One of the spectacular examples of a complex system is the financial market, which displays rich correlation structures among price returns of different assets. The eigenvalue decomposition of a correlation matrix into partial correlations - market, group and random modes, enables identification of dominant stocks or "influential leaders" and sectors or "communities". The correlation-based network of leaders and communities changes with time, especially during market events like crashes , bubbles, etc. Using the eigen-entropy measure , computed from the eigen-centralities (ranks) of different stocks in the correlation network, we extract information about the "disorder " (or randomness) in the market correlation and its different modes. The relative-entropy measures computed for these modes enable us to construct a "phase space", where the different market events undergo "phase-separation" and display " order-disorder " transitions, as observed in critical phenomena in physics. We choose the US S P-500 and Japanese Nikkei-225 financial markets, over a 32-year period, and study the evolution of the cross-correlation matrices and their corresponding eigen-entropies. One of the relative entropy measures displays "universal scaling " behaviorwith respect to the mean market correlation. Further, a functional of the relative entropy measure acts as a good gauge for the "market fragility" (minimum risk of the market portfolio) and the "market fear" (volatility index). This new methodology helps us to better understand market dynamics and characterize the events in different phases as anomalies, bubbles, crashes, etc. that display intriguing phase separation and universal scaling behavior .
Financial markets, being spectacular examples of complex systems, display rich correlation structures among price returns of different assets. The correlation structures change drastically, akin to phase transitions in physical phenomena, as do the influential stocks (leaders) and sectors (communities), during market events like crashes . It is crucial to detect their signatures for timely intervention or prevention. Here we use eigenvalue decomposition and eigen-entropy , computed from eigen-centralities of different stocks in the cross-correlation matrix, to extract information about the disorder in the market . We construct a `phase space', where different market events (bubbles, crashes, etc.) undergo phase separation and display order-disorder transitions. An entropy functional exhibits scaling behavior. We propose a generic indicator that facilitates the continuous monitoring of the internal structure of the market -- important for managing risk and stress-testing the financial system. Our methodology would help in understanding and foreseeing tipping points or fluctuation patterns in complex systems .
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eigenvalue decomposition of a correlation matrix into partial correlations - market, group and random modes, enables identification of dominant stocks or "influential leaders" and sectors or "communities". The correlation-based network of leaders and communities changes with time, especially
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correlation and its different modes. The relative-entropy measures computed for these modes enable us to construct a "phase space", where the
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undergo "phase-separation" and display "
(bubbles, crashes, etc.) undergo phase separation and display
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" transitions, as observed in critical phenomena in physics. We choose the US S
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P-500 and Japanese Nikkei-225 financial markets, over a 32-year period, and study the evolution of the cross-correlation matrices and their corresponding eigen-entropies. One of the relative entropy measures displays "universal scaling " behaviorwith respect to the mean market correlation. Further, a functional of the relative entropy measure acts as a good gauge for the "market fragility" (minimum risk
transitions. An entropy functional exhibits scaling behavior. We propose a generic indicator that facilitates the continuous monitoring of the internal structure
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portfolio) and the "market fear" (volatility index). This new methodology helps us to better understand market dynamics and characterize the events in different phases as anomalies, bubbles, crashes, etc. that display intriguing phase separation and universal scaling behavior
-- important for managing risk and stress-testing the financial system. Our methodology would help in understanding and foreseeing tipping points or fluctuation patterns in complex systems