Efficiency Forces the Shape: Farmer, Gerig, Lillo and Waelbroeck on Why Impact Is Concave

By the early 2010s the market impact literature had accumulated a set of stylised facts that everyone accepted and nobody could derive.

Impact is concave in order size. Impact decays after you stop trading. The market partially reverts once your metaorder ends. These facts are robust, they show up everywhere, and they were mostly explained by appeals to the mechanics of the order book: liquidity is thin, books refill, and so on.

Doyne Farmer, Austin Gerig, Fabrizio Lillo and Henri Waelbroeck take a different route. They argue that you can derive the shape of market impact from the requirement that prices be unpredictable, with almost no assumptions about order books at all. Efficiency, not plumbing, is what does the work.

The problem: why should impact be concave, really?

The mechanical explanations for concave impact are fine as far as they go, but they have a slightly unsatisfying flavour. They say impact is concave because the order book has a particular shape, and the order book has that shape because of how liquidity providers behave, and liquidity providers behave that way because... at some point the chain of explanation runs out and you are just describing the market rather than explaining it.

The authors want something sturdier. They want to show that any market in which prices are unpredictable must exhibit concave, decaying impact, regardless of the details of the exchange.

The key idea via analogy: the market is a Bayesian detective

Here is the setup, and it is worth going slowly because the logic is genuinely clever.

Order flow is highly persistent. This is the long-memory fact: if the last trade was a buy, the next one is likely to be a buy, and this persists for hours. It happens because large traders split their orders into thousands of pieces.

So the market is playing a guessing game. Every liquidity provider watching the tape sees a stream of buys and is trying to answer one question: is this a big buyer who will keep going, or is this order about to finish?

Think of the market as a detective. Each buy is a clue. Early in a long buy programme, the detective sees a handful of buys and thinks: this could be anyone, probably a small order, I will not move my prices much. But as the buys keep coming, the detective updates. Fifty buys in a row. This is a serious buyer. This is information. By this point the detective has already marked prices up substantially.

Now here is the crucial step. If the price is to be unpredictable, the detective's guesses must be right on average. That is what efficiency means: you cannot systematically predict where the price goes next. The price must, at every moment, equal the market's best estimate of value given everything observed so far.

The authors show that when you impose this condition, in a market where order flow has the observed long-memory structure and traders are trying to infer the size of the metaorder in progress, concavity falls out automatically.

The intuition: the first few shares of a large order are highly informative, because they update the market from "nothing is happening" to "somebody might be buying." That is a big shift in belief, so it produces a big price move. The thousandth share is barely informative at all, because the market has already concluded that a large buyer is present and has already priced that in. Its marginal effect is tiny.

Big price move per share early, small price move per share late. That is exactly what concavity means.

The same logic explains the decay. When the metaorder ends, the market's expectation of continued buying has to be revised downward. It was expecting more buys, and they did not come. That downward revision pulls the price partially back, which is precisely the observed post-order reversion. The market overshot because it was pricing in a continuation that never arrived, and now it must correct.

So both stylised facts, concavity and reversion, emerge from a single principle: the market is continually and rationally updating its guess about how much more you have left to trade, and prices must be unpredictable given those guesses.

Why it mattered

• It grounds impact in efficiency rather than plumbing. This is a genuinely different kind of explanation. It says the shape of impact is not an accident of how any particular exchange is built, but a logical consequence of prices being unpredictable in a world where large orders are split up. That predicts universality, which is what the data shows.
• It reframes impact as information, not friction. A great deal of trading practice treats impact as a cost to be minimised, a tax on trading. This paper says impact is largely the market correctly learning from your order flow. The price moves against you because you are telling the market something true, and the market is listening.
• It ties together strands that had been separate. Long memory in order flow, concave impact, and post-order reversion had all been documented independently. This work shows they are three consequences of one mechanism.
• It sharpens what "permanent impact" actually means. The residual price move that survives after everything settles is, in this framework, exactly the information content of your trading. That is a much more meaningful definition than "the bit that did not decay."

The honest limitations

• The market's inference is idealised. Everyone in the model is a perfect Bayesian, correctly updating beliefs from the order flow. Real liquidity providers are running heuristics under latency constraints, not solving inference problems.
• It assumes the market cannot tell who you are. The whole argument turns on the market being uncertain about how much you have left. If a large trader is identifiable, or if their execution schedule is predictable, the story changes considerably.
• The efficiency assumption is doing an enormous amount of work. It is imposed, not derived. Whether real markets are efficient enough at the relevant time scales to force this structure is exactly the sort of question you cannot settle from inside the model.
• It sits alongside, not obviously above, the mechanical explanations. The latent order book story from Toth and colleagues explains the same facts through a completely different mechanism, with no information or inference required. Both fit the data. Deciding between an informational and a mechanical account of impact is one of the genuinely open questions in the field, and this paper argues one side rather than settling the argument.
• It does not give you a calibrated cost model. The paper explains why impact has the shape it does. It does not hand a trading desk a formula with numbers in it.

The one-line takeaway

Farmer, Gerig, Lillo and Waelbroeck argue that concave, decaying market impact is not a quirk of order book plumbing but a logical consequence of price efficiency: the market is constantly guessing how much more you have left to buy, your first shares are far more informative than your last, and when your order ends the market's overshoot must be corrected.