IJPAM: Volume 42, No. 2 (2008)

VERY SLOWLY DECAYING AUTO-CORRELATIONS:
APPLICATION TO VOLATILITY IN FINANCIAL MARKETS

H.E. Roman$^1$, M. Porto$^2$
$^1$Department of Physics
University of Milano-Bicocca
Piazza della Scienza 3, Milano, 20126, ITALY
e-mail: roman@mib.infn.it
$^2$Institut für Festkörperphysik
Technische Universität Darmstadt
Hochschulstr. 8, Darmstadt, 64289, GERMANY
e-mail: porto@fkp.tu-darmstadt.de


Abstract.We discuss empirical data associated to the Dow-Jones Industrial Average stock market index over a daily basis during the period 1928-2007. We study the statistical properties of the logarithmic daily variations of the index, such as the probability distribution function and autocorrelation functions, for both log-returns and their absolute values. We consider in particular the effect of detrending the empirical data with polynomial functions of different degrees and find that stationary results are obtained for degrees larger than about four. A model based on an autoregressive scheme with long-time memory is briefly discussed.

Received: August 17, 2007

AMS Subject Classification: 60G50, 62M10, 62P20, 65Q05

Key Words and Phrases: long-time auto-correlations, fractional Brownian motion, financial markets, volatility

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 42
Issue: 2