In this thesis, we provide several applications of Gram-Charlier expansions in derivative pricing. We first give an exposition on how to calculate swaption prices under the the CIR2 model. Then we extend this method to CIR2++ model. We also develop a procedure to calculate European call options under Heston’s model of stochastic volatility by Gram-Charlier Expansions.

In this thesis, we provide several applications of Gram-Charlier expansions in derivative pricing. We first give an exposition on how to calculate swaption prices under the the CIR2 model. Then we extend this method to CIR2++ model. We also develop a procedure to calculate European call options under Heston’s model of stochastic volatility by Gram-Charlier Expansions.

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Black-Scholes Model is a widely used mathematical model for stock price behaviors, of which the return is assumed to be normally distributed. But this 'normally distributed' assumption is doubted and proved to be not true by realistric data. The main goal of this thesis is to explore polynomial-normal distribution, and use this distribution in the stock return, as a non-normal extension of the Black-Scholes Model. We will develop the properties of polynomial-normal distribtuion in the thesis, and also give the European call and put option price formulas under this model, and show how to use this model to estimate real stock returns.

Neste trabalho propomos a utilização do método da função característica empírica (ECF - empirical characteristic function), para estimação do modelo de duração condicional estocástica (SCD - stochastic conditional duration). Para determinação das variáveis latentes do processo utilizamos três alternativas: um filtro de Kalman, um filtro obtido por integração numérica e um filtro baseado na expansão de Gram-Charlier até 4ª ordem. Os resultados são então aplicados em séries de duração da GE, Microsoft e USD/EUR. / We propose the use of the empirical characteristic function (ECF) method to estimate the parameters of the stochastic conditional duration (SCD) model. In order to estimate the latent variables we propose the use of three alternatives: a Kalman filter, a filter based on numerical integration (quadrature) and a filter based on the 4th-order Gram-Charlier expansion. The results are applied to the estimation of the parameters of the duration process for GE, Microsoft and USD/EUR.

Neste trabalho propomos a utilização do método da função característica empírica (ECF - empirical characteristic function), para estimação do modelo de duração condicional estocástica (SCD - stochastic conditional duration). Para determinação das variáveis latentes do processo utilizamos três alternativas: um filtro de Kalman, um filtro obtido por integração numérica e um filtro baseado na expansão de Gram-Charlier até 4ª ordem. Os resultados são então aplicados em séries de duração da GE, Microsoft e USD/EUR. / We propose the use of the empirical characteristic function (ECF) method to estimate the parameters of the stochastic conditional duration (SCD) model. In order to estimate the latent variables we propose the use of three alternatives: a Kalman filter, a filter based on numerical integration (quadrature) and a filter based on the 4th-order Gram-Charlier expansion. The results are applied to the estimation of the parameters of the duration process for GE, Microsoft and USD/EUR.

Située au centre du triangle formé par la France, les Pays-Bas et l’Allemagne, la Belgique est au cœur du réseau de diffusion des tendances conceptuelles dans les années 1970. La scène artistique belge est cependant sous-représentée dans les études consacrées à ce sujet. Il s’agit donc de montrer la place occupée par les travaux réalisés par Jacques Charlier, Jacques Lizène, Schwind et Alain D’Hooghe entre 1969 et 1974, en les analysant à la lumière des œuvres, mais aussi des textes qui ont contribué à la définition de l’art conceptuel. Cette approche permet de voir qu’ils en utilisent les codes afin de mettre en évidence leur position extérieure à ce réseau, tout en démontrant leur connaissance des débats provoqués par cette nouvelle manière de définir l’art. Au moyen des stratégies subversives puisées dans l’histoire du surréalisme belge, ils y participent en testant dans la pratique les limites de cette définition. Reconsidérer leur situation par rapport aux centres amène ainsi à réévaluer la position occupée par les travaux de ces artistes au sein des conceptualismes européens / Located in the centre of the geographical triangle formed by France, the Netherlands and Germany, Belgium is a merging point in the network that developed the conceptual art trends in the Seventies. The Belgian art scene is, however, under-represented in the critical and academic studies on this topic. The aim of this PhD thesis is therefore to show the relevance of the works by outstanding Belgian artists - Jacques Charlier, Jacques Lizène, Schwind and Alain D'Hooghe - between 1969 and 1974 in the light of those of conceptual art, but also in relation to the essays that have contributed to define it. Although their awareness of the international debate and even if they made use of the conceptual art codes, this comparative approach allows to underline how the above mentioned Belgian artists positioned their practices and how they operated independently from the main conceptual art network. Through subversive strategies inherited from Belgian surrealism, their works contributed to challenge the limit of the notion of conceptual art. To reconsider their stance leads to reassess the position of their work in the scope of the European conceptualisms

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In praktischen Aufgabenstellungen können zur Modellierung zufälliger Einflüsse, welche sich durch schwache Abhängigkeiten auszeichnen, schwach korrelierte zufällige Funktionen genutzt werden. Die nähere Untersuchung von Funktionalen schwach korrelierter zufälliger Funktionen ist durch die Gestalt der Lösungen von praktischen Fragestellungen motiviert. Die stochastischen Charakteristiken dieser Lösungen lassen sich im Allgemeinen nicht exakt bestimmen, so dass auf Approximationsverfahren zurückgegriffen werden muss. Diese stehen im Mittelpunkt der Dissertation. Zu Beginn werden Entwicklungen von Momenten und Kumulanten der betrachteten linearen Integralfunktionale schwach korrelierter Prozesse nach der Korrelationslänge des Prozesses hergeleitet und eine Vermutung über die exakte Darstellung der Kumulanten formuliert. Für Integralfunktionale von schwach korrelierten Simulationsprozessen, welche aus der Interpolation von Moving-Average-Prozessen entstehen, werden die definierten Charakteristiken hergeleitet. Außerdem steht die Approximation der unbekannten Dichtefunktion im Fokus der Arbeit. Es werden verschiedene Zugänge genutzt. Eine alternative Herleitung zur bereits in der Literatur untersuchten Gram-Charlier-Entwicklung wird in Form der Edgeworth-Entwicklung angegeben. Des Weiteren werden die Sattelpunkt-Approximation und die Maximum-Entropie-Methode untersucht und anhand von Simulationsergebnissen für Integralfunktionale von Simulationsprozessen miteinander verglichen. / In engineering applications stochastic influences which are characterized by weak dependencies can be modelled, among others, by weakly correlated random functions. The solutions of such problems shape up as integral functionals of weakly correlated random functions which motivates more detailed investigations. In general the exact calculation of stochastic characteristics of such integral functionals is impossible so that we have to be content with approximation methods this thesis focuses on. At the beginning expansions of moments and cumulants of linear integral functionals of weakly correlated random processes with respect to the correlation length are considered and an explicit formula of cumulants is conjectured. For integral functionals of weakly correlated random simulation processes, defined as interpolations of moving average processes, the required expansion coefficients are derived. Furthermore the approximation of the unknown probability density is requested. In the thesis there are different approaches used. First we state an alternative way to achieve the already known Gram Charlier approximation by means of Edgeworth expansion. Then we study two further methods, namely the saddlepoint approximation and the maximum entropy method and compare them on the basis of simulation results for integral functionals of simulation processes.

In praktischen Aufgabenstellungen können zur Modellierung zufälliger Einflüsse, welche sich durch schwache Abhängigkeiten auszeichnen, schwach korrelierte zufällige Funktionen genutzt werden. Die nähere Untersuchung von Funktionalen schwach korrelierter zufälliger Funktionen ist durch die Gestalt der Lösungen von praktischen Fragestellungen motiviert. Die stochastischen Charakteristiken dieser Lösungen lassen sich im Allgemeinen nicht exakt bestimmen, so dass auf Approximationsverfahren zurückgegriffen werden muss. Diese stehen im Mittelpunkt der Dissertation. Zu Beginn werden Entwicklungen von Momenten und Kumulanten der betrachteten linearen Integralfunktionale schwach korrelierter Prozesse nach der Korrelationslänge des Prozesses hergeleitet und eine Vermutung über die exakte Darstellung der Kumulanten formuliert. Für Integralfunktionale von schwach korrelierten Simulationsprozessen, welche aus der Interpolation von Moving-Average-Prozessen entstehen, werden die definierten Charakteristiken hergeleitet. Außerdem steht die Approximation der unbekannten Dichtefunktion im Fokus der Arbeit. Es werden verschiedene Zugänge genutzt. Eine alternative Herleitung zur bereits in der Literatur untersuchten Gram-Charlier-Entwicklung wird in Form der Edgeworth-Entwicklung angegeben. Des Weiteren werden die Sattelpunkt-Approximation und die Maximum-Entropie-Methode untersucht und anhand von Simulationsergebnissen für Integralfunktionale von Simulationsprozessen miteinander verglichen. / In engineering applications stochastic influences which are characterized by weak dependencies can be modelled, among others, by weakly correlated random functions. The solutions of such problems shape up as integral functionals of weakly correlated random functions which motivates more detailed investigations. In general the exact calculation of stochastic characteristics of such integral functionals is impossible so that we have to be content with approximation methods this thesis focuses on. At the beginning expansions of moments and cumulants of linear integral functionals of weakly correlated random processes with respect to the correlation length are considered and an explicit formula of cumulants is conjectured. For integral functionals of weakly correlated random simulation processes, defined as interpolations of moving average processes, the required expansion coefficients are derived. Furthermore the approximation of the unknown probability density is requested. In the thesis there are different approaches used. First we state an alternative way to achieve the already known Gram Charlier approximation by means of Edgeworth expansion. Then we study two further methods, namely the saddlepoint approximation and the maximum entropy method and compare them on the basis of simulation results for integral functionals of simulation processes.