  By applying the multifractal detrended fluctuation analysis to the
high-frequency tick-by-tick data from Deutsche B\"orse both in the price and in
the time domains, we investigate multifractal properties of the time series of
logarithmic price increments and inter-trade intervals of time. We show that
both quantities reveal multiscaling and that this result holds across different
stocks. The origin of the multifractal character of the corresponding dynamics
is, among others, the long-range correlations in price increments and in
inter-trade time intervals as well as the non-Gaussian distributions of the
fluctuations. Since the transaction-to-transaction price increments do not
strongly depend on or are almost independent of the inter-trade waiting times,
both can be sources of the observed multifractal behaviour of the fixed-delay
returns and volatility. The results presented also allow one to evaluate the
applicability of the Multifractal Model of Asset Returns in the case of
tick-by-tick data.
