The purpose of this thesis is to give an introductory overview on operational risk, and in particular some applications of extreme value theory and methods using ruin probabilities. The each approach is studied in detail, based on the article [DIK08].
The thesis consists of 5 chapters.
In chapter 1 we have present some tools that are often used in operational risk and in ruin theory. We recollect some important probability distributions and their properties. Then we review the risk measures, Value-at-risk and expected shortfall. Next we introduce the basics of the theory of copulas and the basic notions of Extreme Value Theory In chapter 2 is give the framework of operational risk. We describe three approaches: the Basic Indicator Approach (BIA), the Standardized Approach (SA) and the Advanced Measurement Approaches (AMA).
Here we give a typical Advanced Measurement Approach solution for calculating of an operational risk charge for one year, using historical losses.
In chapter 3 we discuss two methods of the Advanced Measurement Approaches: Extreme Value Theory in Operational risk and Ruin theory in operational risk. In chapter 4 contained the main results of the thesis. In this chapter we have given the Kaishev-Dimitrov formula in two cases: for discrete claim distribution and continuous claim distribution. We have calculated the probability of non-ruin in a finite time x for each of these two cases.
In chapter 5 we have considered three alternative distributions of consecutive losses: Logarithmic, Exponential and Pareto. We have given solutions with graphs. The solutions are presented with Mathematica programs. For calculation purposes we also use a formula from Picard -Lefevre