Dependence in extreme events: an analysis based on Copula function

Abstract

Our initial goal for this master thesis was to understand the dynamics of assets co-movement, particularly during market downturns. Recognizing the non-linear dynamics, fat tails, and volatility clustering inherent in financial markets, especially in the stock market, we identified the need for a more precise analysis. This led us to explore copulas as a solution to address the challenges posed by non-linear dependence issues. Beginning with a theoretical exploration, we challenged the conventional assumption of normality that has traditionally underpinned classic finance. As we deepened our analysis, we uncovered the limitations of Normality in capturing the complexities of financial markets, prompting a shift towards alternative methodologies. Then, our focus shifted to understanding the behavior of tails in random variables within the financial landscape. EVT emerged as a crucial analytical tool for studying rare events with the potential to disrupt and threaten market stability. We applied EVT using the Block Maxima Method and Peaks Over Threshold approaches to assess the impact of extreme events on risk management. Applying Extreme Value theory in risk management allows to investigate what can be the impact of a relative rare event and model the possible outcomes. For example, by using risk measures such as Value at Risk (VaR) and Expected Shortfall (ES) and showcasing their application using real return time series data from GOOGLE stock. In the last theory paragraph, we explained Copulas. This kind of distribution offer a useful way to measure the dependence structure between two or more random variables, while considering nonlinear dependence. In this paragraph we are going to analyze different copula types, starting from their distribution. The element we are focusing on is the possible application of copulas in the context of Extreme Value Theory. In the final chapter we are going to analyze two European main stock indexes: CAC 40 and FITSE MIB. These two stock indexes represent respectively French and Italian top 40 listed companies. The analysis is based on what we have seen in theory chapter: verify normality assumption, extreme value theory and copula application. In the last paragraph we are going to see risk measures related to tail estimation as previously discussed.