This thesis studies binomial and trinomial lattice approximations in Black-Scholes type option pricing models. Also, it covers the basics of these models, derivations of model parameters by several methods under different kinds of distributions. Furthermore, the convergence of binomial model to normal distribution, Geometric Brownian Motion and Black-Scholes model isdiscussed. Finally, the connections and interrelations between discrete random variables under the Lattice approach and continuous random variables under models which follow Geometric Brownian Motion are discussed, compared and contrasted.

A new technique is evolved to give operator representation of certain polynomials.

[pt] O resgate automático, mais conhecido por autocall, de produtos estruturados vem se tornado uma prática usual no mercado internacional no que se refere à indexação dos contratos que visam a alavancagem de capital por meio de estratégias de financiamento envolvendo derivativos. O objetivo focal deste trabalho é a análise do apreçamento por meio da dinâmica do autocall em uma emissão de notas de cupom com barreira baseadas em ADRs (data inicial em 02/04/2011 e final em 02/04/2012) da maior companhia de mineração e metais do país, a Vale S.A. (VALE).Neste estudo, pretende-se explicar detalhadamente a dinâmica do autocall: como ele se estrutura; como estes são afetados pelo tempo nas datas de resgate; a probabilidade de serem resgatados (callables) em cada data; a determinação do pagamento devido na data de cupom e como esses pagamentos são dependentes de instrumentos característicos e os principais produtos estruturados com autocall. O modelo numérico de análise será pautado em uma modificação do modelo de Árvore Trinomial. Nesta variação proposta, adicionaram-se as condicionais (autocall, barreira e knock-in) e a modificação para cálculo do valor presente do produto estruturado. Logo, por meio deste estudo, busca-se colocar em prática o desenvolvimento de um novo cálculo de apreçamento envolvendo autocall, a fim de fazer a avaliação de produtos estruturados (com uma tipologia similar a esta emissão da VALE), e introduzir este assunto no meio acadêmico brasileiro já que nenhum outro trabalho lidou com este tema antes. / [en] The automatic redemption, best known as autocall, of structured products has become an usual practice in the international market with regard to the indexation of contracts that aim capital leverage through financing strategies involving derivatives. The focal goal in this work is the pricing analysis by autocall’s dynamic in an issue of coupon barrier notes based in ADRs (starts on 02/04/2011 and ends on 04/02/2012) from the largest mining and metal company in the country, Vale S.A. (VALE). This study intend to explain in details the dynamics of autocall: how it is structured, how they are affected by the time in the redemption dates; the probability of being rescued (callables) on each date; the determination of the due payment in the coupon date and how these payments are dependents from these characteristic instruments and the main structured products structured with autocall. The numerical analysis model will be based on a modification of the trinomial tree model. In this proposal variation, were added the conditionals (autocall, barrier and knock-in) and the modification to calculate the present value of the structured product. Thus, through this article, the idea is to put a new development of calculus that involves autocall pricing in practice in order to make the valuation of structured products (with a similar typology to this issue of VALE), and introduce this subject in the Brazilian academic field as any other article dealt with this subject before.

[pt] Esta dissertação visa analisar como o modelo de apreçamento de opções, utilizando o conceito de árvore trinomial implícita, pode ser aplicado no mercado acionário brasileiro, com resultados mais consistentes, se comparado ao modelo de Black-Scholes. Esse modelo incorpora o conceito de volatilidade implícita, sendo consideradas as expectativas futuras em relação ao preço de um ativo. A volatilidade implícita apresenta diferentes valores para diferentes preços de exercício ao longo do tempo. A denominação sorriso de volatilidade deve-se ao formato da curva da volatilidade implícita em função do preço de exercício. O formato do sorriso varia de acordo com o ativo-objeto da opção. Assim, a volatilidade varia ao longo tempo no cálculo da árvore, pois leva em considerando as oscilações do mercado, o que, conseqüentemente, impacta no preço do ativo e sua opção. / [en] This Paper aims to analyze how the option pricing model, using the concept of Implied Trinomial Trees can be applied to the Brazilian stock market, achieving more accurate results, if compared to the Black-Scholes model. This model includes the Implied Volatility concept, which means that future expectations are considered to price an asset. It presents different values for different Strike Prices through time. The volatility smile is named this way because of the shape of the Implied Volatility x Strike Price curve, which reminds a smile. Its shape changes according to the asset to be priced. Thus, as volatility varies with time, the option pricing using Implied Trinomial Trees is affected by the market’s oscillations, whose consequences can be observed in the asset’s price and its option price, consequently.

No description available.

Let p be a prime, p a power of p and 𝔽q the finite field with q elements. Any function φ: 𝔽q → 𝔽q can be unqiuely represented by a polynomial, 𝔽φ of degree < q. If the map x ↦ Fφ(x) induces a permutation on the underlying field we say Fφ is a permutation polynomial. Permutation polynomials have applications in many diverse fields of mathematics. In this dissertation we are generally concerned with the following question: Given a polynomial f, when does the map x ↦ F(x) induce a permutation on 𝔽q. In the second chapter we are concerned the permutation behavior of the polynomial gn,q, a q-ary version of the reversed Dickson polynomial, when the integer n is of the form n = qa - qb - 1. This leads to the third chapter where we consider binomials and trinomials taking special forms. In this case we are able to give explicit conditions that guarantee the given binomial or trinomial is a permutation polynomial. In the fourth chapter we are concerned with permutation polynomials of 𝔽q, where q is even, that can be represented as the sum of a power function and a linearized polynomial. These types of permutation polynomials have applications in cryptography. Lastly, chapter five is concerned with a conjecture on monomial graphs that can be formulated in terms of polynomials over finite fields.

No description available.

[pt] As decisões estratégicas das empresas são afetadas pelas oportunidades de investimento e as ações das suas concorrentes. Imai e Watanabe propõem um modelo de opções reais para determinar a decisão de investimento ótima de uma empresa, considerando um jogo de múltiplos estágios com duas firmas sob um processo trinomial multiperíodo em um modelo discreto. Utilizamos o modelo de Imai e Watanabe para determinar o momento estratégico ótimo para investimento em uma nova tecnologia em função da variação do custo de investimento e da demanda inicial, considerando duas empresas concorrentes no mercado brasileiro de telecomunicações, onde uma empresa é líder (L) e a outra é seguidora (S). Considerando que ambas empresas já atuam no mercado e pretendem investir em uma nova tecnologia que permitirá a expansão dos seus negócios, determinamos a curva de gatilho do custo do investimento e da demanda inicial dos serviços que delimitam a estratégia de investimento ótima da empresa líder. / [en] Corporate strategic decisions are affected by investment opportunities and actions of rival firms. Imai and Watanabe suggest a real option and game theory model to determine optimal investment decision considering a two firms multistage game following a multiperiod trinomial process in a discret model. Imai & Watanabe model is used to define this optimal time to invest in a new tecnology as a function of the cost of investment and initial demand. We consider two competing firms in the Brazilian telecommunication market where one firm is leader (L) and the other is the follower (S). We assume both firms are already active in this market and intend to invest in a new technology that will allow them to expand their business. We define a trigger curve of cost of investment and initial demand of the services that define optimal investment strategy for the leading firm.

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Barrier options are becoming more popular, mainly due to the reduced cost to hold a barrier option when compared to holding a standard call/put options, but exotic options are difficult to price since the payoff functions depend on the whole path of the underlying process, rather than on its value at a specific time instant. It is a path dependent option, which implies that the payoff depends on the path followed by the price of the underlying asset, meaning that barrier options prices are especially sensitive to volatility. For basic exchange traded options, analytical prices, based on the Black-Scholes formula, can be computed. These prices are influenced by supply and demand. There is not always an analytical solution for an exotic option. Hence it is advantageous to have methods that efficiently provide accurate numerical solutions. This study gives a literature overview and compares implementation of some available numerical methods applied to barrier options. The three numerical methods that will be adapted and compared for the pricing of barrier options are: • Binomial Tree Methods • Monte-Carlo Methods • Finite Difference Methods / Thesis (MSc (Applied Mathematics))--North-West University, Potchefstroom Campus, 2013