Characterization of catastrophic instabilities: Market crashes as paradigm Anirban Chakraborti1,*, Kiran Sharma1, Hirdesh K. Pharasi2, Sourish Das3, Rakesh Chatterjee2,4, and Thomas H. Seligman2,5 1School of Computational and Integrative Sciences, Jawaharlal Nehru University, New Delhi-110067, India 2Instituto de Ciencias F´ısicas, Universidad Nacional Autonoma´ de Mexico,´ Cuernavaca-62210, Mexico´ 3Chennai Mathematical Institute, Chennai-603103, India 4School of Mechanical Engineering, Tel Aviv University, Israel 5Centro Internacional de Ciencias, Cuernavaca-62210, Mexico´ *
[email protected] ABSTRACT Catastrophic events, though rare, do occur and when they occur, they have devastating effects. It is, therefore, of utmost importance to understand the complexity of the underlying dynamics and signatures of catastrophic events, such as market crashes. For deeper understanding, we choose the US and Japanese markets from 1985 onward, and study the evolution of the cross-correlation structures of stock return matrices and their eigenspectra over different short time-intervals or “epochs”. A slight non-linear distortion is applied to the correlation matrix computed for any epoch, leading to the emerging spectrum of eigenvalues. The statistical properties of the emerging spectrum display: (i) the shape of the emerging spectrum reflects the market instability, (ii) the smallest eigenvalue may be able to statistically distinguish the nature of a market turbulence or crisis – internal instability or external shock, and (iii) the time-lagged smallest eigenvalue has a statistically significant correlation with the mean market cross-correlation. The smallest eigenvalue seems to indicate that the financial market has become more turbulent in a similar way as the mean does. Yet we show features of the smallest eigenvalue of the emerging spectrum that distinguish different types of market instabilities related to internal or external causes.