What the Tokyo data cornucopia reveals about market impact

New research confirms universality of one of the most non-intuitive concepts in quant finance

Key points Summary Q&A

• Classical models predict a linear relationship between price impact and traded volume.
• But over the past two decades, evidence has accumulated for a square‑root dependence of impact on traded volume.
• Latest research now finds this square‑root law of price impact holds for all individual stocks and for individual decision‑makers, whether informed or uninformed.
• It also casts a new light on the latent‑liquidity theory, suggesting current models need major revision. 

These summaries are generated by an LLM, but edited by humans.

A study by Japanese academics, based on a vast dataset of trades from the Tokyo Stock Exchange, gives an intriguing insight into the relationship between price impact and the square-root law.

But first, some background. Actively buying or selling an asset moves its price up or down. This is an empirical fact with far‑reaching consequences. It is the transmission belt through which private information is incorporated into prices – but by the same token it also allows random noise (and all sorts of behavioural biases) to seep into those prices. As I will argue in my next column, this may generate long‑lived deviations between market prices and fundamental value.

But impact also generates trading costs that can completely wipe out any ‘paper alpha’ a quantitative researcher may come up with. In particular, mis‑specifying how these costs depend on traded quantity can be lethal to trading strategies. Over the past two decades, evidence has accumulated for a square‑root dependence of impact on traded volume. This square‑root law (SQL) has been reported for stocks, futures, currencies, crypto assets, and even implied volatility, by different research groups using both broker and proprietary datasets.

It is no longer acceptable to be content with models that are merely internally consistent and mathematically convenient

This dependence is highly non‑intuitive. Classical models predict a linear relationship between impact and traded volume. If a trader assumes linear impact when it is in fact square‑root, they will trade too large when signals are weak, leading to negative P&L after costs. Conversely, if they assume square‑root impact when it is actually linear, they will trade too aggressively when signals are strong, again leading to negative P&L after costs.

So it is important to be sure that the square‑root law is real. This means: (a) carefully ruling out data biases that might spuriously mimic a square‑root dependence when the true relation is linear, and (b) proposing a convincing theory for why impact should be a concave function of traded volume. In fact, although the SQL is widely accepted by practitioners, some researchers still question its validity and argue it is an artefact with no predictive value.

One possible obfuscating scenario is the following: imagine a trader endowed with a trading signal able to predict (on average) a certain fraction f of the volatility σ of the asset over time horizon T. Assuming the signal is positive, the trader will buy a quantity Q proportional to T (say Q = γT), and because the trade is informed the price will move up by

f σ √T = f σ √(Q/γ)

Et voilà: the SQL would not be at all a ‘mechanical’ reaction of the market to incoming order flow but a mere reflection of the information content of trades.

Now, let’s go back to the Japanese study. The authors, Yuki Sato and Kiyoshi Kanazawa from Kyoto University, obtained access to a treasure trove: a dataset containing all trades in all stocks on the Tokyo Stock Exchange between 2012 and 2018, with tags that allow one to associate each trade with an anonymised trader. They were thereby able to reconstruct all ‘metaorders’, ie, sequences of trades with the same direction (buy/sell) initiated by the same decision‑maker. Their careful analysis shows unambiguously that the SQL holds not only for all individual stocks, but more importantly for all individual decision‑makers – whether informed or uninformed, professional traders, hedge funds, pension funds, or retail investors.

Related

The cost of mis-specifying price impact 

Expected returns can be significantly affected by the wrong use of impact models

This result, based on vastly more data than any previous study, confirms the conjectured universality of the SQL and explains its ubiquitous appearance in earlier work. The absence of any strong dependence on the putative information content of trades is direct evidence that the square‑root law is mechanical (ie, arising from the flow itself) rather than informational in origin.

The dataset is so rich that it also allows one to probe more deeply the ‘latent liquidity’ interpretation of the SQL that I and others have been advocating for 15 years. In our picture, as a buyer pushes the price up, more and more sellers emerge from the sidelines and act as buffers against further price increases – implying that impact should be a concave function of executed volume Q. If the latent liquidity profile is linear close to the current price (as our model predicts), then impact should cross over from linear to square‑root as Q increases.

However, the orders of magnitude do not match at all: the crossover occurs at values of Q two or three orders of magnitude smaller than the theory predicts. In fact, the data suggests the SQL already holds at the level of single transactions, ie, for metaorders with a single child order. Although the latent liquidity idea may still be qualitatively correct, its detailed mathematical incarnation seems to require a complete overhaul – or perhaps we have been heading in the wrong direction altogether, and a new theory must be devised from scratch.

This is what makes quantitative finance so exciting today: the amount and quality of data now available allows us to test models with a level of accuracy comparable to that of the natural sciences. It is no longer acceptable to be content with models that are merely internally consistent and mathematically convenient. As Feynman famously said: “It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are. If it doesn’t agree with experiment, it’s wrong.”

Editing by Alex Krohn