The Liquidity You See Is Not the Liquidity There Is: Toth, Bouchaud and the Square-Root Law

Ask any execution trader how much a large order costs, and if they know their business they will tell you it goes roughly with the square root of the size. Buy four times as much, pay twice the impact. This is the single most reliable quantitative rule in trading, it holds across stocks, futures, currencies and decades, and every serious cost model has it baked in.

Now ask why, and watch the room go quiet.

The square-root law is deeply strange. It is not what any standard model predicts. Kyle's model, the foundational theory of price impact, says impact should be linear in order size. And yet the data insists, stubbornly and universally, on the square root.

Toth, Lemperiere, Deremble, de Lataillade, Kockelkoren and Bouchaud's 2011 paper is the most influential attempt to explain why, and the explanation says something genuinely unsettling about what liquidity actually is.

The problem: the visible book is far too small

Here is the fact that should bother you, and that motivates the entire paper.

Look at a real limit order book. Add up all the shares resting on it at every price level. For a typical liquid stock, that total is a tiny fraction of a single day's trading volume, often well under one percent.

Now think about what that means. Institutions trade enormous orders every day, orders vastly larger than the entire visible book. If the visible book were all the liquidity that existed, a single large institutional order would obliterate it, sweeping the book clean and moving the price catastrophically.

That does not happen. Large orders get executed all the time, with impact that is significant but nothing like the devastation the visible book implies.

So where is the liquidity coming from?

The key idea via analogy: the iceberg and the funnel

The authors' answer is that the visible order book is not the liquidity. It is the tip of it.

Most of the true supply and demand in a market is latent: it exists as intentions in the minds of traders and in the parameters of algorithms, not as orders resting on the book. A pension fund that would happily buy at a five percent discount does not post that order. It waits. An algorithm that will supply liquidity if the price comes to it does not show its hand. It watches.

So real liquidity is a hidden reservoir, and the book we see is the thin surface film where a little of that reservoir has been made visible.

Now comes the crucial structural claim, and it is the heart of the paper. The authors argue that this latent liquidity has a specific shape: it is V-shaped and it vanishes at the current price.

Picture a funnel. Far from the current price, there is a lot of latent interest: plenty of people would buy if it fell ten percent. As you get closer to the current price, the latent liquidity thins out. And right at the current price, it goes essentially to zero.

Why must it be zero there? This is the elegant part of the argument. Because if it were not, the price could not move the way it does. Prices are approximately diffusive, they wander like a random walk. If there were a thick wall of liquidity sitting exactly at the current price, the price would be pinned there, unable to wander. The fact that prices do diffuse tells you that liquidity right at the current price must be vanishingly thin. The liquidity gets eaten away at precisely the point where trading is happening.

And from this V-shaped, vanishing-at-the-middle liquidity profile, the square-root law falls out.

Here is the intuition. As you buy, you eat into the funnel. But the funnel is thin near the price and gets thicker as you push further out. So the first shares you buy meet almost no resistance and move the price easily. As you keep pushing, you reach fatter parts of the funnel, and each additional share moves the price less than the one before. That is concavity, and when you work through the geometry of a V-shaped profile, the concavity comes out as precisely the square root.

The word "anomalous" in the title, and the word "critical," carry real weight. In physics, a system is critical when it sits exactly at the boundary between two phases, poised, hypersensitive, where small perturbations produce disproportionate responses. The authors argue markets are exactly this: liquidity is critically thin at the current price, which is why small orders have anomalously large effects and why markets are fragile. The market is not comfortably liquid with a buffer. It is balanced on a knife edge, continuously.

Why it mattered

• It gave the industry's most important empirical rule a theory. The square-root law was a fact in search of an explanation. This paper supplied the most compelling one available, and it explains not just the exponent but why the exponent is what it is.
• It reframed liquidity itself. The distinction between displayed liquidity and latent liquidity is now central to how sophisticated execution is thought about. The book on your screen is not the market. It is a rendering of a small part of it, and reasoning about execution from displayed depth alone is a beginner's error. This paper is the clearest statement of why.
• It explains why markets are fragile. If liquidity is critically thin at the price at all times, then markets do not have a comfortable safety margin. They are always on the edge. Flash crashes, sudden gaps, and liquidity evaporation stop being anomalies requiring special explanation and become the natural behaviour of a system poised at criticality. That is a sobering and important reframing.
• It connects the pieces. The paper explains, in one framework, why order flow has long memory (because large orders must be shredded and dribbled out to navigate the thin funnel), why impact is concave, and why prices diffuse. Three separate empirical puzzles turn out to be three faces of the same thing.

The honest limitations

• The latent liquidity is unobservable by construction. The entire theory rests on a hidden reservoir that, by definition, cannot be measured directly. The V-shape is inferred from its consequences, not seen. That is legitimate science, but it means the theory is hard to falsify directly, and one should be honest that it is a story consistent with the data rather than a directly verified structure.
• The square root is approximate, and the exponent wobbles. Empirically, the impact exponent is not exactly one half. Estimates across markets and studies range around it, often somewhere between 0.4 and 0.7. The theory predicts exactly one half, which is either an impressive achievement or a slightly awkward over-precision, depending on your temperament.
• It says little about the decay of impact. The law describes the impact of a metaorder while it is being executed. What happens afterwards matters enormously to a trader: how much of the impact is permanent, how much relaxes back, and how fast. That question is central to profitability, and this framework addresses it much less completely than it addresses the impact itself.
• The assumptions are idealized. The derivation assumes a fairly stylized order flow and a stationary market. Real markets have regimes, news, and coordinated behaviour that the model does not contain.
• Rival explanations exist. Other frameworks derive concave or square-root impact from different starting points, including models based on the informational content of order flow. The Toth and Bouchaud account is the most influential and the most mechanistically satisfying, but it is not the only one on the table.

The one-line takeaway

Toth, Bouchaud and colleagues explained the square-root law of market impact by arguing that the visible order book is a sliver of a much larger hidden reservoir of latent liquidity, which is V-shaped and vanishes right at the current price, so a large order pushes into progressively thicker liquidity as it goes, and the market as a whole sits permanently at a critical, hypersensitive knife edge.