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published_or_final_version / abstract / toc / Mathematics / Master / Master of Philosophy

published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

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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 Mathematics / Nathan Albin / The term stochastic is defined as having a random probability distribution or pattern that may be analyzed statistically but may not be predicted precisely. A stochastic model attempts to estimate outcomes while allowing a random variation in one or more inputs over time. These models are used across a number of fields from gene expression in biology, to stock, asset, and insurance analysis in finance. In this thesis, we will build up the basic probability theory required to make an ``optimal estimate", as well as construct the stochastic integral. This information will then allow us to introduce stochastic differential equations, along with our overall model. We will conclude with the "optimal estimator", the Kalman Filter, along with an example of its application.

The goal of this paper is to extend the results of Bayraktar and Young (2006) on minimizing an individual's probability of lifetime ruin; i.e. the probability that the individual goes bankrupt before dying. We consider a scenario in which the individual is allowed to invest in both a domestic bank account and a foreign bank account, with the exchange rate between the two currencies being modeled by geometric Brownian motion. Additionally, we impose the restriction that the individual is not allowed to borrow money, and assume that the individual's wealth is consumed at a constant rate. We derive formulas for the minimum probability of ruin as well as the individual's optimal investment strategy. We also give a few numerical examples to illustrate these results.

In this dissertation, we investigate a very interesting class of quasi-linear stochastic partial differential equations. The main purpose of this article is to prove an existence result for such type of stochastic differential equations with non-standard growth conditions. The main difficulty in the present problem is that the existence cannot be easily retrieved from the well known results under Lipschitz type of growth conditions [42]. / Dissertation (MSc)--University of Pretoria, 2010. / Mathematics and Applied Mathematics / unrestricted

Within the context of rank-based interacting particle systems, we study the fluctuations in a symmetric ensemble around its stable distribution. This system is inspired by the classic Atlas model but represents its opposite pole since both the highest- and lowest-ranked particles will have non-zero drifts. In the first part of the dissertation, we derive a fine asymptotic analysis that includes a Law of Large Numbers. The lack of monotonicity of the ensemble requires that we develop alternative tools to those traditionally used in the analysis of the Atlas model. In the second part of the dissertation, we characterize the system’s fluctuations and show that, as the number of particles goes to infinity, they converge weakly to the mild solution of the Additive Stochastic Heat Equation on the real line with a symmetric initial condition. To establish this result, we use the technique proposed by Dembo and Tsai, 2017, where the Empirical Measure Process is used as a proxy for the ensemble’s fluctuations. We expect that a combination of our work, and the available knowledge about the Atlas model, could help draw a full picture of how a finite rank-based interacting particle system with a general drift structure fluctuates around its stationary distribution as the number of particles goes to infinity, a long-standing open question in the field.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI/Dissertation Abstracts International.