One Curve for Every Stock: Lillo, Farmer and Mantegna on Market Impact

Here is a question that costs the asset management industry an enormous amount of money every year: if I buy this much stock, how much will I push the price up?

The answer, obviously, depends on the stock. Buying a million dollars of a tiny company will move it violently. Buying a million dollars of Apple will move it not at all. Every trader knows this, and every trader has learned, painfully and by experience, roughly what to expect for the names they trade.

Lillo, Farmer and Mantegna asked a physicist's question about this mess: is there any order underneath it? Is the relationship between trade size and price impact fundamentally different for a big stock and a small one, or is it the same relationship, just stretched to a different scale?

Their answer, published in Nature in 2003, is one of the more elegant results in empirical finance.

The problem: every stock seems to have its own rules

Plot price impact against trade size for a set of stocks, and you get a mess. Small caps have steep curves. Large caps have shallow ones. Different companies, different curves, apparently no common structure.

You could resign yourself to this. Fit a separate impact model to every stock, recalibrate constantly, accept that liquidity is idiosyncratic.

Or you could suspect that the differences are cosmetic, and that some deeper regularity is hiding underneath.

The key idea via analogy: the same wave in a different sized pool

Physicists have a habit that finance people should steal. When two systems look different, they ask whether the difference is just a matter of units.

Think about dropping a stone into a pond versus dropping a stone into a bathtub. The splashes look completely different in absolute terms. But if you measure the splash relative to the size of the container and the stone relative to the volume of water, you might find that the physics is identical, and the two situations differ only in scale.

That is the move. Rescale, and see if the curves collapse.

Lillo, Farmer and Mantegna took a large body of stock data, sorted the stocks into groups by market capitalization, and computed the average price impact for trades of different sizes in each group. As expected, the raw curves were all over the place, one per size class.

Then they rescaled. They divided the trade size by a measure of how much that class of stock can absorb, and divided the price move by a corresponding scale factor. In effect they stopped asking "how many shares did you buy" and started asking "how big was your trade relative to what this stock can normally digest?"

And the curves collapsed onto one.

Not approximately. Not for a couple of stocks. All the size classes, across four different years, landed on a single common curve. Once you measure size in the right units, a small cap and a mega cap respond to a trade in exactly the same way.

The technical name for this is a data collapse, and in physics it is a very strong signal that you have found a real law rather than a coincidence. If the underlying relationship were genuinely different across stocks, no rescaling would make them line up. The fact that a single rescaling works means there is one mechanism, operating at different scales.

The shape of that universal curve is concave: impact grows with trade size, but at a decreasing rate. The second million shares hurts less than the first million did. This is the same concavity Hasbrouck had found in his 1991 analysis, and it is the empirical fact that would later crystallize into the famous square-root law of market impact, that impact grows roughly with the square root of size rather than in proportion to it.

Why it mattered

• It made market impact a science rather than a craft. The finding that impact obeys a universal law, once you normalize for liquidity, means impact is predictable from a small number of observable quantities. That is the entire foundation of modern transaction cost analysis and pre-trade cost estimation.
• It is the cornerstone of the square-root law. The concave, universal impact curve documented here is the empirical bedrock on which Bouchaud, Toth and collaborators built the square-root impact law, which is now the industry standard model of what a large order costs. Every execution desk's cost model is, in some form, a descendant of this curve.
• It changed how big orders get traded. Concavity has a huge practical consequence. If impact were linear in size, splitting an order into ten pieces would cost exactly the same as trading it in one go, and there would be no point slicing. Because impact is concave, and because the cost of pushing the price scales sub-linearly, the economics of order execution, of how fast to trade and how much to show, all hinge on the exact shape of this curve. Getting the shape right is worth billions.
• It demonstrated that markets have universal laws. This is the deeper claim, and it is what got the paper into Nature. The authors argue that the collapse suggests fluctuations from supply-demand equilibrium in many financial assets are governed by the same statistical rule. That is a strong, almost physical statement about markets, and it is one of econophysics's genuine successes.

The honest limitations

• This is the impact of individual trades, not of your order. The paper measures the average price move following single transactions. But what a real trader cares about is the impact of a metaorder: a large parent order sliced into hundreds of children and executed over hours. Those are different objects, and the mapping between them is subtle and contested. A lot of subsequent work exists precisely to sort this out, and the square-root law as practitioners use it is a statement about metaorders.
• Correlation is not causation here either. The curve shows that big trades are associated with big price moves. It does not establish that the trade caused the move. A large buy might occur precisely because information has arrived that would have moved the price anyway. Disentangling the mechanical impact of the order from the information that motivated it is genuinely hard, and this measurement does not do it.
• The rescaling requires estimating the scale factors. The collapse is only as good as the normalization, and the normalization depends on choices about how to measure a stock's liquidity. Different choices produce different degrees of collapse.
• Universality claims invite scepticism. Data collapses are seductive, and the human eye is generous when looking at log-log plots. The result has held up across many markets and many subsequent studies, which is reassuring, but the strength of the "universal law" claim is stronger than the strength of any single test of it.
• It describes, it does not explain. The paper documents a universal shape. It does not say why impact is concave. The theoretical explanation, which involves the latent, hidden nature of real liquidity, came later, most notably from Toth, Bouchaud and collaborators.

The one-line takeaway

Lillo, Farmer and Mantegna showed that price impact looks different for every stock only because we were measuring in the wrong units: rescale trade size by what the stock can absorb, and every impact curve, for every size of company, collapses onto one universal concave shape, which is the empirical seed of the square-root law of market impact.