Dependencies and non-stationarity in financial time series
The growing interest of physicists in economic problems has led to the emergence of a new interdisciplinary field called econophysics. It applies methods and concepts of statistical physics to study economic and financial phenomena. This thesis contributes to the research activities in econophysics, focusing mainly on the study of financial markets. Financial markets can be viewed as complex systems which exhibit highly non-stationary behavior. We concentrate in particular on the non-stationarity of correlations between companies. We begin by introducing an approach to model the non-stationarity of correlations. This approach is based on concepts from random matrix theory and allows us to construct a correlation averaged multivariate normal distribution which takes the non-stationary correlations into account. We perform an empirical study to verify the random matrix approach for financial returns. We consider two applications of the random matrix approach. First, we study the effect of non-stationary correlations on portfolios, and in particular on the distribution of portfolio returns. We derive an average portfolio return distribution which we compare with the portfolio returns of randomly selected portfolios. Second, we investigate the stability of the correlation structure of market states. The random matrix approach reduces the complexity of the financial market to a single parameter which characterizes the correlation fluctuations due to non-stationarity. This allows us to estimate the fluctuation strength of correlations directly from the empirical return distributions and thus to assess the stability of the correlation structure. We further extend our market states analysis by studying the empirical de- pendence structure of market states. We estimate and study empirical pairwise copulas for each market state. In addition, we derive a copula which arises from the correlation averaged multivariate normal distribution and compare it with the empirical pairwise copulas of each state. The results confirm the consistency of our random matrix model with financial data once again. Finally, we focus on the extreme value statistics of correlated random variables. We derive a distribution for the maximum of correlated normal random variables and extend the result to the non-normal case. We verify our findings in numerical simulations.