Made available in DSpace on 2016-04-27T16:58:29Z (GMT). No. of bitstreams: 1 Nelson Dias Leme.pdf: 5327513 bytes, checksum: 4de15af5883da758e73a3351fa8b55d1 (MD5) Previous issue date: 2007-10-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This work aims to contribute to the investigation of the impact of a constructionist approach to the use of electronic spreadsheets on the teaching and learning of topics related to Financial Mathematics. To this end, a teaching experiment was designed which involved students in the construction of their own formulas, using spreadsheets, for calculating interests and future values of investments, under regimes of both simple and compound rates. The constructionist conceptions of Papert provided a theoretical base for the development of the activities. The methodology adopted for the study was modelled according to the design-based research methodologies. Two phases of experimentation were elaborated. The first phase involves students initiating their studies in Financial Mathematics in working on a set of activities with spreadsheets, and then completing a series of paper and pencil tasks. In the second phase, to provide a basis for comparing the approach adopted with the more usual practice of giving students previously defined formulae for calculating interest, the paper and pencil task were also administered to a group of students who has previously studied Financial Mathematics. The analysis of data followed the cycle of description-execution-reflectiondebugging- description described by Valente. According to these analyses, the formulae constructed by the students and implemented on the computer served as computational models providing feedback and enabling simulations of various possible situations. This in turn allowed students to engage in a cycle of expression, evaluation and reflection of the mathematical domain in question / O objetivo deste trabalho é colaborar na investigação do impacto da abordagem construcionista e das potencialidades das planilhas eletrônicas no ensino-aprendizagem de conteúdos da Matemática Financeira. Para alcançar o objetivo proposto, foi elaborado um experimento de ensino envolvendo alunos na construção de suas próprias fórmulas, usando planilhas eletrônicas, para o cálculo dos juros e do montante, nos regimes dos juros simples e compostos. Para o desenvolvimento das atividades buscamos referência na concepção construcionista de Seymour Papert (1994). A metodologia empregada neste trabalho está baseada no design-based research methodologies . Metodologia de Pesquisa Baseada em Design. Foram desenvolvidas duas fases de experimentação. A primeira fase envolveu um grupo de alunos iniciantes seus em estudos de Matemática Financeira, em um conjunto de atividades com planilhas eletrônicas e, em uma série de tarefas em papel e lápis. Na segunda fase para comparar com o desempenho do primeiro grupo, foram aplicadas as atividades com papel e lápis a um grupo de alunos que já concluiu seus estudos de Matemática Financeira e que vivenciou uma abordagem de ensino onde as fórmulas não foram construídas. A análise empregou o ciclo descrição-execução-reflexão-depuraçãodescrição de Valente (2002). Segundo nossas análises, as fórmulas deduzidas e implementadas no computador são modelos computacionais que possibilitam o feedback e a simulação, favorecendo o envolvimento dos aprendizes no ciclo básico de expressão, avaliação e reflexão sobre o domínio considerado

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Humanities, School of Education, 2013 / Mathematics-for-teaching (MfT) is complex, multi-faceted and topic-specific. In this study, a Financial Mathematics course for pre-service secondary mathematics teachers provides a revelatory case for investigating MfT. The course was designed and taught by the author to a class of forty-two students at a university in South Africa. Eight students, forming a purposive sample, participated as members of two focus tutorial groups and took part in individual and group interviews. As an instance of insider research, the study makes use of a qualitative methodology that draws on a variety of data sources including lecture sessions and group tutorials, group and individual interviews, students’ journals, a test and a questionnaire. The thesis is structured in two parts. The first part explores revisiting of school mathematics with particular focus on compound interest and the related aspects of percentage change and exponential growth. Four cases are presented, in the form of analytic narrative vignettes which structure the analysis and provide insight into opportunities for learning MfT of compound interest. The evidence shows that opportunities may be provided to learn a range of aspects of MfT through revisiting school mathematics. The second part focuses on obstacles experienced by students in learning annuities, their time-related talk, as well as their use of mathematical resources such as timelines and spreadsheets. A range of obstacles are identified. Evidence shows that students use timelines in a range of non-standard ways but that this does not necessarily determine or reflect their success in solving annuities problems. Students’ use of spreadsheets reveals that spreadsheets are a powerful tool for working with annuities. A key finding with regard to teachers’ mathematical knowledge, and which cuts across both parts of the thesis, is the importance of being able to move between compressed and decompressed forms of mathematics. The study makes three key contributions. Firstly, a framework for MfT is proposed, building on existing frameworks in the literature. This framework is used as a conceptual tool to frame the study, and as an analytic tool to explore opportunities to learn MfT as well as the obstacles experienced by. A second contribution is the theoretical and empirical elaboration of the notion of revisiting. Thirdly, a range of theoretical constructs related to teaching and learning introductory financial mathematics are introduced. These include separate reference landscapes for the concepts of compound interest and annuities

Derivative pricing, and in particular the pricing of options, is an important area of current research in financial mathematics. Experts debate on the best method of pricing and the most appropriate model of a price process to use. In this thesis, a ``Switching Black-Scholes'' model of a price process is proposed. This model is based on the standard geometric Brownian motion (or Black-Scholes) model of a price process. However, the drift and volatility parameters are permitted to vary between a finite number of possible values at known times, according to the state of a hidden Markov chain. This type of model has been found to replicate the Black-Scholes implied volatility smiles observed in the market, and produce option prices which are closer to market values than those obtained from the traditional Black-Scholes formula. As the Markov chain incorporates a second source of uncertainty into the Black-Scholes model, the Switching Black-Scholes market is incomplete, and no unique option pricing methodology exists. In this thesis, we apply the methods of mean-variance hedging, Esscher transforms and minimum entropy in order to price options on assets which evolve according to the Switching Black-Scholes model. C programs to compute these prices are given, and some particular numerical examples are examined. Finally, filtering techniques and reference probability methods are applied to find estimates of the model parameters and state of the hidden Markov chain. / Thesis (Ph.D.)--Applied Mathematics, 2003.

<p>In this thesis the double exponential jump-diffusion model is considered and the Laplace transform is used as a method for pricing both plain vanilla and path-dependent options. The evolution of the underlying stock prices are assumed to follow a double exponential jump-diffusion model. To invert the Laplace transform, the Euler algorithm is used. The thesis includes the programme code for European options and the application to the real data. The results show how the Kou model performs on the NASDAQ OMX Stockholm Market in the case of the SEB stock.</p>

In this thesis the double exponential jump-diffusion model is considered and the Laplace transform is used as a method for pricing both plain vanilla and path-dependent options. The evolution of the underlying stock prices are assumed to follow a double exponential jump-diffusion model. To invert the Laplace transform, the Euler algorithm is used. The thesis includes the programme code for European options and the application to the real data. The results show how the Kou model performs on the NASDAQ OMX Stockholm Market in the case of the SEB stock.

In the first part of our work we focus on the model of the optimal consumption with a random income. We provide the three dimensional equation for this model, demonstrate the reduction to the two dimensional case and provide for two different utility functions the full point-symmetries' analysis of the equations. We also demonstrate that for the logarithmic utility there exists a unique and smooth viscosity solution the existence of which as far as we know was never demonstrated before. In the second part of our work we develop the concept of the empirical liquidity measure. We provide the retrospective view of the works on this issue, discuss the proposed definitions and develop our own empirical measure based on the intuitive mathematical model and comprising several features of the definitions that existed before. Then we verify the measure provided on the real data from the market and demonstrate the advantages of the proposed value for measuring the illiquidity.

One of the major problem faced by banks is how to manage the risk exposure in large portfolios. According to Basel II regulation banks has to measure the risk using Value-at-Risk with confidence level 99%. However, this regulation does not specify the way to calculate Valueat- Risk. The easiest way to calculate Value-at-Risk is to assume that portfolio returns are normally distributed. Altough, this is the most common way to calculate Value-at-Risk, there exists also other methods. The previous crisis shows that the regular methods are unfortunately not always enough to prevent bankruptcy. This paper is devoted to compare the classical methods of estimating risk with other methods such as Cornish-Fisher Expansion (CFVaR) and assuming generalized hyperbolic distribution. To be able to do this study, we estimate the risk in a large portfolio consisting of ten stocks. These stocks are chosen from the NASDAQ 100-list in order to have highly liquid stocks (bluechips). The stocks are chosen from different sectors to make the portfolio welldiversified. To investigate the impact of dependence between the stocks in the portfolio we remove the two most correlated stocks and consider the resulting eight stock portfolio as well. In both portfolios we put equal weight to the included stocks. The results show that for a well-diversified large portfolio none of the risk measures are violated. However, for a portfolio consisting of only one highly volatile stock we prove that we have a violation in the classical methods but not when we use the modern methods mentioned above.

The problem of managing the portfolio provisions is of very high importance for any financial institution. In this paper we provide both static and dynamic models of provisions estimation for the case when the decision about provisions is made at the first moment of time subject to the absence of information and for the case of complete and incomplete information. Also the hedging strategy for the case of the defaultable market is presented in this work as another tool of reducing the risk of default. The default time is modelled as a first-passage time of a standard Brownian motion through a deterministic barrier. Some methods of numerical provision estimation are also presented.

Derivative pricing, and in particular the pricing of options, is an important area of current research in financial mathematics. Experts debate on the best method of pricing and the most appropriate model of a price process to use. In this thesis, a ``Switching Black-Scholes'' model of a price process is proposed. This model is based on the standard geometric Brownian motion (or Black-Scholes) model of a price process. However, the drift and volatility parameters are permitted to vary between a finite number of possible values at known times, according to the state of a hidden Markov chain. This type of model has been found to replicate the Black-Scholes implied volatility smiles observed in the market, and produce option prices which are closer to market values than those obtained from the traditional Black-Scholes formula. As the Markov chain incorporates a second source of uncertainty into the Black-Scholes model, the Switching Black-Scholes market is incomplete, and no unique option pricing methodology exists. In this thesis, we apply the methods of mean-variance hedging, Esscher transforms and minimum entropy in order to price options on assets which evolve according to the Switching Black-Scholes model. C programs to compute these prices are given, and some particular numerical examples are examined. Finally, filtering techniques and reference probability methods are applied to find estimates of the model parameters and state of the hidden Markov chain. / Thesis (Ph.D.)--Applied Mathematics, 2003.

Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / The computation of the greeks of an option is an important aspect of financial mathematics. The information gained from knowing the value of a greek of an option can help investors decide whether or not to hold on to or to sell their options to avoid losses or gain a profit. However, there are technical difficulties that arise from having to do this. Among them is the fact that the mathematical formula for the value some options is complex in nature and evaluating their greeks may be cumber- some. On the other hand the greek might have to be numerically estimated if the option does not posses an explicit evaluation formula. This could be a computationally expensive undertaking. Malliavin calculus offers us a solution to these problems. We can find formula that can be used in combination with Monte Carlo simulations to give results quickly and which are not computationally expensive to obtain and hence give us an degree of accuracy higher that non Malliavin calculus techniques. This thesis will develop the Malliavin calculus tools that will enable us to develop the tools which we will then use to compute the greeks of some known options.