The <i>q</i> -dependent detrended cross-correlation analysis of stock market

Abstract

Abstract Properties of the q -dependent cross-correlation matrices of the stock market have been analyzed by using random matrix theory and complex networks. The correlation structures of the fluctuations at different magnitudes have unique properties. The cross-correlations among small fluctuations are much stronger than those among large fluctuations. The large and small fluctuations are dominated by different groups of stocks. We use complex network representation to study these q -dependent matrices and discover some new identities. By utilizing those q -dependent correlation-based networks, we are able to construct some portfolios of those more independent stocks which consistently perform better. The optimal multifractal order for portfolio optimization is around q = 2 under the mean-variance portfolio framework, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>q</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>6</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mstyle> </mml:math> under the expected shortfall criterion. These results have deepened our understanding regarding the collective behavior of the complex financial system.