The volatility model approach to forecasting Value at Risk is complemented with modelling of Expected Shortfalls using an extreme value approach. Using three models from the GARCH family (GARCH, EGARCH and GJR-GARCH) and assuming two conditional distributions, normal Gaussian and Student t’s distribution, to make predictions of VaR, the forecasts are used as a threshold for assigning losses to the distribution tail. The Expected Shortfalls are estimated assuming that the violations of VaR follow the Generalized Pareto distribution, and the estimates are evaluated. The results indicate that the most efficient model for making predictions of VaR is the asymmetric GJR-GARCH, and that assuming the t distribution generates conservative forecasts. In conclusion there is evidence that the commodities are characterized by asymmetry and conditional normality. Since no comparison is made, the EVT approach can not be deemed to be either superior or inferior to standard approaches to Expected Shortfall modeling, although the data intensity of the method suggest that a standard approach may be preferable.

This study employs Extreme Value Theory and several univariate methods to compare their Value-at-Risk and Expected Shortfall predictive performance. We conduct several out-of-sample backtesting procedures, such as uncondi- tional coverage, independence and conditional coverage tests. The dataset in- cludes five different stock markets, PX50 (Prague, Czech Republic), BIST100 (Istanbul, Turkey), ATHEX (Athens, Greece), PSI20 (Lisbon, Portugal) and IBEX35 (Madrid, Spain). These markets have different financial histories and data span over twenty years. We analyze the global financial crisis period sep- arately to inspect the performance of these methods during the high volatility period. Our results support the most common findings that Extreme Value Theory is one of the most appropriate risk measurement tools. In addition, we find that GARCH family of methods, after accounting for asymmetry and fat tail phenomena, can be equally useful and sometimes even better than Extreme Value Theory based method in terms of risk estimation. Keywords Extreme Value Theory, Value-at-Risk, Expected Shortfall, Out-of-Sample Backtesting Author's e-mail ozturkdurmus@windowslive.com Supervisor's e-mail ies.avdulaj@gmail.com

The master's thesis deals with modeling Value at Risk model adjusted by liquid- ity. For this purpose we use quantile regression analysis and liquidity proxies. We find out that Garman-Klass volatility estimator can be very useful in pe- riod 2000-2008 for the small and mid-size semiconductor companies but not in period 2008-2015. The NASDAQ composite Garman-Klass volatility is useful for all semiconductor companies for period 2008-2015. We might conclude that from the outbreak of the crisis returns of all semiconductor companies might depend on movement of NASDAQ composite index. We use Amihud and Roll measures as the liquidity proxies but the results are not persuasive regardless or size of companies and period we analyzed. JEL Classification G11, G14, G17, G18, G32 Keywords liquidity, value at risk, quantile regression Author's e-mail michalnd@gmail.com Supervisor's e-mail barunik@utia.cas.cz Abstrakt Diplomová práce se zabývá modelováním hodnoty v risku upravenou o likvid- itu. Pro tuto analýzu jsme použili kvantilovou regresi a proměnné indikující likviditu. Došli jsme k závěru, že Garman-Klass volatility estimator je velmi užitečný pro malé a středně velké firmy operující na trhu s polovodiči a to v ob- dobí 2000-2007, nikoliv však období 2008-2015. NASDAQ composite...

[pt] Diversos estudos têm procurado uma variável de risco que empiricamente tenha uma relação positiva e significativa com os retornos condicionais de mercado. Na maior parte dos casos a escolha recai sobre novas abordagens envolvendo a variância condicional dos retornos. Neste trabalho substituímos a variância pelo Value at Risk (VaR) para analisar se no mercado brasileiro existe o trade-off entre risco e retorno. O VaR é estimado paramétrica e não parametricamente com base em janelas de dados de um a seis meses. Os resultados mostram que em nosso mercado não há relação positiva e significativa entre o VaR e os retornos mensais. A causa mais aparente para essa divergência é que o prêmio de risco de mercado é negativo em 114 dos 217 meses que compõem a série temporal da variável dependente, impactando os coeficientes do VaR nas regressões. Quando utilizados retornos com frequência diária, os resultados mostram que em períodos mais recentes há relação positiva e significativa entre esses retornos e o VaR paramétrico. / [en] Several studies have searched a risk variable with an empirically positive and significant relation with excess market returns. At the most part of the cases the choices are new approaches of conditional variance of the returns. In this paper we substitute the variance for the Value at Risk (VaR) to analyze whether in the Brazilian market there is relation between risk and returns. The VaR is estimated in parametric and nonparametric ways, considered the precedents intervals of time from one to six months. The results show that in our market there is not a positive and significant relation between VaR and the monthly market returns. The most obvious cause that supports our results is that the market premium risk is negative on 114 of 217 total monthly observations that form the temporal series of the dependent variable, impacting the VaR coefficients in the regressions. When used daily frequency returns, the results show a positive and significant relation between these results and parametric VaR in recent periods.

[pt] Um dos principais desafios no mercado financeiro é simular preços mantendo a estrutura de correlação entre os inúmeros ativos de um portfólio. Análise de Componentes Principais emerge como uma solução para este último problema. Além disso, dada a incerteza presente nos mercados de commodities de derivados de petróleo, o investidor quer proteger seus ativos de perdas potenciais. Como uma alternativa a esse problema, a otimização de várias medidas de risco, como Value-at-risk, Conditional Value-at-risk e medida Ômega, são ferramentas financeiras importantes. Além disso, o backtest é amplamente utilizado para validar e analisar o desempenho do método proposto. Nesta dissertação, trabalharemos com um portfólio de commodities de petróleo. Vamos unir diferentes técnicas e propor uma nova metodologia que consiste na diminuição da dimensão do portfólio proposto. O passo seguinte é simular os preços dos ativos na carteira e, em seguida, otimizar a alocação do portfólio de commodities de derivados do petróleo. Finalmente, vamos usar técnicas de backtest, a fim de validar nosso método. / [en] One of the main challenges in the financial market is to simulate prices keeping the correlation structure among numerous assets. Principal Component Analysis emerges as solution to the latter problem. Also, given the uncertainty present in commodities markets, an investor wants to protect his/her assets from potential losses, so as an alternative, the optimization of various risk measures, such as Value-at-risk, Conditional Value-at-risk and Omega Ratio, are important financial tools. Additionally, the backtest is widely used to validate and analyze the performance of the proposed methodology. In this dissertation, we will work with a portfolio of oil commodities. We will put together different techniques and propose a new methodology that consists in the (potentially) decrease the dimension of the proposed portfolio. The following step is to simulate the prices of the assets in the portfolio and then optimize the allocation of the portfolio of oil commodities. Finally, we will use backtest techniques in order to validate our method.

In order to evaluate and manage portfolio risk, we separated this project into three sections. In the first section we constructed a portfolio with 15 different stocks and six options with different strategies. The portfolio was implemented in Interactive Brokers and rebalanced weekly through five holding periods. In the second section we modeled the loss distribution of the whole portfolio with normal and student-t distributions, we computed the Value-at-Risk and expected shortfall in detail for the portfolio loss in each holding week, and then we evaluated differences between the normal and student-t distributions. In the third section we applied the ARMA(1,1)-GARCH(1,1) model to simulate our assets and compared the polynomial tails with Gaussian and t-distribution innovations.

This report is for MA575: Market and Credit Risk Models and Management, given by Professor Marcel Blais. In this project, three different methods for estimating Value at Risk (VaR) and Expected Shortfall (ES) are used, examined, and compared to gain insightful information about the strength and weakness of each method. In the first part of this project, a portfolio of underlying assets and vanilla options were formed in an Interactive Broker paper trading account. Value at Risk was calculated and updated weekly to measure the risk of the entire portfolio. In the second part of this project, Value at Risk was calculated using semi-parametric model. Then the weekly losses of the stock portfolio and the daily losses of the entire portfolio were both fitted into ARMA(1,1)-GARCH(1,1), and the estimated parameters were used to find their conditional value at risks (CVaR) and the conditional expected shortfalls (CES).

In order to evaluate and manage portfolio risk, we separated this project into three sections. In the first section we constructed a portfolio with 15 different stocks and six options with different strategies. The portfolio was implemented in Interactive Brokers and rebalanced weekly through five holding periods. In the second section we modeled the loss distribution of the whole portfolio with normal and student-t distributions, we computed the Value-at-Risk and expected shortfall in detail for the portfolio loss in each holding week, and then we evaluated differences between the normal and student-t distributions. In the third section we applied the ARMA(1,1)-GARCH(1,1) model to simulate our assets and compared the polynomial tails with Gaussian and t-distribution innovations.

Since the introduction univariate GARCH models number of available models have grown rapidly and has been extended to the multivariate area. This paper compares three different multivariate GARCH models and they are evaluated using out of sample Value at Risk of dif- ferent portfolios. Sector portfolios are used with different market capitalization. The models compared are the DCC,CCC and the GO-Garch model. The forecast horizon is 1-day, 5-day and 10-day ahead forecast of the estimated VaR limit. The DCC performs best with regards to both conditional anc unconditional violations of the VaR estimates.

Cette thèse étudie la gestion et la couverture du risque en s’appuyant sur la Value-at-Risk (VaR) et la Value-at-Risk Conditionnelle (CVaR), comme mesures de risque. La première partie propose un modèle d’évolution de prix que nous confrontons à des données réelles issues de la bourse de Paris (Euronext PARIS). Notre modèle prend en compte les probabilités d’occurrence des pertes extrêmes et les changements de régimes observés sur les données. Notre approche consiste à détecter les différentes périodes de chaque régime par la construction d’une chaîne de Markov cachée et à estimer la queue de distribution de chaque régime par des lois puissances. Nous montrons empiriquement que ces dernières sont plus adaptées que les lois normales et les lois stables. L’estimation de la VaR est validée par plusieurs backtests et comparée aux résultats d’autres modèles classiques sur une base de 56 actifs boursiers. Dans la deuxième partie, nous supposons que les prix boursiers sont modélisés par des exponentielles de processus de Lévy. Dans un premier temps, nous développons une méthode numérique pour le calcul de la VaR et la CVaR cumulatives. Ce problème est résolu en utilisant la formalisation de Rockafellar et Uryasev, que nous évaluons numériquement par inversion de Fourier. Dans un deuxième temps, nous nous intéressons à la minimisation du risque de couverture des options européennes, sous une contrainte budgétaire sur le capital initial. En mesurant ce risque par la CVaR, nous établissons une équivalence entre ce problème et un problème de type Neyman-Pearson, pour lequel nous proposons une approximation numérique s’appuyant sur la relaxation de la contrainte / This thesis studies the risk management and hedging, based on the Value-at-Risk (VaR) and the Conditional Value-at-Risk (CVaR) as risk measures. The first part offers a stocks return model that we test in real data from NSYE Euronext. Our model takes into account the probability of occurrence of extreme losses and the regime switching observed in the data. Our approach is to detect the different periods of each regime by constructing a hidden Markov chain and estimate the tail of each regime distribution by power laws. We empirically show that powers laws are more suitable than Gaussian law and stable laws. The estimated VaR is validated by several backtests and compared to other conventional models results on a basis of 56 stock market assets. In the second part, we assume that stock prices are modeled by exponentials of a Lévy process. First, we develop a numerical method to compute the cumulative VaR and CVaR. This problem is solved by using the formalization of Rockafellar and Uryasev, which we numerically evaluate by Fourier inversion techniques. Secondly, we are interested in minimizing the hedging risk of European options under a budget constraint on the initial capital. By measuring this risk by CVaR, we establish an equivalence between this problem and a problem of Neyman-Pearson type, for which we propose a numerical approximation based on the constraint relaxation