Beyond the Square Root: When the Industry's Favourite Formula Breaks

The square-root law of market impact is close to gospel. Trade four times as much and you move the price twice as far. It is used by trading desks everywhere to estimate the cost of large orders, it has a beautiful theoretical justification from the latent order book literature, and it fits the data.

Elia Zarinelli, Michele Treccani, Doyne Farmer and Fabrizio Lillo went and checked how well it fits, over how wide a range, on a large set of real institutional orders. Their answer is uncomfortable: the square-root law fits well over about two orders of magnitude in order size, and a logarithm fits well over nearly five.

The problem: a good fit over a narrow window is not a law

There is a trap in empirical work that is easy to fall into. You collect data, you plot it, you fit a curve, and it looks great. What you sometimes forget to ask is: over how wide a range did I test it?

Most tests of the square-root law were run on order sizes spanning a fairly narrow band. Typical institutional orders. And over that band, the square root really does fit nicely. But lots of functions look alike over a narrow range. A square root and a logarithm, both concave, both slowly growing, are extremely hard to tell apart if you only look at a small window of sizes. You need to zoom way out before they separate.

So the authors did the zooming. They took a large database of metaorders, real parent orders executed in the US equity market, and examined how impact behaves across an enormous span of order sizes, from tiny to very large. When you plot across five orders of magnitude rather than two, the two candidate curves pull apart, and you can finally see which one the data prefers.

The data prefers the logarithm.

The key idea via analogy: two curves that look identical until they do not

Imagine two runners. One slows down according to a square root, the other according to a logarithm. Watch them for a hundred metres and you cannot tell them apart. Watch them over a marathon and they end up in completely different places.

That is the situation here. Both a square root and a logarithm are concave: they rise, but with diminishing returns, so the first shares you trade cost you far more than the last. Over the middle of the range they are nearly indistinguishable. But they say very different things at the extremes.

• A square root keeps growing without bound, just slowly. Trade a hundred times more and impact grows by a factor of ten. There is no ceiling.
• A logarithm grows far more slowly still, and it grows ever more slowly. Each successive multiplication of the order size adds the same fixed amount of impact, not a multiplied amount.

The logarithmic finding says, in effect, that the market is even more forgiving of very large orders than the square-root law suggests. The marginal cost of adding more size to an already-large order flattens out dramatically. Very large orders are proportionally much cheaper than the square root predicts.

The paper's second contribution is arguably more useful for practitioners than the first. Impact does not depend on one variable, it depends on two: how big your order is and how aggressively you work it. You can trade a million shares over an hour or over a week, and those are very different trades.

So the authors construct what they call an impact surface: impact as a function of both the size of the metaorder and the participation rate, the fraction of market volume you consume while you are active. This is the object a trading desk actually wants, because a desk does not just choose the size, it chooses the schedule. And when they map that surface, the same logarithmic character shows up in the participation-rate direction too.

Why it mattered

• It put a serious empirical dent in a near-universal assumption. The square-root law is used everywhere and is often treated as settled. A careful study on a large, direction-labelled dataset arguing that a different function fits substantially better across a wider range is a genuinely important challenge.
• The practical consequences are large where it matters most. Where the square root and the logarithm disagree most is at very large order sizes, which is exactly where a trading desk has the most money at stake. Getting the shape wrong at the top end can mean badly misestimating the cost of the trades that matter.
• The impact surface is the right object. Recognising that impact is a two-dimensional surface over size and participation rate, rather than a one-dimensional curve, is a modelling advance that maps directly onto how execution decisions are actually made.
• It is a good methodological lesson. Fitting a curve over a narrow range and declaring a universal law is a mistake made all over quantitative finance. This paper is a clean demonstration of why you must test across the widest range you can get.

The honest limitations

• This does not settle the argument. The square-root law has a theoretical derivation behind it, from the latent order book and criticality ideas. The logarithm, in this paper, is primarily an empirical finding without a comparably deep first-principles story. A well-motivated theory that fits slightly worse is not automatically beaten by a better-fitting curve with no theory.
• The data comes from a specific broker's flow in a specific period. These are US equity metaorders from 2007 to 2009, a window that includes the financial crisis. Whether the conclusion generalises across markets, asset classes and calmer regimes is a fair question.
• The largest orders are the least reliable data points. The whole argument turns on behaviour at the extremes of the size distribution, and those extremes are, by definition, where you have the fewest observations and the noisiest estimates.
• Selection effects are hard to purge. Very large orders that get fully executed are not a random sample. Traders abandon orders that are going badly, and that censoring biases the impact you measure on the survivors.
• Both functions are concave, and that is the finding that matters most. The headline disagreement can obscure the fact that the two camps agree on the big thing: impact is strongly concave, and linear impact models are wrong.

The one-line takeaway

Zarinelli and colleagues showed that the square-root law of market impact fits well only across a narrow range of order sizes, and that when you widen the lens across five orders of magnitude a logarithmic law fits far better, implying that very large orders are considerably cheaper at the margin than the industry's standard formula assumes.