (ODEs) to approximate the underlying CTMC. This thesis presents an improved mapping from PEPA to ODEs based on the numerical representation scheme, which extends the class of PEPA models that can be subjected to fluid approximation. Furthermore, we have established the fundamental characteristics of the derived ODEs, such as the existence, uniqueness, boundedness and nonnegativeness of the solution. The convergence of the solution as time tends to infinity for several classes of PEPA models, has been proved under some mild conditions. For general PEPA models, the convergence is proved under a particular condition, which has been revealed to relate to some famous constants of Markov chains such as the spectral gap and the Log-Sobolev constant. This thesis has established the consistency between the fluid approximation and the underlying CTMCs for PEPA, i.e. the limit of the solution is consistent with the equilibrium probability distribution corresponding to a family of underlying density dependent CTMCs. These developments and investigations for PEPA have been applied to both qualitatively and quantitatively evaluate the large scale content adaptation system proposed by the Mobile VCE. These analyses provide an assessment of the current design and should guide the development of the system and contribute towards efficient working patterns and system optimisation.

This article investigates several variance reduction techniques in Monte Carlo simulation applied in option pricing. It first shows how Monte Carlo simulation could be leveraged in the field of option pricing by demonstrating the quality of Monte Carlo methods and properties of stock options. Then the articles simulate stock price trajectories to infer the optimal option price by averaging the payoff at maturity. The article shows in depth the effect of control variates and antithetic variates, and importance sampling in reducing variance. The last part of the article shows how the same variance reduction techniques could be used in more exotic options such as Asian and Bermuda options. In these cases, their closed-form expressions are more difficult to derive compared to the European options, and thus simulation is widely practiced in the industry.

Adaptive radiation therapy re-optimizes treatment plans based on updated tumor geometries from magnetic resonance imaging scans. However, the imaging process is costly in labor and equipment. In this study, we develop a mathematical model that describes tumor evolution based on a Markov assumption. We then extend the model to predict tumor evolution with any level of information from a new patient: weekly MRI scans are used to estimate transition probabilities when available, otherwise historical MRI scans are used. In the latter case, patients in the historical data are clustered into two groups, and the model relates the new patient's behavior to the existing two groups. The models are evaluated with 33 cervical cancer patients from Princess Margaret Cancer Centre. The result indicates that our models outperform the constant volume model, which replicates the current clinical practice.

Adaptive radiation therapy re-optimizes treatment plans based on updated tumor geometries from magnetic resonance imaging scans. However, the imaging process is costly in labor and equipment. In this study, we develop a mathematical model that describes tumor evolution based on a Markov assumption. We then extend the model to predict tumor evolution with any level of information from a new patient: weekly MRI scans are used to estimate transition probabilities when available, otherwise historical MRI scans are used. In the latter case, patients in the historical data are clustered into two groups, and the model relates the new patient's behavior to the existing two groups. The models are evaluated with 33 cervical cancer patients from Princess Margaret Cancer Centre. The result indicates that our models outperform the constant volume model, which replicates the current clinical practice.