Multiscale behaviour of volatility autocorrelations in a financial market

It is well known that stock market returns are uncorrelated on lags larger than a single day, in agreement with the hypothesis of efficient market. On the contrary, absolute returns have memory for longer times; this phenomenon is known in financial literature as clustering of volatility. In ARCH-GARCH models [1–3], volatility memory is longer than a single time step but it decays exponentially while empirical evidence is for hyperbolic correlations [4–9]. In this paper, we perform a scaling analysis of the standard deviation of a new class of observables, the generalized cumulative absolute returns. This analysis clearly shows that volatility correlations are powerlaws on a time range from one day to one year and, more important, that the exponent is not unique. This kind of multiscale behaviour is known to be relevant in the theory of dynamical systems, of fully developed turbulence and in the statistical mechanics of disordered systems (see [10] for a review) while it is a new concept for financial modeling. We consider the daily New York Stock Exchange (NYSE) index, from January 1966 to June 1998, for a total of N = 8180 working days. The quantity we consider is the (de-meaned) daily return, defined as