# Are trading invariants really invariant? Trading costs matter

## TL;DR

This paper empirically tests the trading invariance hypothesis using a large dataset and finds that while the 3/2 scaling holds, the invariance does not, but new invariants related to trading costs show promise.

## Contribution

The study challenges the invariance hypothesis and introduces new cost-based invariants with reduced variance, supported by extensive empirical analysis.

## Key findings

- The 3/2 scaling between risk and number of bets is robust.
- The original invariant I is highly correlated with trading costs.
- New invariants based on cost ratios have significantly lower variance.

## Abstract

We revisit the trading invariance hypothesis recently proposed by Kyle and Obizhaeva by empirically investigating a large dataset of bets, or metaorders, provided by ANcerno. The hypothesis predicts that the quantity $I:=\ri/N^{3/2}$, where $\ri$ is the exchanged risk (volatility $\times$ volume $\times$ price) and $N$ is the number of bets, is invariant. We find that the $3/2$ scaling between $\ri$ and $N$ works well and is robust against changes of year, market capitalisation and economic sector. However our analysis clearly shows that $I$ is not invariant. We find a very high correlation $R^2>0.8$ between $I$ and the total trading cost (spread and market impact) of the bet. We propose new invariants defined as a ratio of $I$ and costs and find a large decrease in variance. We show that the small dispersion of the new invariants is mainly driven by (i) the scaling of the spread with the volatility per transaction, (ii) the near invariance of the distribution of metaorder size and of the volume and number fractions of bets across stocks.