Extreme value theory with applications in finance

Abstract

The development and implementation of extreme value theory models has been very significant as they demonstrate an application of statistics that is very much needed in the analysis of extreme events in a wide range of industries, and more recently the cryptocurrency industry. The crypto industry is booming as the phenomenon of cryptocurrencies is spreading worldwide and constantly drawing the attention of investors, the media, as well as financial institutions. Cryptocurrencies are highly volatile assets whose price fluctuations continually lead to the loss of millions in a variety of currencies in the market. In this thesis, the extreme behaviour in the tail of the distribution of returns of Bitcoin will be examined. High-frequency Bitcoin data spanning periods before as well as after the COVID-19 pandemic will be utilised. The Peaks-over-Threshold method will be used to build models based on the generalised Pareto distribution, and both positive returns and negative returns will be modelled. Several techniques to select appropriate thresholds for the models are explored and the goodness-offit of the models assessed to determine the extent to which extreme value theory can model Bitcoin returns sufficiently. The analysis is extended and performed on Bitcoin data from a different crypto exchange to ensure model robustness is achieved. Using Bivariate extreme value theory, a Gumbel copula is fitted by the method of maximum likelihood with censored data to model the dynamic relationship between Bitcoin returns and trading volumes at the extreme tails. The extreme dependence and correlation structures will be analysed using tail dependence coefficients and the related extreme correlation coefficients. All computations are executed in R and the results are recorded in tabular and graphical formats. Tail-related measures of risk, namely Value-at-Risk and Expected Shortfall, are estimated from the extreme value models. Backtesting procedures are performed on the results from the risk models. A comparison between the negative returns of Bitcoin and those of Gold is carried out to determine which is the less risky asset to invest in during extreme market conditions. Extreme risk is calculated using the same extreme value approach and the results show that Bitcoin is riskier than Gold.