| Abstract: |
The probability distribution of log-returns for financial time series, sampled
at high frequency, is the basis for any further developments in quantitative
finance. In this letter, we present experimental results based on a large set
of time series on futures. We show that the t-distribution with $\nu \simeq 3$
gives a nice description of almost all data series considered for a time scale
$\Delta t$ below 1 hour. For $\Delta t \ge 8$ hours, the Gaussian regime is
reached. A particular focus has been put on the DAX and Euro futures. This
appears to be a quite general result that stays robust on a large set of
futures, but not on any data sets. In this sense, this is not universal. A
technique using factorial moments defined on a sequence of returns is
described and similar results for time scales are obtained. Let us note that
from a fundamental point of view, there is no clear reason why DAX and Euro
futures should present similar behavior with respect to their return
distributions. Both are complex markets where many internal and external
factors interact at each instant to determine the transaction price. These
factors are certainly different for an index on a change parity (Euro) and an
index on stocks (DAX). Thus, this is striking that we can identify universal
statistical features in price fluctuations of these markets. This is really
the advantage of micro-structure analysis to prompt unified approaches of
different kinds of markets. Finally, we examine the relation of power law
distribution of returns with another scaling behavior of the data encoded into
the Hurst exponent. We have obtained $H=0.54 \pm 0.04$ for DAX and $H=0.51 \pm
0.03$ for Euro futures. |