Statistical Analysis and Modeling of Extreme Values in Financial Data

Abstract

This thesis studies extreme value theory (EVT) and the statistical methods and tools you can use to describe extremes. We first study the theory used to model independent and identically distributed data, the blocks and Peaks over threshold methods are described alongside the possible types of extreme value distributions. The first section primarily has a focus on the Peaks over threshold method, which is used to fit generalized Pareto distributions to our data. We then extend the theory to allow for some degree of dependence in our data, this is done with the introduction of the Extremal index, which is used to adjust the estimates found in i.i.d. case, so they fit the data under the assumption of dependence. Finally, we introduce the statistical tool known as the extremogram, which is used to analyze the dependency structure within a dataset, but also the correlation between two different sets of data. More specifically it has the ability to assign a probability to another extreme event occurring h time lags after an initial extreme event. Throughout the thesis the theory and methods are applied to real-life data in the form of the Microsoft stock and the Nasdaq index. We estimate the different statistical models and utilizes this to estimate some financial risk measures