Electricity prices in competitive markets are extremely volatile with salient features such as mean-reversion and jumps and spikes. Modeling electricity spot prices is essential for asset and project valuation as well as risk management. I introduce the mean-reversion feature into a classical variance gamma model to model the electricity price dynamics as a mean-reverting variance gamma (MRVG) process. Derivative pricing formulae are derived through transform analysis and model parameters are estimated by the generalized method of moments and the Markov Chain Monte Carlo method. A real option approach is proposed to value a tolling contract incorporating operational characteristics of the generation asset and contractual constraints. Two simulation-based methods are proposed to solve the valuation problem. The effects of different electricity price assumptions on the valuation of tolling contracts are examined. Based on the valuation model, I also propose a heuristic scheme for hedging tolling contracts and demonstrate the validity of the hedging scheme through numerical examples. Autoregressive Conditional Heteroscedasticity (ARCH) and Generalized ARCH (GARCH) models are widely used to model price volatility in financial markets. Considering a GARCH model with heavy-tailed innovations for electricity price, I characterize the limiting distribution of a Value-at-Risk (VaR) estimator of the conditional electricity price distribution, which corresponds to the extremal quantile of the conditional distribution of the GARCH price process. I propose two methods, the normal approximation method and the data tilting method, for constructing confidence intervals for the conditional VaR estimator and assess their accuracies by simulation studies. The proposed approach is applied to electricity spot price data taken from the Pennsylvania-New Jersey-Maryland market to obtain confidence intervals of the empirically estimated Value-at-Risk of electricity prices. Several directions that deserve further investigation are pointed out for future research.

The current study empirically analyzes the extreme behavior and the impact of deregulation policies as well as financial turmoil on the extreme behavior of changes of Taiwan short term interest rate. A better knowledge of short-term interest rate properties, such as heavy tails, asymmetry, and uneven tail fatness between right and left tails, provide an insight to the extreme behavior of short-term interest rate as well as a more accurate estimation of interest risk. The predicting performances of filtered and unfiltered VaR (Value at risk) models are also examined to suggest the proper models for management of interest rate risk. By applying Extreme Value theory (EVT), tail behavior is analyzed and tested and the VaR based on parametric and non-parametric EVT models are calculated.The empirical findings show that, first, the distribution of change of rate are heavy-tailed indicating that the actual risk would be underestimated based on normality assumption. Second, the unconditional distribution is consistent with the heavier-tailed distributions such as ARCH process or Student¡¦t. Third, the right tail of distribution of change of rate are significantly heavier than the left one pointing out that the probabilities and magnitudes of rise in rate could be higher than those of drop in rate. Fourth, the amount of tail-fatness in tail of distribution of change of rate increase after 1999 and the vital factors to cause structural break in tail index are the interest rate policies taken by central bank of Taiwan instead of the deregulation policies in money market. Fifth, based on the two break points found in tail index of right and left tail, long sample of CP rates should not be treated as samples from a single distribution. Sixth, the dependent and heteroscedastic properties of data series should be considered in applying EVT to improve accuracy of VaR forecasts. Finally, EVT models predict VaR accurately before 2001 and the benchmark model, HS and GARCH, generally are superior to EVT models after 2001. Among EVT models, MRE and CHE are relative consistent and reliable in VaR prediction.

Au cours des dernières années, des changements importants dans le domaine des assurances et des finances attirent de plus en plus l’attention sur la nécessité d’élaborer un cadre normalisé pour la mesure des risques. Récemment, il y a eu un intérêt croissant de la part des experts en assurance sur l’utilisation de l’espérance conditionnelle des pertes (CTE) parce qu’elle partage des propriétés considérées comme souhaitables et applicables dans diverses situations. En particulier, il répond aux exigences d’une mesure de risque “cohérente”, selon Artzner [2]. Cette thèse représente des contributions à l’inférence statistique en développant des outils, basés sur la convergence des intégrales fonctionnelles, pour l’estimation de la CTE qui présentent un intérêt considérable pour la science actuarielle. Tout d’abord, nous développons un outil permettant l’estimation de la moyenne conditionnelle E[X|X > x], ensuite nous construisons des estimateurs de la CTE, développons la théorie asymptotique nécessaire pour ces estimateurs, puis utilisons la théorie pour construire des intervalles de confiance. Pour la première fois, l’approche de bootstrap non paramétrique est explorée dans cette thèse en développant des nouveaux résultats applicables à la valeur à risque (VaR) et à la CTE. Des études de simulation illustrent la performance de la technique de bootstrap.

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Electricity prices in competitive markets are extremely volatile with salient features such as mean-reversion and jumps and spikes. Modeling electricity spot prices is essential for asset and project valuation as well as risk management. I introduce the mean-reversion feature into a classical variance gamma model to model the electricity price dynamics as a mean-reverting variance gamma (MRVG) process. Derivative pricing formulae are derived through transform analysis and model parameters are estimated by the generalized method of moments and the Markov Chain Monte Carlo method. A real option approach is proposed to value a tolling contract incorporating operational characteristics of the generation asset and contractual constraints. Two simulation-based methods are proposed to solve the valuation problem. The effects of different electricity price assumptions on the valuation of tolling contracts are examined. Based on the valuation model, I also propose a heuristic scheme for hedging tolling contracts and demonstrate the validity of the hedging scheme through numerical examples. Autoregressive Conditional Heteroscedasticity (ARCH) and Generalized ARCH (GARCH) models are widely used to model price volatility in financial markets. Considering a GARCH model with heavy-tailed innovations for electricity price, I characterize the limiting distribution of a Value-at-Risk (VaR) estimator of the conditional electricity price distribution, which corresponds to the extremal quantile of the conditional distribution of the GARCH price process. I propose two methods, the normal approximation method and the data tilting method, for constructing confidence intervals for the conditional VaR estimator and assess their accuracies by simulation studies. The proposed approach is applied to electricity spot price data taken from the Pennsylvania-New Jersey-Maryland market to obtain confidence intervals of the empirically estimated Value-at-Risk of electricity prices. Several directions that deserve further investigation are pointed out for future research.

<p>In this thesis are extreme value theory used to estimate the probability that large insuranceclaims are exceeding a certain threshold. The expected claim size, given that the claimhas exceeded a certain limit, are also estimated. Two different models are used for thispurpose. The first model is based on maximum domain of attraction conditions. A Paretodistribution is used in the other model. Different graphical tools are used to check thevalidity for both models. Länsförsäkring Kronoberg has provided us with insurance datato perform the study.Conclusions, which have been drawn, are that both models seem to be valid and theresults from both models are essential equal.</p> / <p>I detta arbete används extremvärdesteori för att uppskatta sannolikheten att stora försäkringsskadoröverträffar en vis nivå. Även den förväntade storleken på skadan, givetatt skadan överstiger ett visst belopp, uppskattas. Två olika modeller används. Den förstamodellen bygger på antagandet att underliggande slumpvariabler tillhör maximat aven extremvärdesfördelning. I den andra modellen används en Pareto fördelning. Olikagrafiska verktyg används för att besluta om modellernas giltighet. För att kunna genomförastudien har Länsförsäkring Kronoberg ställt upp med försäkringsdata.Slutsatser som dras är att båda modellerna verkar vara giltiga och att resultaten ärlikvärdiga.</p>

In this thesis are extreme value theory used to estimate the probability that large insuranceclaims are exceeding a certain threshold. The expected claim size, given that the claimhas exceeded a certain limit, are also estimated. Two different models are used for thispurpose. The first model is based on maximum domain of attraction conditions. A Paretodistribution is used in the other model. Different graphical tools are used to check thevalidity for both models. Länsförsäkring Kronoberg has provided us with insurance datato perform the study.Conclusions, which have been drawn, are that both models seem to be valid and theresults from both models are essential equal. / I detta arbete används extremvärdesteori för att uppskatta sannolikheten att stora försäkringsskadoröverträffar en vis nivå. Även den förväntade storleken på skadan, givetatt skadan överstiger ett visst belopp, uppskattas. Två olika modeller används. Den förstamodellen bygger på antagandet att underliggande slumpvariabler tillhör maximat aven extremvärdesfördelning. I den andra modellen används en Pareto fördelning. Olikagrafiska verktyg används för att besluta om modellernas giltighet. För att kunna genomförastudien har Länsförsäkring Kronoberg ställt upp med försäkringsdata.Slutsatser som dras är att båda modellerna verkar vara giltiga och att resultaten ärlikvärdiga.

This thesis discusses the phenomenon of currency crises, in particular it is devoted to empirical identification of crisis periods. As a crisis indicator, we aim to utilize an exchange market pressure index which has been revealed as a very powerful tool for the exchange market pressure quantification. Since enumeration of the exchange market pressure index is crucial for further analysis, we pay special attention to different approaches of its construction. In the majority of existing literature on exchange market pressure models, a currency crisis is defined as a period of time when the exchange market pressure index exceeds a predetermined level. In contrast to this, we incorporate a probabilistic approach using the extreme value theory. Our goal is to prove that stochastic methods are more accurate, in other words they are more reliable instruments for crisis identification. We illustrate the application of the proposed method on a selected sample of four central European countries over the period 1993 - 2012, or 1993 - 2008 respectively, namely the Czech Republic, Hungary, Poland and Slovakia. The choice of the sample is motivated by the fact that these countries underwent transition reforms to market economies at the beginning of 1990s and therefore could have been exposed to speculative attacks on their newly arisen currencies. These countries are often assumed to be relatively homogeneous group of countries at similar stage of the integration process. Thus, a resembling development of exchange market pressure, particularly during the last third of the estimation period, would not be surprising.

The concern of this diploma thesis is extreme value distributions. The first part formulates and proves the limit theorem for distribution of maximum. Further there are described basic properties of class of extreme value distributions. The key role of this thesis is on non-parametric estimations of extreme value index. Primarily, Hill and moment estimator are derived, for which is, based on the results of mathematical analysis, suggested an alternative choice of optimal sample fraction using a bootstrap based method. The estimators of extreme value index are compared based on simulations from proper chosen distributions, being close to distribution of given rain-fall data series. This time series is recommended a suitable estimator and suggested choice of optimal sample fraction, which belongs to the most difficult task in the area of extreme value theory.

[pt] Valor em Risco -VaR- já é parte das ferramentas habituais que um analista financeiro utiliza para estimar o risco de mercado. Na implementação do VaR é necessário que seja estimados quantis de baixa probabilidade para a distribuição condicional dos retornos dos portfólios. A metodologia tradicional para o cálculo do VaR requer a estimação de um modelo tipo GARCH com distribuição normal. Entretanto, a hipótese de normalidade condicional nem sempre é adequada, principalmente quando se deseja estimar o VaR em períodos atípicos, caracterizados pela ocorrência de eventos extremos. Nesta situações a distribuição condicional deve apresentar excesso de curtose. O uso de distribuições derivadas do Teorema do Valor Extremos -TVE-, conhecidas coletivamente como GEV,associadas aos modelos tipo GARCH, tornou possível o cálculo do VaR nestas situações.Um parâmetro chave nas distribuições da família GEV é o índice de cauda, o qual pode ser estimado através do estimador de Hill. Entretanto este estimador apresenta muita sensibilidade em termos de variância e viés com respeito à fração amostral utilizada na sua estimação. O objetivo principal desta dissertação foi fazer uma comparação entre três métodos de escolha da fração amostral, recentemente sugeridos na literatura: o método bootstrap duplo Danielsson, de Haan, Peng e de Vries 1999, o método threshold Guillou e Hall 2001 e o Hill plot alternativo Drees, de Haan e Resnick 2000. A avaliação dos métodos foi feita através do teste de cobertura condicional de Christoffersen 1998, o qual foi aplicado às séries de retornos dos índices: NASDAQ, NIKKEY,MERVAL e IBOVESPA. Os nossos resultados indicam que os três métodos apresentam aproximadamente o mesmo desempenho, com uma ligeira vantagem dos métodos bootstrap duplo e o threshold sobre o Hill plot alternativo, porque este ultimo tem um componente normativo na determinação do índice de cauda ótimo. / [en] Value at Risk -VaR- is already part of the toolkit of financial analysts assessing market risk. In order to implement VaR it is needed to estimate low quantiles of the portfolio returns distribution. Traditional methodologies combine a normal conditional distribution together with ARCH type models to accomplish this goal. Albeit well succeed in evaluating risk for typical periods, this methodology has not been able to accommodate events that occur with very low probabilities. For these situations one needs conditional distributions with excess of kurtosis. The use of distributions derived from the ExtremeValue Theory -EVT-, collectively known as Generalized Extreme Value distribution -GEV-, together with ARCH type models have made it possible to address this problem ina proper framework. A key parameter in the GEV distribution is the tail index, which can be estimated by Hill s estimator. Hill s estimator is very sensible, in terms of bias and RMSE, to the sample fraction that is used in its estimation. The objective of this dissertation is to compare three recently suggested methods presented in the statistical literature: the double bootstrap method Danielsson, de Haan, Peng and de Vries 1999,the threshold method Guillou and Hall 2001 and the alternative Hill plot Drees, de Haan and Resnick 2000. The methods have been evaluated with respect to the conditional coverage test of Christoffersen 1998, which has been applied to the followingreturns series : NASDAQ, NIKKEY, MERVAL e IBOVESPA. Our empirical findings suggests that, overall the three methods have the same performance, with some advantage of the bootstrap and threshold methods over the alternative Hill plot, which has a normative component in the determination of the optimal tail index.