On The Cross-correlation Of Financial Time Series

The financial system is a very typical complex system.The detailed investigation about its property is extremely important for both theoretical and practical aspects.Statistical physics and complex network theory are quite important methodologies for the analysis of complex system.Thus the application of those two theories to the financial system is very straightforward.The research about the collective dynamics of financial system is an important topic of econophysics from which lots of fruitful theoretical and empirical progresses have been achieved.Those results have great implications to modern financial engineering.In this thesis,the random matrix theory and complex network theory have been utilized to analyze the cross-correlation among financial market.First,we employ the complex network representation to analyze the cross-correlation matrix of financial time series.We uncover the structural variation of financial market dur-ing financial crisis by using the topological parameters of correlation-based network.We find that the correlation structure of the financial market has global expansion and local clus-tering characteristics.Meanwhile,we also analyze the sector and community structures of the financial market.We then use the structural variation of the correlation-based network as market panic index.The structural variation can serve as a prediction signal of the market panic.The multifractal detrended fluctuation analysis is a very effective tool for the financial time series analysis.Here we use this method to analyze the multifractal behavior of con-tinuous phase transition.The multifractal behavior of the magnetization time series of two dimensional Ising model has very distinct variation around critical temperature.The shape parameters of singularity spectrum can be used as early-waring signals for phase transition.At the same instant,the topological parameters of visibility graph transformed from those magnetization time series can be used to analyze the geometrical structure of the magneti-zation time series.Thus the multifractal detrended fluctuation analysis and visibility graph methods are very effective ways to detect the early-warning signals of critical transition.As a very useful tool to analyze the long-range correlation of non-stationary time se-ries,the multifractal detrended fluctuation analysis has been generalized to the multifractal detrended cross-correlation analysis.Then we employ this cross-correlation to study the nonlinear cross-correlation of financial system.By analyzing the cross-correlation matrices for different multifractal order,we find that the correlation structures of financial markets among different magnitude of fluctuations have completely different identities.The corre-lation strength of small fluctuation is significantly stronger than that of large fluctuation.Whilst,the random matrix analysis shows that the for different magnitudes of fluctuations,the sector contributions are different.Based on the multifractal detrended cross-correlation matrices,we construct some correlation-based networks.The correlation-based networks for small fluctuations are very heterogeneous with some hub stocks.The topological quan-tities of those correlation-based networks have also revealed the structure difference for different magnitude of fluctuations.Moreover,considering the temporal evolution property of financial market,the temporal network theory has been used to study the time evolving structures of the correlation-based networks for three major stock markets.The common patterns of time evolution for worldwide stock markets have been explored by using the topological parameters of those time evolving correlation-based networks.We then propose three portfolio construction strategies based on the above empirical results.We construct some portfolios based on the compound centrality of multifractal detrended correlation-based network,the temporal centrality of temporal correlation-based networks and the community structures of correlation-based networks.We find that three portfolio construction strategies can be used to effectively construct some portfolios which have consistent high performance with lower risk and higher return under both Markowiz and expected shortfall frameworks.To sum up,we have devoted our efforts to investigate the linear and nonlinear cross-correlations of complex financial systems.We also generalize the multifractal methodology for time series analysis to the critical phase transition and financial markets.Whilst,the portfolio construction strategies proposed here should be very instructive for the portfolio optimization theory and financial risk management.