Extreme Value Theory and Application to VaR Calculation
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Original Italian title:
Extreme Value Theory e applicazione al calcolo del VaR
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Giuseppe Cascarino, Università degli Studi di Napoli - Federico II, 2010-11
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ABSTRACT: This thesis discusses the theoretical foundations of Extreme Value Theory and application problems in the financial field in the calculation of Value-at-Risk (VaR), on the daily and multiperiodale horizon, as well as the use of ARMA and GARCH models in support of this calculation. It also studies the performance of the model developed on the series of an equity index (Nasdaq Composite Index), performing back testing and showing how, in comparison with the traditional methods, it provides results largely above. All the functions specially developed for MATLAB are set out in full.

BRIEF INTRODUCTION: The problem of the evaluation and control of market risk associated with a financial position has gained over the years in importance and increasing visibility, because the financial crisis of 2008-2009 and the crisis in the sovereign debt of the countries of Europe in 2011. The tool most used by financial institutions and more than required by the supervisory authorities and the assessment of Value-at-Risk (VaR). In the last fifteen years the international scientific community has provided numerous contributions to the development of more refined methodologies for the estimation of the VaR. THE models to estimate the VaR so-called traditional, based on non-parametric methods such as the historical simulation or on parametric methods are still linked to the assumption of normal distribution of returns have proved inadequate. The first significant weaknesses exist from a technical point of view and require statistical unrealistic conditions to provide reliable results, while the latter are based on assumptions inconsistent with the characteristics of financial series highlighted by empirical studies, in particular the increased frequency of extreme events (excess kurtosis) than is provided by case gaussian. One of the approaches developed to improve the precision of the estimates of the VaR and due to the Extreme Value Theory (EVT). The theory of extreme values provides statistical models to estimate only the form of code of a distribution, apart from the hypothesis on the distribution of the random variable. The estimate of a specific model for each queue allows you to obtain more precise estimates of the quantiles extremes of the distribution and to take account of empirical characteristics of the historical series which the excess kurtosis and skewness negative, using generalized distributions that are flexible and fit the empirical distribution of the historical data (instead of using distributions of form itself, albeit at tailed). Between the different methods based on EVT, attention is given, the implementation part of the work, the method POT (peaks over threshold). This method will be applied jointly with the use of linear models WEAPON-GARCH for the prediction of the mean and variance packed to filter the data, and will be carried out through the application of procedures for backtesting, the comparison with the effectiveness of alternative models based on distributions of fixed form and the assessment of the capacity of absolute model POT to estimate correctly the VaR.

The work is structured as follows: In the first chapter introduces the concept of VaR, if nor evaluate strengths and weaknesses as a measure of risk, and describes a brief taxonomy of the methods used for his esteem. In the second chapter is offering a theoretical description of linear models WEAPON and GARCH for the estimation of the historical series and modeling eteroschedasticita conditioning and describes how they are used for the estimation of forecasts and for the purposes of calculating the VaR. The third chapter is the body of theoretical central thesis: are exposed the fundamental theory of extreme values and methods for the estimation of code on it based (Block Maxima, POT). In the fourth chapter the methodologies described in the previous chapters are applied jointly for the estimation of the VaR in a day and the esteem of the VaR multiperiodale. We describe some fundamental test for assessing the goodness of VaR models, and carries out the backtesting models packed and not conditioned, reasonable and not on EVT, on historical series of an equity index. Applications are to be conducted through the use of the software MATLAB ® ; the functions specially developed for the purposes of work are fully described
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