An option, a type of a financial derivative, is a contract that creates an opportunity for a market player to avoid risks involved in investing, especially in equities. An investor desires to know the accurate value of an option before entering into a contract to buy/sell the underlying asset (stock). There are various techniques that try to simulate real market conditions in order to price or evaluate an option. However, most of them achieved limited success due to high uncertainty in price behavior of the underlying asset. In this study, I propose two new Firefly variant algorithms to compute accurate worth for European and American option contracts and compare them with popular option pricing models (such as Black-Scholes-Merton, binomial lattice, Monte-Carlo, etc.) and real market data. In my study, I have first modelled the option pricing as a multi-objective optimization problem, where I introduced the pay-off and probability of achieving that pay-off as the main optimization objectives. Then, I proposed to use a latest nature-inspired algorithm that uses the bioluminescence of Fireflies to simulate the market conditions, a first attempt in the literature. For my thesis, I have proposed adaptive weighted-sum based Firefly algorithm and non-dominant sorting Firefly algorithm to find Pareto optimal solutions for the option pricing problem. Using my algorithm(s), I have successfully computed complete Pareto front of option prices for a number of option contracts from the real market (Bloomberg data). Also, I have shown that one of the points on the Pareto front represents the option value within 1-2 % error of the real data (Bloomberg). Moreover, with my experiments, I have shown that any investor may utilize the results in the Pareto fronts for deciding to get into an option contract and can evaluate the worth of a contract tuned to their risk ability. This implies that my proposed multi-objective model and Firefly algorithm could be used in real markets for pricing options at different levels of accuracy. To the best of my knowledge, modelling option pricing problem as a multi-objective optimization problem and using newly developed Firefly algorithm for solving it is unique and novel. / October 2016

In this dissertation we first formulate the Black-Scholes equation with a tensor (or matrix) diffusion coefficient into a conservative form and present a convergence analysis for the two-dimensional Black-Scholes equation arising in the Hull-White model for pricing European options with stochastic volatility. We formulate a non-conforming Petrov-Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems defined on element edges. We show that the bilinear form of the finite element method is coercive and continuous and establish an upper bound of order O(h) on the discretization error of method, where h denotes the mesh parameter of the discretization. We then present a finite volume method for the resulting equation, based on a fitting technique proposed for a one-dimensional Black-Scholes equation. We show that the method is monotone by proving that the system matrix of the discretized equation is an M-matrix. Numerical experiments, performed to demonstrate the usefulness of the method, will be presentd.

Option is an asymmetric contract between two parties with future payoff derived from the price of underlying asset. Methods of pricing di erent types of options under more or less general assumptions have been extensively studied since the Nobel price winning works of Black and Scholes [1] and Merton [12] were published in 1973. A new way of pricing options with the use of Mellin transforms have been recently introduced by Panini and Srivastav [15] in 2004. This thesis offers a brief introduction to option pricing with Mellin transforms and a revision of some of the recent research in this field.

A closed-form pricing formula of European options is obtained by Fischer Black and Myron Scholes (1973). In such a European option, the payoff depends `linearly' on the underlying asset price at the expiration time. An power option has a payoff which depends nonlinearly on the underlying asset price at the expiration time by raising a certain exponent. In the Black-Scholes model, a closed-form formula of a power option is obtained by Esser (2004). This paper extends Esser's result to the generalized Black- Scholes model. That is, we derive a closed-form pricing formula of a power option in the case when both the interest rate and the stock volatility are time-dependent.

Esta dissertação revisa a literatura acadêmica existente sobre a teoria de opções utilizando os modelos de precificação com saltos. Os conceitos foram equalizados, a nomenclatura foi padronizada, sendo gerado um material de referência sobre o assunto. O pressuposto de lognormalidade com volatilidade constante não é aceito pelo mercado financeiro. É freqüente, no meio acadêmico, a busca de modelos que reproduzam os fenômenos observados de leptocurtose ou assimetria dos log-retornos financeiros e que possuam a mesma robustez e facilidade para manipulação analítica do consagrado modelo de Black-Scholes. Os modelos com saltos são uma alternativa para esse problema. Avaliou-se o modelo de Kou no mercado acionário brasileiro composto por um componente de difusão que segue um movimento browniano geométrico e um componente de saltos que segue um processo de Poisson com intensidade do salto descrito por uma distribuição duplamente exponencial. A simulação histórica do modelo aponta, em geral, uma superioridade preditiva do modelo, porém as dificuldades de calibração dos parâmetros e de hedge em mercados incompletos são as principais deficiências para o uso dos modelos com saltos. / This master dissertation reviews the academic literature about option pricing and hedging with jumps. The theory was equalized and the notation was standardized, becoming this document a reference document about this subject. The log-normality with constant volatility is not accepted by the market. Academics search consistent models with the same analytical capabilities like Black-Scholes? model which can support the observed leptokurtosis or asymmetry of the financial daily log-returns behavior. The jump models are an alternative to these issues. The Kou?s model was evaluated and this one consists of two parts: the first part being continuous and following a geometric Brownian motion and the second being a jump process with its jump intensity defined by a double exponential distribution. The model backtesting showed a better predictive performance of the Kou´s model against other models. However, there are some handicaps regarding to the parameters calibration and hedging.

[pt] Muitas empresas apresentam exposições, como dívidas emitidas em outras moedas, diferente de sua moeda funcional. As empresas podem se proteger contra a desvalorização de sua moeda funcional através da contratação de instrumentos financeiros para este fim. Os estudos realizados apresentam uma estratégia de tratamento a exposição de um grupo de moedas com um custo menor quando comparado a estratégias de tratamento de cada exposição de forma individual. A estratégia apresentada é a utilização de opção de uma cesta de moedas, embora a cesta possa ser de vários tipos, como cesta de índices e ações. O objetivo do trabalho é realizar um estudo da precificação de uma alternativa de proteção à exposição a um conjunto de moedas. Os resultados obtidos com a estratégia proposta são comparados com a alternativa de se realizar proteção com uma opção de compra simples, bem como é apresentado um estudo de evolução do prêmio da opção dada correlação entre as moedas que compõem a cesta de opções. Os resultados obtidos do modelo de precificação mostram que a opções para uma cesta de moedas são um instrumento adequado para tratamento de exposição à moedas, pois apresenta um custo menor quando comparado com o custo de opções de compra para cada moeda a que está exposta. / [en] Many companies have exposures, such as debts issued in other currencies that are not their functional currency. Companies can protect themselves against the devaluation of their functional currency by contracting financial instruments for this purpose. The studies carried out present a treatment strategy the exposure of a group of currencies with a lower cost when compared to treatment strategies of each exposure individually. The strategy presented is the use of a basket option of currencies, although the basket can be of several types, such as baskets option of indexes and basket option of stocks. The purpose is to carry out a study of the pricing of an alternative to protect exposure to a set of currencies. The results obtained with the proposed strategy are compared with the alternative of realizing protection with a call vanilla, as well as a study of the evolution of the premium of the option given correlation between the currencies that belong to the basket of options. The results obtained from the pricing model show that the options for a basket of currencies are an appropriate instrument for treatment of currency exposure because it presents a lower cost when compared to the cost of call options for each currency to which it is exposed.

[pt] As opções financeiras são instrumentos derivativos cada dia mais usados na gestão de risco de mercado das empresas e dos investidores. Dependendo do tipo e das características da opção escolhida, geralmente não existem soluções analíticas ao problema de apreçamento do instrumento. A simulação de Monte- Carlo é um método que, aplicado ao problema de apreçamento, possibilita uma grande flexibilidade na integração das variáveis de cálculo e uma precisão que depende do número de simulações efetuadas. As opções exóticas têm características especiais e seus valores podem ser estimados com precisão aplicando as técnicas de simulação. Esta dissertação propõe uma abordagem e aplica técnicas de cálculo no apreçamento das opções exóticas mais freqüentemente encontradas nos mercados de capitais. Os algoritmos desenvolvidos podem ser usados no estudo e valoração de casos reais. / [en] Financial options are derivatives tools each day more and more used in market and enterprise risk control systems. Depending on the option type used, it doesn`t have an analytical solution for the pricing problem. A Monte-Carlo simulation is a very flexible method, which applied to the pricing problem, allows very-easy new variable implementation and accuracy increase with the number of simulation done. Exotics options have special features and pricing them by this method gives accurate results. Thus, this study explores a pricing solution and applied techniques of quite common exotics options traded on the market. The algorithms developed can be used for pricing real cases.

Esta dissertação revisa a literatura acadêmica existente sobre a teoria de opções utilizando os modelos de precificação com saltos. Os conceitos foram equalizados, a nomenclatura foi padronizada, sendo gerado um material de referência sobre o assunto. O pressuposto de lognormalidade com volatilidade constante não é aceito pelo mercado financeiro. É freqüente, no meio acadêmico, a busca de modelos que reproduzam os fenômenos observados de leptocurtose ou assimetria dos log-retornos financeiros e que possuam a mesma robustez e facilidade para manipulação analítica do consagrado modelo de Black-Scholes. Os modelos com saltos são uma alternativa para esse problema. Avaliou-se o modelo de Kou no mercado acionário brasileiro composto por um componente de difusão que segue um movimento browniano geométrico e um componente de saltos que segue um processo de Poisson com intensidade do salto descrito por uma distribuição duplamente exponencial. A simulação histórica do modelo aponta, em geral, uma superioridade preditiva do modelo, porém as dificuldades de calibração dos parâmetros e de hedge em mercados incompletos são as principais deficiências para o uso dos modelos com saltos. / This master dissertation reviews the academic literature about option pricing and hedging with jumps. The theory was equalized and the notation was standardized, becoming this document a reference document about this subject. The log-normality with constant volatility is not accepted by the market. Academics search consistent models with the same analytical capabilities like Black-Scholes? model which can support the observed leptokurtosis or asymmetry of the financial daily log-returns behavior. The jump models are an alternative to these issues. The Kou?s model was evaluated and this one consists of two parts: the first part being continuous and following a geometric Brownian motion and the second being a jump process with its jump intensity defined by a double exponential distribution. The model backtesting showed a better predictive performance of the Kou´s model against other models. However, there are some handicaps regarding to the parameters calibration and hedging.

Les marchés financiers occupent une place prépondérante dans l’économie. La future évolution des législations dans le domaine de la finance mondiale va rendre inévitable l’introduction de frictions pour éviter les mouvements spéculatifs des capitaux, toujours menaçants d’une crise. C’est pourquoi nous nous intéressons principalement, ici, aux modèles de marchés financiers avec coûts de transaction.Cette thèse se compose de trois chapitres. Le premier établit un critère d’absence d’opportunité d’arbitrage donnant l’existence de systèmes de prix consistants, i.e. martingales évoluant dans le cône dual positif exprimé en unités physiques, pour une famille de modèles de marchés financiers en temps continu avec petits coûts de transaction.Dans le deuxième chapitre, nous montrons la convergence des ensembles de sur-réplication d’une option européenne dans le cadre de la convergence topologique des ensembles. Dans des modèles multidimensionnels avec coûts de transaction décroissants a l’ordre n−1/2, nous donnons une description de l’ensemble limite pour des modèles particuliers et en déduisons des inclusions pour les modèles généraux (modèles de KABANOV). Le troisième chapitre est dédié a l’approximation du prix d’options européennes pour des modèles avec diffusion très générale (sans coûts de transaction). Nous étudions les propriétés des pay-offs pour pouvoir utiliser au mieux l’approximation du processus de prix du sous-jacent par un processus intuitif défini par récurrence grâce aux itérations de PICARD / Financial markets play a prevailing role in the economy. The future legislation development in the field of globalfinance will unavoidably lead to friction to prevent speculative capital movements, always threatening with crisis. Thatis why we are interested in the financial market models with transaction costs.This thesis consists of three chapters. The first one establishes a criterion of absence of arbitrage opportunitiesgiving the existence of consistent price systems, i.e. martingale evolving in the dual cone expressed in physical units.The criterion holds for a family of financial market models in continuous time with small transaction costs.In the second chapter, we show the convergence of super-replication sets for a European option in the contextof the topological convergence of sets. In multivariate models with transaction costs decreasing at rate n-1/2, we give adescription of the limit set for specific models. We deduce inclusions for general models (KABANOV's models).The third chapter is dedicated to the approximation of the European option price for models with very generaldiffusion (without transaction costs). We study properties of the pay-off to make best use of the approximation of theunderlying asset price, based on PICARD iterations.

Several existing pricing models of financial derivatives as well as the effects of volatility risk are analyzed. A new option pricing model is proposed which assumes that stock price follows a diffusion process with square-root stochastic volatility. The volatility itself is mean-reverting and driven by both diffusion and compound Poisson process. These assumptions better reflect the randomness and the jumps that are readily apparent when the historical volatility data of any risky asset is graphed. The European option price is modeled by a homogeneous linear second-order partial differential equation with variable coefficients. The case of underlying assets that pay continuous dividends is considered and implemented in the model, which gives the capability of extending the results to American options. An American option price model is derived and given by a non-homogeneous linear second order partial integro-differential equation. Using Fourier and Laplace transforms an exact closed-form solution for the price formula for European call/put options is obtained.