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 parameters 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 “contrarians”, the index jumps to anti-persistence that is a far-from-equilibrium state in which market crashes are likely to occur.
Key words: Gaussian processes; Mixed fractional Brownian motion; Hurst exponent; Local self-similarity; Persistence; Anti-persistence; Definiteness of covariance functions; Dissipative dynamic systems