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.

No description available.

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.