Mixed fractional Brownian motion, short and long-term Dependence and economic conditions: the case of the S&P-500 Index

Abstract

The Kolmogorov-Mandelbrot-van Ness Process is a zero mean Gaussian process indexed by the Hurst Parameter (H). When it models financial data, a controversy arises as to whether or not financial data exhibit short or long-range dependence. This paper argues that the Mixed Fractional Brownian is a more suitable tool for the purpose as it leaves no room for controversy. It is used here to model the S&P-500 Index, sampled daily over the period 1950-2011. The main results are as follows: The S&P-500 Index is characterized by both short and long-term dependence. More explicitly, it is characterized by at least 12 distinct scaling pa-rameters that are, ex hypothesis, determined by investors’ approach to the market. When the market is dominated by “blue-chippers” or ‘long-termists’, or when bubbles are ongoing, the index is persistent; and when the market is dominated by “con-trarians”, the index jumps to anti-persistence that is a far-from-equilibrium state in which market crashes are likely to occur.Gaussian Processes; Mixed Fractional Brownian Motion; Hurst Exponent; Local Self-similarity, Persistence; Anti-persistence; Definiteness of covariance Functions; Dissipative dynamic systems