In this thesis, we present and analyze three algorithms that are designed to make computer simulation more efficient, valid, and/or applicable. The first algorithm uses simulation cloning to enhance efficiency in transient simulation. Traditional simulation cloning is a technique that shares some parts of the simulation results when simulating different scenarios. We apply this idea to transient simulation, where multiple replications are required to achieve statistical validity. Computational savings are achieved by sharing some parts of the simulation results among several replications. We improve the algorithm by inducing negative correlation to compensate for the (undesirable) positive correlation introduced by sharing some parts of the simulation. Then we identify how many replications should share the same data, and provide numerical results to analyze the performance of our approach. The second algorithm chooses a set of best systems when there are multiple candidate systems and multiple objectives. We provide three different formulations of correct selection of the Pareto optimal set, where a system is Pareto optimal if it is not inferior in all objectives compared to other competing systems. Then we present our Pareto selection algorithm and prove its validity for all three formulations. Finally, we provide numerical results aimed at understanding how well our algorithm performs in various settings. Finally, we discuss the estimation of input distributions when theoretical distributions do not provide a good fit to existing data. Our approach is to use a quasi-empirical distribution, which is a mixture of an empirical distribution and a distribution for the right tail. We describe an existing approach that involves an exponential tail distribution, and adapt the approach to incorporate a Pareto tail distribution and to use a different cutoff point between the empirical and tail distributions. Then, to measure the impact, we simulate a stable M/G/1 queue with a known inter-arrival and unknown service time distributions, and estimate the mean and tail probabilities of the waiting time in queue using the different approaches. The results suggest that if we know that the system is stable, and suspect that the tail of the service time distribution is not exponential, then a quasi-empirical distribution with a Pareto tail works well, but with a lower bound imposed on the tail index.

In this thesis, the credit worthiness of a company is modelled using a stochastic process. Two credit models are considered; Merton's model, which models the value of a firm's assets using geometric Brownian motion, and the distance to default model, which is driven by a two factor jump diffusion process. The probability of default and the default time are simulated using Monte Carlo and the number of scenarios needed to obtain convergence in the simulations is investigated. The simulations are performed using the probability matrix method (PMM), which means that a transition probability matrix describing the process is created and used for the simulations. Besides this, two variance reduction techniques are investigated; importance sampling and antithetic variates. / I den här uppsatsen modelleras kreditvärdigheten hos ett företag med hjälp av en stokastisk process. Två kreditmodeller betraktas; Merton's modell, som modellerar värdet av ett företags tillgångar med geometrisk Brownsk rörelse, och "distance to default", som drivs av en två-dimensionell stokastisk process med både diffusion och hopp. Sannolikheten för konkurs och den förväntade tidpunkten för konkurs simuleras med hjälp av Monte Carlo och antalet scenarion som behövs för konvergens i simuleringarna undersöks. Vid simuleringen används metoden "probability matrix method", där en övergångssannolikhetsmatris som beskriver processen används. Dessutom undersöks två metoder för variansreducering; viktad simulering (importance sampling) och antitetiska variabler (antithetic variates).

日前由於金融海嘯的發生，導致全球金融市場對於結構債此種收益較高的商品存在眾多疑點，然而究其原因乃投資人無法正確地了解到自身的風險屬性，盲目地追求高收益率，反而造成投資結果不盡理想。本文將應用模型來推導商品的理論價格，並深入分析結構債可能帶來的風險與報酬。 本文兩個案商品之連動標的分別為利率與股權。第一個個案商品為英國勞埃德TSB銀行所發行之十年期累積計息利率連動債，在評價上將採用LIBOR市場模型，利用市場上既有的資料校準模型所需參數與期初遠期利率；此外由於本商品具有提前贖回特性，故將應用最小平方法蒙地卡羅來找出該商品發行之期初價格，並分別就發行機構探討其避險策略及投資人應注意的風險作深入分析。 第二個個案商品為摩根大通公司所發行之附部分保本之股權連結式自動贖回債券，利用對數常態股價模型及蒙地卡羅法評價出其理論價格，並針對發行機構可能面對之風險與避險策略作完整探討，最後就投資人之報酬及風險層面作詳盡地剖析。

Les fichiers qui accompagnent mon document ont été réalisés avec le logiciel Latex et les simulations ont été réalisés par Splus(R). / Dans ce mémoire, nous travaillons sur une méthode de simulation de Monte-Carlo qui utilise des variables antithétiques pour estimer un intégrale de la fonction f(x) sur un intervalle (0,1] où f peut être une fonction monotone, non-monotone ou une autre fonction difficile à simuler. L'idée principale de la méthode qu'on propose est de subdiviser l'intervalle (0,1] en m sections dont chacune est subdivisée en l sous intervalles. Cette technique se fait en plusieurs étapes et à chaque fois qu'on passe à l'étape supérieure la variance diminue. C'est à dire que la variance obtenue à la kième étape est plus petite que celle trouvée à la (k-1)ième étape ce qui nous permet également de rendre plus petite l'erreur d’estimation car l'estimateur de l'intégrale de f(x) sur [0,1] est sans biais. L'objectif est de trouver m, le nombre optimal de sections, qui permet de trouver cette diminution de la variance. / In this master thesis, we consider simulation methods based on antithetic variates for estimate integrales of f(x) on interval (0,1] where f is monotonic function, not a monotonic function or a function difficult to integrate. The main idea consists in subdividing the (0,1] in m sections of which each one is subdivided in l subintervals. This method is done recursively. At each step the variance decreases, i.e. The variance obtained at the kth step is smaller than that is found at the (k-1)th step. This allows us to reduce the error in the estimation because the estimator of integrales of f(x) on interval [0,1] is unbiased. The objective is to optimize m.

[pt] Nesta tese, consideramos um modelo de espaço de estado bivariado para dados de contagem. A abordagem usada para resolver integrais não-analíticas que se apresentam no modelo é uma natural extensão da metodologia proposta por Durbin e Koopman - (DK), no sentido de que o Modelo Gaussiano Aproximador deve possuir algumas matrizes de covariâncias diagonais. Esta modificação traz a vantagem de viabilizar o uso do tratamento univariado para séries multivariadas com as recursões de Kalman, o qual, como se sabe, é mais eficiente do que o tratamento usual e facilita o uso de inicializações exatas destas mesmas recursões. O vetor de estado do modelo proposto é definido usando-se abordagem estrutural, onde os elementos do vetor de estado têm interpretação direta como tendência e sazonalidade. Apresentamos exemplos simulados e reais para ilustrar o modelo. / [en] In this thesis we consider a state space model for bivariate observations of count data. The approach used to solve the non analytical integrals that appears as the solution of the resulting non-Gaussian filter is a natural extension of the methodology advocated by Durbin and Koopman (DK). In our approach the aproximated Gaussian Model (AGM), has a diagonal Covariance matrix, while in the original DK, this is a full matrix. This modification make it possible to use univariate Kalman recursoes to construct the AGM, resulting in a computationally more efficient solution for the estimation of a Bivariate Poisson model. This also facilitates the use of exact initialization of those recursions. The state vector is specified using the structural approach, where the state elements are components which have direct interpretation, such as trend and seasonals. In our bivariate set up the dependence between the bivariate vector of time series is accomplished by use of common components which drive both series. We present both simulation and real life examples illustrating the use of our model.

Dans ce mémoire, nous travaillons sur une méthode de simulation de Monte-Carlo qui utilise des variables antithétiques pour estimer un intégrale de la fonction f(x) sur un intervalle (0,1] où f peut être une fonction monotone, non-monotone ou une autre fonction difficile à simuler. L'idée principale de la méthode qu'on propose est de subdiviser l'intervalle (0,1] en m sections dont chacune est subdivisée en l sous intervalles. Cette technique se fait en plusieurs étapes et à chaque fois qu'on passe à l'étape supérieure la variance diminue. C'est à dire que la variance obtenue à la kième étape est plus petite que celle trouvée à la (k-1)ième étape ce qui nous permet également de rendre plus petite l'erreur d’estimation car l'estimateur de l'intégrale de f(x) sur [0,1] est sans biais. L'objectif est de trouver m, le nombre optimal de sections, qui permet de trouver cette diminution de la variance. / In this master thesis, we consider simulation methods based on antithetic variates for estimate integrales of f(x) on interval (0,1] where f is monotonic function, not a monotonic function or a function difficult to integrate. The main idea consists in subdividing the (0,1] in m sections of which each one is subdivided in l subintervals. This method is done recursively. At each step the variance decreases, i.e. The variance obtained at the kth step is smaller than that is found at the (k-1)th step. This allows us to reduce the error in the estimation because the estimator of integrales of f(x) on interval [0,1] is unbiased. The objective is to optimize m. / Les fichiers qui accompagnent mon document ont été réalisés avec le logiciel Latex et les simulations ont été réalisés par Splus(R).

次級房貸風暴使得包裝複雜的衍生性金融商品紛紛遭受波及後，目前結構型金融商品的條款設計將朝簡單化和透明化的趨勢發展，有助於全球金融市場的效率性、完整性與穩定性。本文從市場上選擇具代表性的利率結構型商品，應用模型來推導商品的價格，並深入分析商品的報酬與風險型態。 本文分析的個案商品為全球知名的匯豐銀行所發行之十年期「固定期限交換利率利差連動債券」，在評價上將採用LIBOR市場模型，利用市場上既有的資料求算出期初遠期利率，並校準模型所需的參數化波動度函數與相關係數函數，建立與市場一致的利率期間結構與利率波動度期間結構。模擬路徑時應用最小平方法蒙地卡羅來求得該商品發行之期初價格，此外，亦採用反向變異法加速收斂效果，並針對商品的條款設計作拆解與分析。最後，本文探討了發行機構發行商品之風險與避險策略，並且從投資人之報酬及風險層面作詳盡地剖析。