Studying the correlation structure based on market geometry
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Studying the correlation structure based on market geometry Chun-Xiao Nie1 Received: 20 January 2019 / Accepted: 10 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Network methods can extract the structure of financial correlation matrices, and market geometry reconstructs the correlation relationship by constructing a vector set in the Euclidean space. This study uses a geometric perspective to analyse financial networks and examine the relationship between correlation structures and geometric conditions. Based on the concept of Euclidean space, we can naturally define geometric concepts such as stock vector and inner product between stocks. The analysis reveals that the structure of the financial correlation network is significantly affected by geometric conditions. We use stock market data to construct networks with different structures, such as a network with a hub node. We find that some stocks with small vector norms have an important effect on changes in network structure. In addition, we define a dimension to describe the correlation information included in the subspace of the market space and find that the dynamics of the dimension are related to the market state. This paper establishes a way to study network structure through market geometry, thereby providing a new method of correlation analysis. Keywords Correlation matrix · Complex network · Stock norm · Financial crisis
1 Introduction To study large-dimensional correlation matrices, how to extract the hidden structure of the matrix is an important issue. Random matrix theory (RMT) and complex network theory provide some effective methods (Laloux et al. 1999; Plerou et al. 1999; Mantegna 1999; Tumminello et al. 2005; Yang and Yang 2008; Tse et al. 2010). RMTbased analysis reveals that the financial correlation matrix includes patterns different from theoretical predictions (Laloux et al. 1999; Plerou et al. 1999). Theoretical and empirical analysis supports the widespread use of RMT in financial data analysis, such
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Chun-Xiao Nie [email protected]; [email protected] School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China
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as constructing portfolios (Plerou et al. 2002; Joël et al. 2017). Furthermore, empirical analysis has shown that the dynamics of the correlation matrix are helpful for understanding the overall pattern of the market, such as market status (Thomas et al. 2009; Ahmet et al. 2013; Junior and Franca 2012; Zheng et al. 2012). In particular, analyzing the relationship between the dynamics of correlation structure and crises, such as constructing indicators of the financial crisis based on correlations (Ahmet et al. 2013; Junior and Franca 2012; Zheng et al. 2012), is an important issue. In addition to directly studying the spectrum or properties of the correlation matrix, an important method used in recent years is converting the correlation matrix into a network and examining its structure (Mantegna 1999; Tumminello et al. 2005
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