Utbetalningsbelopp för skadeersättning studeras. Datamaterialet uppvisar skevheter och ''tjocka svansar'' vilket motiverar att modellera med en mer flexibel fördelning än normalfördelning. I detta arbete undersöks om NIG (normal invers gaussisk) fördelning passar för det. / The purpose of this essay is to study if NIG (Normal Inverse Gaussian) distribution is suitable for modelling stochastic payment in certain insurance activities.

<p>Utbetalningsbelopp för skadeersättning studeras. Datamaterialet uppvisar skevheter och ''tjocka svansar'' vilket motiverar att modellera med en mer flexibel fördelning än normalfördelning. I detta arbete undersöks om NIG (normal invers gaussisk) fördelning passar för det.</p> / <p>The purpose of this essay is to study if NIG (Normal Inverse Gaussian) distribution is suitable for modelling stochastic payment in certain insurance activities.</p>

This study attempts to capture the behavior of volatility in the commodity futures market by importing the normal mixture GARCH Model and the NIG mixture GARCH model (Normal-inverse Gaussian Mixture GARCH Model). Normal mixture GARCH Model (what follows called NM-GARCH Model) is a model mixed by two to several normal distributions with a specific weight portfolio, and its variance abide by GAECH process. The ability of capturing the financial data with leptokurtosis and fat-tail of NM-GARCH Model is better than Normal GARCH Model and Student¡¦s t GARCH Model.¡CAlso¡AThe Variance of the factor with lower weight in NM-GARCH Model usually higher, and the volatility of the factor with higher weight is lower, which explains the situation happens in the real market that the probability of large fluctuations (shocks) is small, and the probability of small fluctuations are higher. Generally, the volatilities which keeping occurring in common cases are respectively flat, and the shocks usually bring large impacts but less frequent. NIG Mixture Distribution is a distribution mixed by two to several weighted distributions, and the distribution of every factor abides by NIG Distribution. Compare to Normal Mixture Distribution, NIG Mixture Distribution takes the advantages of NIG Distribution into account, which can not only explain leptokurtosis and the deviation of data, but describe the fat-tail phenomenon more complete as well, because of the both tails of NIG Distribution decreasing slowly. This study will apply the NM GARCH Model and NIG GARCH Model to the Volatility forecasting of the return rates in the crude oil futures market, and infer the predictive abilities of this two kinds of models are significantly better than other volatility model by implementing parameter estimation, forecasting, loss function and statistic significant test.

La thèse porte sur une décomposition explicite de Föllmer-Schweizer d'une classe importante d'actifs conditionnels lorsque le cours du sous-jacent est un processus à accroissements indépendants ou une exponentielle de tels processus. Ceci permet de mettre en oeuvre un algorithme efficace pour établir des stratégies optimales dans le cadre de la couverture quadratique. Ces résultats ont été implémentés dans le cas du marché de l'électricité.

<p>We model Normal Inverse Gaussian distributed log-returns with the assumption of stochastic volatility. We consider different methods of parametrization of returns and following the paper of Lindberg, [21] we</p><p>assume that the volatility is a linear function of the number of trades. In addition to the Lindberg’s paper, we suggest daily stock volumes and amounts as alternative measures of the volatility.</p><p>As an application of the models, we perform Value-at-Risk and Expected Tail Loss predictions by the Lindberg’s volatility model and by our own suggested model. These applications are new and not described in the</p><p>literature. For better understanding of our caluclations, programmes and simulations, basic informations and properties about the Normal Inverse Gaussian and Inverse Gaussian distributions are provided. Practical applications of the models are implemented on the Nasdaq-OMX, where we have calculated Value-at-Risk and Expected Tail Loss</p><p>for the Ericsson B stock data during the period 1999 to 2004.</p>

We model Normal Inverse Gaussian distributed log-returns with the assumption of stochastic volatility. We consider different methods of parametrization of returns and following the paper of Lindberg, [21] we assume that the volatility is a linear function of the number of trades. In addition to the Lindberg’s paper, we suggest daily stock volumes and amounts as alternative measures of the volatility. As an application of the models, we perform Value-at-Risk and Expected Tail Loss predictions by the Lindberg’s volatility model and by our own suggested model. These applications are new and not described in the literature. For better understanding of our caluclations, programmes and simulations, basic informations and properties about the Normal Inverse Gaussian and Inverse Gaussian distributions are provided. Practical applications of the models are implemented on the Nasdaq-OMX, where we have calculated Value-at-Risk and Expected Tail Loss for the Ericsson B stock data during the period 1999 to 2004.

Zur Approximation der Dichte von linearen Integralfunktionalen schwach korrelierter Prozesse mit Korrelationslänge wurden bisher Gram-Charlier-Reihen benutzt. In diesem Artikel werden weitere Verfahren zur Dichteapproximation "integraler" Zufallsgrößen beschrieben und untersucht, ob sie sinnvoll auf das Dichteapproximationsproblem bei linearen Integralfunktionalen angewendet werden können.

<p>The work is focused on the Value at Risk and the Expected Shortfallcalculation. We assume the returns to be based on two pillars - the white noise and the stochastic volatility. We assume that the white noise follows the NIG distribution and the volatility is modeled using the nGARCH, NIG-GARCH, tGARCH and the non-parametric method. We apply the models into the stocks of three Banks of the Nordic market. We consider the daily and the intraday returns with the frequencies 5, 10, 20 and 30 minutes. We calculate the one step ahead VaR and ES for the daily and the intraday data. We use the Kupiec test and the Markov test to assess the correctness of the models. We also provide a new concept of improving the daily VaR calculation by using the high frequency returns. The results show that the intraday data can be used to the one step ahead VaR and the ES calculation. The comparison of the VaR for the end of the following trading day calculated on the basis of the daily returns and the one computed using the high frequency returns shows that using the intraday data can improve the VaR outcomes.</p>

The work is focused on the Value at Risk and the Expected Shortfallcalculation. We assume the returns to be based on two pillars - the white noise and the stochastic volatility. We assume that the white noise follows the NIG distribution and the volatility is modeled using the nGARCH, NIG-GARCH, tGARCH and the non-parametric method. We apply the models into the stocks of three Banks of the Nordic market. We consider the daily and the intraday returns with the frequencies 5, 10, 20 and 30 minutes. We calculate the one step ahead VaR and ES for the daily and the intraday data. We use the Kupiec test and the Markov test to assess the correctness of the models. We also provide a new concept of improving the daily VaR calculation by using the high frequency returns. The results show that the intraday data can be used to the one step ahead VaR and the ES calculation. The comparison of the VaR for the end of the following trading day calculated on the basis of the daily returns and the one computed using the high frequency returns shows that using the intraday data can improve the VaR outcomes.

本文延伸Hull and White (2010)之跨池因子繫聯結構模型中違約相關性之描述，藉由納入Normal Inverse Gaussian分配並允許其帶有狀態轉換之特性，我們探究信用結構式商品清償順位結構中，影響次順位信用保護層(subordination level)之因素。我們以房屋抵押擔保貸款債權憑證(MBS CDO)為例，分析資產違約相關性、資產池微粒化程度、跨池違約相關性等結構性變數如何影響分券評等之合理性及風險特徵。本文的研究結果呼應Azzalini and Capitanio(2003)中所提及採用Gaussian因子繫聯結構模型之於評價信用結構商品的缺失。我們發現增進信用資產池損失分配的之厚尾性描述，得以改善高估或低估分券信用價差的情況。 / By incorporating the Normal Inverse Gaussian distribution and allowing for regime shifts in the correlation structure of the multi-pool factor copula of Hull and White (2010), in this thesis we explorer the factors constituenting the subordination levels of credit-structured products. Using MBS CDOs as an example, we examine how model-embedded variables, such as default correlation, reference-portfolio granularity, and cross-pool correlation, affect the risk profiles of MBS CDO tranches. Our numerical results echo the findings of Azzalini and Capitanio(2003) in that correlation structure obtained under the Gaussian factor copula model may be inadequate in capturing the fact-tailed characteristic of the reference-pool loss distribution, thus can result in over/under-estimation of CDO tranche spreads.