# The amazing power of dimensional analysis: Quantifying market impact

## TL;DR

This paper uses dimensional analysis to demonstrate that market impact of large trades depends primarily on four key variables, leading to a universal square-root law of impact, with broader implications for market microstructure modeling.

## Contribution

It extends Kyle and Obizhaeva's work by applying dimensional analysis to derive a unique functional form for market impact based on key variables.

## Key findings

- Market impact is proportional to the square-root of trade size.
- Dimensional analysis constrains the form of impact functions.
- Broader variable sets lead to more complex impact relations.

## Abstract

This note complements the inspiring work on dimensional analysis and market microstructure by Kyle and Obizhaeva [18]. Following closely these authors, our main result shows by a similar argument as usually applied in physics the following remarkable fact. If the market impact of a meta-order only depends on four well-defined and financially meaningful variables, then -- up to a constant -- there is only one possible form of this dependence. In particular, the market impact is proportional to the square-root of the size of the meta-order. This theorem can be regarded as a special case of a more general result of Kyle and Obizhaeva. These authors consider five variables which might have an influence on the size of the market impact. In this case one finds a richer variety of possible functional relations which we precisely characterize. We also discuss the analogies to classical arguments from physics, such as the period of a pendulum.