Theoretical and empirical analysis of trading activity

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 tradesN, the traded volumeV, the asset priceP, the squared volatility sigma 2 the bid-ask spreadSand the cost of tradingC. 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 similar to sigma 2 More sophisticated relations are the so called 3/2-lawN3/2 similar to sigma PV/Cand the intriguing scalingN similar to(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.