Multiple-Point Correlation Approach to Stock Market Index Series

The original concept of multiple-point correlation is extended to reveal the fine structure of correlation in mono-variate time series. Specifically, we display the patterns of the three-point correlation in daily volatility series for five stock markets distributed across the world and compare them with those for increment series generated by the fractional Brownian motion model(fBm). For the fBm increment series, with the increase of persistence (Hurst exponent) the macroscopic patterns occur and contribute more and more to the correlation pattern, and the original three-point correlation has a fractal structure whose fractal dimension increases with the Hurst exponent. For the empirical series, whose Hurst exponents turn out to be 0.79 (SSE), 0.78 (HSI), 0.73 (N225), 0.81 (NSDQ), and 0.72 (FTSE), a simple comparison with the fBm increment series generated with identical Hurst exponents shows that the fractal structures in the original correlations can be reproduced by the fBm model. However, the fBm model can not reproduce the behaviors of the widely distributed contributions for the first principal components and the characteristics of the topological structures for the original and reconstructed three-point correlations in empirical records. Hence, the three-point correlation can tell us the fine-scale difference between the fBm model and the real dynamical processes for stock markets.