Neste trabalho, apresentamos um método de simulação Monte Carlo para o cálculo do hedging dinâmico de opções do tipo europeia em mercados multidimensionais do tipo Browniano e livres de arbitragem. Baseado em aproximações martingales de variação limitada para as decomposições de Galtchouk-Kunita-Watanabe, propomos uma metodologia factível e construtiva que nos permite calcular estratégias de hedging puras com respeito a qualquer opção quadrado integrável em mercados completos e incompletos. Uma vantagem da abordagem apresentada aqui é a flexibilidade de aplicação do método para os critérios quadráticos de minimização do risco local e de variância média de forma geral, sem a necessidade de se considerar hipóteses de suavidade para a função payoff. Em particular, a metodologia pode ser aplicada para calcular estratégias de hedging quadráticas multidimensionais para opções que dependem de toda a trajetória dos ativos subjacentes em modelos de volatilidade estocástica e com funções payoff descontínuas. Ilustramos nossa metodologia, fornecendo exemplos numéricos dos cálculos das estratégias de hedging para opções vanilla e opções exóticas que dependem de toda a trajetória dos ativos subjacentes escritas sobre modelos de volatilidade local e modelos de volatilidade estocástica. Ressaltamos que as simulações são baseadas em aproximações para os processos de preços descontados e, para estas aproximações, utilizamos o método numérico de Euler-Maruyama aplicado em uma discretização aleatória simples. Além disso, fornecemos alguns resultados teóricos acerca da convergência desta aproximação para modelos simples em que podemos considerar a condição de Lipschitz e para o modelo de volatilidade estocástica de Heston. / In this work, we present a Monte Carlo simulation method to compute de dynamic hedging of european-type contingent claims in a multidimensional Brownian-type and arbitrage-free market. Based on bounded variation martingale approximations for the Galtchouk-Kunita- Watanabe decomposition, we propose a feasible and constructive methodology which allows us to compute pure hedging strategies with respect to any square-integrable contingent claim in complete and incomplete markets. An advantage of our approach is the exibility of quadratic hedging in full generality without a priori smoothness assumptions on the payoff function. In particular, the methodology can be applied to compute multidimensional quadratic hedgingtype strategies for fully path-dependent options with stochastic volatility and discontinuous payoffs. We illustrate our methodology, providing some numerical examples of the hedging strategies to vanilla and exotic contingent claims written on local volatility and stochastic volatility models. The simulations are based in approximations to the discounted price processes and, for these approximations, we use an Euler-Maruyama-type method applied to a simple random discretization. We also provide some theoretical results about the convergence of this approximation in simple models where the Lipschitz condition is satisfied and the Heston\'s stochastic volatility model.

<p>In the Black-Scholes model, the volatility considered being deterministic and it causes some</p><p>inefficiencies and trends in pricing options. It has been proposed by many authors that the</p><p>volatility should be modelled by a stochastic process. Heston Model is one solution to this</p><p>problem. To simulate the Heston Model we should be able to overcome the correlation</p><p>between asset price and the stochastic volatility. This paper considers a solution to this issue.</p><p>A review of the Heston Model presented in this paper and after modelling some investigations</p><p>are done on the applet.</p><p>Also the application of this model on some type of options has programmed by MATLAB</p><p>Graphical User Interface (GUI).</p>

In the Black-Scholes model, the volatility considered being deterministic and it causes some inefficiencies and trends in pricing options. It has been proposed by many authors that the volatility should be modelled by a stochastic process. Heston Model is one solution to this problem. To simulate the Heston Model we should be able to overcome the correlation between asset price and the stochastic volatility. This paper considers a solution to this issue. A review of the Heston Model presented in this paper and after modelling some investigations are done on the applet. Also the application of this model on some type of options has programmed by MATLAB Graphical User Interface (GUI).

In this paper, we investigate the problem of estimating the autocorrelation of squared returns modeled by diffusion processes with data observed at non-equi-spaced discrete times. Throughout, we will suppose that the stock price processes evolve in continuous time as the Heston-type stochastic volatility processes and the transactions arrive randomly according to a Poisson process. In order to estimate the autocorrelation at a fixed delay, the original non-equispaced data will be synchronized. When imputing missing data, we adopt the previous-tick interpolation scheme. Asymptotic property of the sample autocorrelation of squared returns based on the previous-tick synchronized data will be investigated. Simulation studies are performed and applications to real examples are illustrated.

Neste trabalho, apresentamos um método de simulação Monte Carlo para o cálculo do hedging dinâmico de opções do tipo europeia em mercados multidimensionais do tipo Browniano e livres de arbitragem. Baseado em aproximações martingales de variação limitada para as decomposições de Galtchouk-Kunita-Watanabe, propomos uma metodologia factível e construtiva que nos permite calcular estratégias de hedging puras com respeito a qualquer opção quadrado integrável em mercados completos e incompletos. Uma vantagem da abordagem apresentada aqui é a flexibilidade de aplicação do método para os critérios quadráticos de minimização do risco local e de variância média de forma geral, sem a necessidade de se considerar hipóteses de suavidade para a função payoff. Em particular, a metodologia pode ser aplicada para calcular estratégias de hedging quadráticas multidimensionais para opções que dependem de toda a trajetória dos ativos subjacentes em modelos de volatilidade estocástica e com funções payoff descontínuas. Ilustramos nossa metodologia, fornecendo exemplos numéricos dos cálculos das estratégias de hedging para opções vanilla e opções exóticas que dependem de toda a trajetória dos ativos subjacentes escritas sobre modelos de volatilidade local e modelos de volatilidade estocástica. Ressaltamos que as simulações são baseadas em aproximações para os processos de preços descontados e, para estas aproximações, utilizamos o método numérico de Euler-Maruyama aplicado em uma discretização aleatória simples. Além disso, fornecemos alguns resultados teóricos acerca da convergência desta aproximação para modelos simples em que podemos considerar a condição de Lipschitz e para o modelo de volatilidade estocástica de Heston. / In this work, we present a Monte Carlo simulation method to compute de dynamic hedging of european-type contingent claims in a multidimensional Brownian-type and arbitrage-free market. Based on bounded variation martingale approximations for the Galtchouk-Kunita- Watanabe decomposition, we propose a feasible and constructive methodology which allows us to compute pure hedging strategies with respect to any square-integrable contingent claim in complete and incomplete markets. An advantage of our approach is the exibility of quadratic hedging in full generality without a priori smoothness assumptions on the payoff function. In particular, the methodology can be applied to compute multidimensional quadratic hedgingtype strategies for fully path-dependent options with stochastic volatility and discontinuous payoffs. We illustrate our methodology, providing some numerical examples of the hedging strategies to vanilla and exotic contingent claims written on local volatility and stochastic volatility models. The simulations are based in approximations to the discounted price processes and, for these approximations, we use an Euler-Maruyama-type method applied to a simple random discretization. We also provide some theoretical results about the convergence of this approximation in simple models where the Lipschitz condition is satisfied and the Heston\'s stochastic volatility model.

Dans cette thèse, on s’intéresse à la combinaison des méthodes de réduction de variance et de réduction de la complexité de la méthode Monte Carlo. Dans une première partie de cette thèse, nous considérons un modèle de diffusion continu pour lequel on construit un algorithme adaptatif en appliquant l’importance sampling à la méthode de Romberg Statistique Nous démontrons un théorème central limite de type Lindeberg Feller pour cet algorithme. Dans ce même cadre et dans le même esprit, on applique l’importance sampling à la méthode de Multilevel Monte Carlo et on démontre également un théorème central limite pour l’algorithme adaptatif obtenu. Dans la deuxième partie de cette thèse,on développe le même type d’algorithme pour un modèle non continu à savoir les processus de Lévy. De même, nous démontrons un théorème central limite de type Lindeberg Feller. Des illustrations numériques ont été menées pour les différents algorithmes obtenus dans les deux cadres avec sauts et sans sauts. / In this thesis, we are interested in studying the combination of variance reduction methods and complexity improvement of the Monte Carlo method. In the first part of this thesis,we consider a continuous diffusion model for which we construct an adaptive algorithm by applying importance sampling to Statistical Romberg method. Then, we prove a central limit theorem of Lindeberg-Feller type for this algorithm. In the same setting and in the same spirit, we apply the importance sampling to the Multilevel Monte Carlo method. We also prove a central limit theorem for the obtained adaptive algorithm. In the second part of this thesis, we develop the same type of adaptive algorithm for a discontinuous model namely the Lévy processes and we prove the associated central limit theorem. Numerical simulations are processed for the different obtained algorithms in both settings with and without jumps.

This master's thesis focuses on the problem area of option pricing under stochastic volatility. The theoretical part includes terms that are essential for understanding the problem area of option pricing and explains particular models for both option pricing under stochastic volatility and those under constant volatility. The application of described models is performed in the practical part of the thesis. After that particular models are compared to the real data.

The thesis focuses on methods of option prices calculations using two different pricing models which are Heston and Black-Scholes models. The first part describes theory of these two models and conlcudes with a comparison of the risk-neutral measures of these two models. In the second part, the relations between input parameters and the option price generated by these models are clarified. This part ends up with an analysis of the market data and it answers the question which model predicts better.

In this thesis, we derive a closed-form approximation to the implied volatility for a European option, assuming that the underlying asset follows the generalized Heston model. A new para- meter is added to the Heston model which constructed the generalized Heston model. Based on the results in Lorig, Pagliarani and Pascucci [11], we obtain implied volatility expansions up to third-order. We conduct numerical studies to check the accuracy of our expansions. More specifically we compare the implied volatilities computed using our expansions to the results by Monte Carlo simulation method. Our numerical results show that the third-order implied volatility expansion provides a very good approximation to the true value.

In this thesis, we consider the pricing problem of an American put option. We introduce a new market model for the evolution of the underlying asset price. Our model adds a new parameter to the well known Heston model. Hence we name our model the extended Heston model. To solve the American put pricing problem we adapt the idea developed by Fouque et al. (2000) to derive the asymptotic formula. We then connect the idea developed by Medvedev and Scaillet (2010) to provide an asymptotic solution for the leading order term P0. We do numerical analysis to gain insight into the accuracy and validity of our asymptotic approximation formula.