Copula models for financial time series

The present thesis has two objectives: a. to provide a clustering methodology with a consequent study of the con tagion that allows an efficient diversification of the risk; b. to build a model for the estimation of market risk that adapts to finan cial data and their peculiarities more efficiently than the usual models built through the use of Normal distribution. A sample of the nine most capitalized stock indices has been studied, such as: Dow Jones, S&P500, Nasdaq 100, FTSE 100, Nikkei 225, SSE Composite, SZSE Component, Euronext 100, HANG SENG. The chosen time horizon is about ten years, more precisely from January 2, 2012 to October 11, 2022. The work is divided into three chapters, of which the first chapter sets out the theoretical foundations on which the analysis under discussion is based, analytically presenting the em pirical properties of financial data, such as: the presence of non-independent and identically distributed yield series; the significant correlation of returns squared; the near zero value of the above conditional expected values; the characteristic leptocurtic tails and the presence of clusters for the extreme values. The importance of an accurate estimate of rare events is underlined and the extreme values theory is dealt with, focusing on the distribution of generalized extreme values (GEV), on the maximum domain of attraction, on the ”Block Maxima” method and over threshold exceedances. We proceed by providing the definition and explaining the mathematical properties of the copula function; we describe, moreover, various types of copulas including the skew t copula, the Vine copula and the elliptic copulas, to this last category belong the Gaussian copula and the Student t copula. In addition, the cali bration methods of the parameters of the function under consideration, such as the Maximum Likelihood Method (ML), the Margin Inference Functions Method (IFM) and the Method of Maximum Canonical Likelihood (CML). A quick description of the sample being analysed is also provided. The second chapter presents an analysis of the behavior of time series in risky scenarios in order to enable the implementation of performing portfolio diversification strategies. Through a methodology that is articulated in four different phases: to find the model that is well adapted to the data in analysis, to measure the dependence of tail, to create the matrix of dissimilarity, to construct of the clusters that allow the study of the phenomenon of the contagion in extreme events. Three different thresholds were chosen to study the correlation of the nine indices in risky scenarios and in stable market conditions. The third and final chapter explains accurately all the operations carried out that have finally led to the construction of the desired model, starting from the daily closing prices of the nine indices constituting the portfolio. The combined use of extreme value theory and copulas: t , skew t and Vine leads to a market risk modelling approach that stands out from traditional risk management models. They assume conditional normality for logarithmic returns on finan cial assets or risk factors despite empirical evidence that yield distributions are characterized by leptocurtic tails. The main objective of this study was to obtain a model consistent with this empirical evidence. Finally, the obtained portfolio index returns are simulated and the Value at Risk is calculated with relative back-testing to test the goodness of the presented model.