Abstract
In econophysics, the achievements of information filtering methods over the past 20 years, such as the minimal spanning tree (MST) by Mantegna and the planar maximally filtered graph (PMFG) by Tumminello et al., should be celebrated. Here, we show how one can systematically improve upon this paradigm along two separate directions. First, we used topological data analysis (TDA) to extend the notions of nodes and links in networks to faces, tetrahedrons, or k-simplices in simplicial complexes. Second, we used the Ollivier-Ricci curvature (ORC) to acquire geometric information that cannot be provided by simple information filtering. In this sense, MSTs and PMFGs are but first steps to revealing the topological backbones of financial networks. This is something that TDA can elucidate more fully, following which the ORC can help us flesh out the geometry of financial networks. We applied these two approaches to a recent stock market crash in Taiwan and found that, beyond fusions and fissions, other non-fusion/fission processes such as cavitation, annihilation, rupture, healing, and puncture might also be important. We also successfully identified neck regions that emerged during the crash, based on their negative ORCs, and performed a case study on one such neck region.
Highlights
At the turn of the 20th century, Bachelier suggested in his PhD thesis that stock prices follow geometric Brownian motions and worked out some of the consequences [1]
The hypergraph approach is difficult to implement starting from pairwise correlations, so we argue that the more promising approach to extract deeper insights into the hierarchical structure in financial markets is through topological data analysis (TDA) [55,56,57,58]
In addition to the TDA, we found another promising approach for extending the information filtering paradigm of minimal spanning tree (MST) and planar maximally filtered graph (PMFG)
Summary
At the turn of the 20th century, Bachelier suggested in his PhD thesis that stock prices follow geometric Brownian motions and worked out some of the consequences [1]. In a recent paper [59], we demonstrated how TDA can be used to understand the topological changes that accompany market crashes For such extreme events in financial markets, one of the key questions not well answered through the use of MSTs or planar maximally filtered graphs (PMFGs) is how the hierarchy of cross correlations between stocks re-organizes itself. We use the OllivierRicci curvature to analyze a sequence of PMFGs obtained from the cross correlations of TWSE stocks in overlapping time windows leading up to the market crash of interest, before ending with a comparative case study of two neck regions.
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