  Understanding the structure of financial markets deals with suitably
determining the functional relation between financial variables. In this
respect, important variables are the trading activity, defined here as the
number of trades $N$, the traded volume $V$, the asset price $P$, the squared
volatility $\sigma^2$, the bid-ask spread $S$ and the cost of trading $C$.
Different reasonings result in simple proportionality relations ("scaling
laws") between these variables. A basic proportionality is established between
the trading activity and the squared volatility, i.e., $N \sim \sigma^2$. More
sophisticated relations are the so called 3/2-law $N^{3/2} \sim \sigma P V /C$
and the intriguing scaling $N \sim (\sigma P/S)^2$. We prove that these
"scaling laws" are the only possible relations for considered sets of variables
by means of a well-known argument from physics: dimensional analysis. Moreover,
we provide empirical evidence based on data from the NASDAQ stock exchange
showing that the sophisticated relations hold with a certain degree of
universality. Finally, we discuss the time scaling of the volatility $\sigma$,
which turns out to be more subtle than one might naively expect.
