Theoretical and empirical analysis of trading activity

TL;DR

This paper combines theoretical dimensional analysis with empirical data to explore and validate scaling laws relating trading activity, volatility, and other market variables, revealing universal patterns in financial markets.

Contribution

It introduces and proves the uniqueness of scaling laws between market variables using dimensional analysis and supports these laws with empirical NASDAQ data.

Findings

Scaling laws like N ∼ σ² and N^{3/2} ∼ σ P V / C hold empirically.

The 3/2-law and related scaling relations exhibit universality across data.

Time scaling of volatility σ is more complex than expected.

Abstract

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.