Modelo de precificação de ativos por cadeias de Markov
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Data
2018-06-15
Autores
Hashioka, Jean Akio Shida
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Editor
Universidade Estadual Paulista (Unesp)
Resumo
Este trabalho consiste em apresentar a abordagem das cadeias de Markov como ferramenta auxiliadora na prática docente da matemática no Ensino Médio, tornando o processo mais tangível à realidade dos alunos. A contextualização dos conteúdos de matrizes, sistemas lineares e probabilidade poderá ser feita com exemplos práticos do cotidiano, considerando o meio social em que vivem os estudantes, resgatando assim o desejo pela aprendizagem e pelas aplicações da matemática. Espera-se desta forma maior receptividade da disciplina por parte dos discentes e, potencialmente, melhor resposta ao aprendizado pretendido. Assim sendo, o estudo aborda, num primeiro momento, a convergência de distribuição de probabilidade de uma cadeia de Markov de dois estados por meio de limites no in nito de uma função de probabilidade. Desta primeira cadeia de Markov de dois estados, é elaborado um roteiro de aula a ser abordado como exemplo a ser trabalhado em sala de aula relacionado às probabilidades de um time de futebol vencer as suas próximas partidas. Prosseguindo, observa-se a aplicabilidade das cadeias de Markov para calcular a distribuição de probabilidades de um jogador estar perdido em diferentes salas de um labirinto para cada tentativa de encontrar a saída. A m de evidenciar outro exemplo de aplicação das cadeias de Markov, há a construção de um modelo de preci cação de ativos com o objetivo de prever os preços de algumas ações de empresas negociadas na BM&FBOVESPA, a bolsa de valores do Brasil. Tal modelo de preci cação de ativos mostrou-se adequado estatisticamente como ferramenta de análise e cálculo dos retornos médios esperados de alguns dos ativos estudados. Por meio do conteúdo apresentado neste estudo, espera-se contribuir com o aprofundamento de alguns recursos e conceitos para a prática docente com aulas sobre cadeias de Markov no Ensino Médio. Esses aspectos direcionam esta pesquisa para um relevante processo de desenvolvimento do raciocínio, senso crítico e tomada de decisões em situações progressivamente mais complexas vividas pelos alunos.
This work presents the Markov chain approach as a useful tool in the teaching practice of mathematics in High School, making the process more tangible to the students' reality. The contextualization of matrix contents, linear systems and probability can be done with practical examples of daily life, considering the social environment in which students live, thus recovering the desire for learning and the applications of mathematics. It is expected in this way more receptivity of the discipline on the part of the students and, potentially, better response to the intended learning. Thus, the study addresses, rst, the convergence of probability distribution of a two-state Markov chain by means of in nite limits of a probability function. From this rst Markov chain of two states, a lesson script is elaborated to be approached as example to be worked in classroom related to the probabilities of a soccer team to win its next matches. Proceeding, we observe the applicability of Markov chains to calculate the probability distribution of a player being lost in di erent rooms of a maze for each attempt to nd the exit. In order to highlight another example of the application of the Markov chains, an asset pricing model is designed to predict the prices of some shares of companies traded on BM&FBOVESPA, the Brazilian stock exchange. Such an asset pricing model proved to be statistically adequate as a tool for analysis and calculation of the expected average returns of some of the assets studied. Through the content presented in this study, it is hoped to contribute with the deepening of some resources and concepts for the teaching practice with classes on Markov chains in High School. These aspects direct this research to a relevant process of development of reasoning, critical sense and decision making in progressively more complex situations experienced by students.
This work presents the Markov chain approach as a useful tool in the teaching practice of mathematics in High School, making the process more tangible to the students' reality. The contextualization of matrix contents, linear systems and probability can be done with practical examples of daily life, considering the social environment in which students live, thus recovering the desire for learning and the applications of mathematics. It is expected in this way more receptivity of the discipline on the part of the students and, potentially, better response to the intended learning. Thus, the study addresses, rst, the convergence of probability distribution of a two-state Markov chain by means of in nite limits of a probability function. From this rst Markov chain of two states, a lesson script is elaborated to be approached as example to be worked in classroom related to the probabilities of a soccer team to win its next matches. Proceeding, we observe the applicability of Markov chains to calculate the probability distribution of a player being lost in di erent rooms of a maze for each attempt to nd the exit. In order to highlight another example of the application of the Markov chains, an asset pricing model is designed to predict the prices of some shares of companies traded on BM&FBOVESPA, the Brazilian stock exchange. Such an asset pricing model proved to be statistically adequate as a tool for analysis and calculation of the expected average returns of some of the assets studied. Through the content presented in this study, it is hoped to contribute with the deepening of some resources and concepts for the teaching practice with classes on Markov chains in High School. These aspects direct this research to a relevant process of development of reasoning, critical sense and decision making in progressively more complex situations experienced by students.
Descrição
Palavras-chave
Cadeias de Markov, Probabilidade, Mercado financeiro, Markov chains, Probability, Financial market