Are trading invariants really invariant? Trading costs matter

Abstract

We revisit the trading invariance hypothesis recently proposed by Kyle, A.S. and Obizhaeva, A.A. [‘Market microstructure invariance: Empirical hypotheses.’ Econometrica, 2016, 84(4), 1345–1404] by empirically investigating a large dataset of metaorders provided by ANcerno. The hypothesis predicts that the quantity $I:=\mathcal{W}/N^{3/2}$, where $\mathcal{W}$ is the daily exchanged risk (volatility × volume × price) and N is the daily number of metaorders, is invariant, either in distribution or in expectation. We find that the 3/2 scaling between $\mathcal{W}$ and N works well and is robust against changes of year, market capitalisation and economic sector. However our analysis shows that I is not invariant, and we find a very high correlation ($R^2>0.8$) between I and the trading cost (spread + market impact costs) of the metaorder. Guided by these results we propose new invariants defined as a ratio of I to the aforementioned trading 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, across stocks, of the shape of the distribution of metaorder size and of the volume and number of metaorders normalised to market volume and number of trades, respectively.

Keywords:

• Trading invariance
• Metaorders
• Trading costs
• Stocks

Acknowledgments

We thank Alexios Beveratos, Laurent Erreca, Antoine Fosset, Charles-Albert Lehalle and Amine Raboun for fruitful discussions.

Data availability statement

The data were purchased from the company ANcerno Ltd (formerly the Abel Noser Corporation) which is a widely recognised consulting firm that works with institutional investors to monitor their equity trading costs. Its clients include many pension funds and asset managers. The authors do not have permission to redistribute them, even in aggregate form. Requests for this commercial dataset can be addressed directly to the data vendor. See www.ancerno.com for details.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Note that here we only explore the daily level, time does not mean the same thing as in Benzaquen et al. (2016) where we varied the time intervals over which the variables were computed.

2 In the following we will make use of such an approximation and use the words ‘metaorder’ instead of ‘bet’.

3 Note that Kyle and Obizhaeva commented on how to modify their invariance principle in an international context. In particular they suggested that ‘invariance relationships can also be applied to an international context in which markets have different currencies or different real exchange rates’ by scaling to ‘the nominal cost of financial services calculated from the productivity-adjusted wages of finance professionals in the local currency of the given market during the given time period’ (Kyle and Obizhaeva 2017).

4 In fact, for example, Kyle and Obizhaeva tackled this problem investigating a proprietary dataset of portfolio transitions (Kyle and Obizhaeva 2016).

5 ANcerno Ltd. (formerly the Abel Noser Corporation) is a widely recognised consulting firm that works with institutional investors to monitor their equity trading costs. Its clients include many pension funds and asset managers. In Kyle and Obizhaeva (2016) the authors claim that the ANcerno database includes more orders than the data set of portfolio transitions they used in their work.

6 We checked that the results discussed in the present work are still valid using other definitions of the daily volatility and of the price in analogy to what done for example in Kyle and Obizhaeva (2016). Specifically, the results are still valid when computing ${\sigma }_{\mathrm{d}}$ with the Rogers-Satchell volatility estimator (Rogers and Satchell 1991, Benzaquen et al. 2016) or as the monthly averaged daily volatility, i.e. ${\overline{\sigma }}_{\mathrm{d}}=\sum _{m=1}^{25}{\sigma }_{d,m}$ and/or defining the price $P_i$ as the closing price of the day before the metaorder's execution.

7 The daily spread is not provided in the ANcerno dataset. We computed it as the time average spread across the day using publicly available market data.

8 The coefficient of variation is the ratio of standard deviation and mean, an indicator of distribution ‘peakedness’.

9 Here $⟨\cdot ⟩$ denotes the average over all days and stocks present in the sample.

10 In analogy, the variance $⟨{\mathcal{W}}_i^2{⟩}_N-⟨{\mathcal{W}}_i^2{⟩}_N$ scales linearly with N, i.e. $⟨{\mathcal{W}}_i^2{⟩}_N-⟨{\mathcal{W}}_i^2{⟩}_N\approx ⟨{\mathcal{W}}_i{⟩}_N^2\sim N$.

Additional information

Funding

This research was conducted within the Econophysics & Complex Systems Research Chair, under the aegis of the Fondation du Risque, the Fondation de l'Ecole polytechnique, the Ecole polytechnique and Capital Fund Management.