  We consider a few quantities that characterize trading on a stock market in a
fixed time interval: logarithmic returns, volatility, trading activity (i.e.,
the number of transactions), and volume traded. We search for the power-law
cross-correlations among these quantities aggregated over different time units
from 1 min to 10 min. Our study is based on empirical data from the American
stock market consisting of tick-by-tick recordings of 31 stocks listed in Dow
Jones Industrial Average during the years 2008-2011. Since all the considered
quantities except the returns show strong daily patterns related to the
variable trading activity in different parts of a day, which are the best
evident in the autocorrelation function, we remove these patterns by detrending
before we proceed further with our study. We apply the multifractal detrended
cross-correlation analysis with sign preserving (MFCCA) and show that the
strongest power-law cross-correlations exist between trading activity and
volume traded, while the weakest ones exist (or even do not exist) between the
returns and the remaining quantities. We also show that the strongest
cross-correlations are carried by those parts of the signals that are
characterized by large and medium variance. Our observation that the most
convincing power-law cross-correlations occur between trading activity and
volume traded reveals the existence of strong fractal-like coupling between
these quantities.
