Dimensional Analysis and Empirical Scaling Laws for Trading Activity

Summary

This paper studies relationships among trade counts, traded volume, price, volatility, spreads, and trading costs. It uses dimensional analysis to show which proportionality relations are possible for selected combinations of these variables. The relations include a proportionality between trade count and squared volatility, a three-halves scaling involving price, volume, volatility, and cost, and a squared scaling involving volatility, price, and spread.

The authors test the more sophisticated relations using NASDAQ stock data and report empirical support with some degree of universality. They also discuss how volatility scales over time, noting that this behavior is more subtle than a simple assumption might suggest. The summary provides no effect sizes or detailed testing procedures, and the reported relationships should be understood within the variable sets and evidence considered in the study.

Key ideas

• Trade count, volume, price, volatility, spread, and cost can be linked through dimensional scaling relations.
• Dimensional analysis identifies the possible forms of scaling laws for selected variable sets.
• The paper examines a three-halves relation connecting activity, volatility, price, volume, and cost.
• NASDAQ data provide empirical support for the sophisticated scaling relations, with some universality.
• Volatility's time scaling requires more care than a naive assumption would imply.

Tags

# Theoretical and empirical analysis of trading activity


# 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 trades $N$, the traded volume $V$, the asset price $P$, the squared volatility $σ^2$, 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., $N \sim σ^2$. More sophisticated relations are the so called 3/2-law $N^{3/2} \sim σP V /C$ and the intriguing scaling $N \sim (σP/S)^2$. 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 $σ$, which turns out to be more subtle than one might naively expect.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.