Minkowski Metric for GARCH (1,1)

In this paper we discuss a stylized fact on long memory process of volatility cluster phenomena by using Minkowski metric for GARCH (1,1). Also presented result of minus sign of volatility in reversed direction of time scale. It is named as dark volatility or hidden risk fear field.

[1]  N. Ayache,et al.  Log‐Euclidean metrics for fast and simple calculus on diffusion tensors , 2006, Magnetic resonance in medicine.

[2]  W. Pauli The Connection Between Spin and Statistics , 1940 .

[3]  M. Marchesi,et al.  Scaling and criticality in a stochastic multi-agent model of a financial market , 1999, Nature.

[4]  D. Sornette,et al.  Predictability of large future changes in major financial indices , 2003, cond-mat/0304601.

[5]  N. Huang,et al.  The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[6]  Robert F. Engle,et al.  Risk and Volatility: Econometric Models and Financial Practice , 2004 .

[7]  Tim Bollerslev,et al.  Glossary to ARCH (GARCH) , 2008 .

[8]  M. Marchesi,et al.  VOLATILITY CLUSTERING IN FINANCIAL MARKETS: A MICROSIMULATION OF INTERACTING AGENTS , 2000 .

[9]  L. Bachelier Théorie de la spéculation ; Théorie mathématique du jeu , 1995 .

[10]  E. Fama EFFICIENT CAPITAL MARKETS: A REVIEW OF THEORY AND EMPIRICAL WORK* , 1970 .

[11]  I. Osorio,et al.  Intrinsic time-scale decomposition: time–frequency–energy analysis and real-time filtering of non-stationary signals , 2007, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[12]  R. Cont Volatility Clustering in Financial Markets: Empirical Facts and Agent-Based Models. , 2005 .

[13]  Gibbons,et al.  Dark matter, time-varying G, and a dilaton field. , 1990, Physical review letters.