Πολυμεταβλητή θεωρία ακραίων τιμών με εφαρμογές στη διαχείριση κινδύνου
Multivariate extreme value theory with applications to risk management
KeywordsΘεωρία ακραίων τιμών
The main objective of this MSc thesis is to present introductory results related to univariate and multivariate extreme value theory with the use of copulas, while a second task is to show their application to the measurement of risk in portfolios with two or more dependent investments. More specifically, the first chapter of this MSc thesis provides introductory concepts and definitions of extreme value theory for the univariate case. The second chapter presents definitions and properties regarding the dependence between random variables with the use of copulas. It also presents results in multivariate extreme value theory using parametric methods such as multivariate block maxima and multivariate peaks over threshold. Chapter three describes various measures of risk, such as Value-at-Risk and Expected Shortfall, and discusses their estimation through various methods, such as historical simulation and variance-covariance method. In addition, a description of the methodology for estimating VaR at the portfolio level is provided. In chapter four, an application of multivariate extreme value theory is carried out with two-dimensional return data using R software. More specifically, bivariate block maxima methods are applied to a set of two-dimensional returns of Bitcoin (BTC) and Cardano (ADA) cryptocurrencies to estimate VaR via the bivariate (joint) extreme value distribution function. Finally, a numerical comparison study is performed with other parametric methods of Value at Risk estimation.