Your Impact Model Might Be a Money Machine: Gatheral on No-Dynamic-Arbitrage

By the late 2000s, the market impact literature had two well-established empirical facts and no idea whether they were compatible.

Fact one: the impact of a trade is concave in its size. Trading twice as much does not move the price twice as far, it moves it less than that. The square-root law is the popular version.

Fact two: impact is transient. It decays. Push the price up with a buy, and the price partially drifts back down over the following minutes.

So the natural thing to do, if you are building an execution model, is to pick a concave impact function you like and a decay function you like, bolt them together, and start optimising. Thousands of people did exactly that.

Jim Gatheral's paper points out that most of the ways you could bolt them together are internally inconsistent, in the strongest possible sense: they imply the existence of a strategy that makes money out of nothing, with certainty, by trading in circles.

The problem: models that print money are wrong models

Here is a test any impact model must pass. Consider a round trip: buy some shares, then sell exactly the same number, ending where you started with zero position.

You have consumed liquidity in both directions. You have paid the spread twice. Your expected profit from a round trip, ignoring any information you might have, must be negative. Trading costs money. If your model says otherwise, your model is broken.

Now, the sneaky part. Nobody sets out to build a model that pays you to trade. But when you combine a concave impact function with a decay function, whether the round trip costs money depends on the interaction between the two shapes, and that interaction is not remotely obvious by inspection.

Gatheral shows that with the wrong pairing, there exist trading trajectories, sometimes rather baroque ones involving trading fast then slow then fast again, that generate positive expected profit. These are called price manipulation strategies, and their existence in your model is fatal. Any optimiser you point at that model will find them, and the "optimal" execution schedule it hands you will be a nonsense strategy trying to harvest a bug in your own assumptions.

The key idea via analogy: the shape of the dent and the speed of the bounce

Think of the price as a mattress and your trades as you pressing down on it.

The impact function describes how deep the dent is when you press with a given force. A concave impact function means the mattress resists more as you press harder: the first bit of pressure makes a big dent, further pressure makes progressively less additional dent.

The decay function describes how the mattress bounces back after you stop pressing. Fast bounce, slow bounce, immediate partial bounce followed by a lingering residual, and so on.

Gatheral's question is: which pairs of dent-shape and bounce-shape are physically coherent?

The answer is that they are not independent. The shape of the impact function constrains the permissible shape of the decay. They must be matched to each other, and the paper derives the relationships that must hold.

The result that most people remember is a sharp one, and it is uncomfortable. Exponential decay of market impact is only compatible with linear market impact. Exponential decay is the modelling assumption everyone reaches for first, because it is mathematically convenient and it is what Obizhaeva and Wang used. Gatheral's result says that if you insist on exponential decay, you are implicitly committed to linear impact, which flatly contradicts the mountain of evidence for concavity. You cannot have your convenient decay and your empirically supported concavity at the same time.

So what does work? Gatheral shows that pairing a power-law impact function with a power-law decay can be made consistent, provided the exponents satisfy certain inequalities. And here is the genuinely striking part: when he checks those inequalities against the exponents people actually measure in real markets, the inequalities are close to being satisfied with equality. The real market appears to sit right at the boundary of what is arbitrage-free. It is as though the market has arranged itself so that no free money is available, but only just.

Why it mattered

• It imposed discipline on a sloppy field. Before this, impact modelling was largely "fit something concave, fit something decaying, ship it." Gatheral supplied a consistency check that every model must pass, and quite a few popular ones failed.
• It killed a favourite shortcut. The exponential-decay assumption is everywhere in the execution literature because it makes the maths tractable. Gatheral showed the price you pay for that tractability is an impact function contradicted by the data.
• It explained a real practical failure. Execution desks had been finding that optimisers run against certain impact models produced bizarre, obviously wrong schedules. This paper explained why: the optimiser was correctly finding the arbitrage the modeller had accidentally built in.
• It suggested markets sit at a critical point. The observation that real-world exponents nearly saturate the no-arbitrage inequalities is a genuinely deep hint about market structure, and it connects to the "criticality of liquidity" ideas that Bouchaud and colleagues were pursuing at the same time.

The honest limitations

• The no-arbitrage principle is heuristic, not a theorem about the world. Gatheral is careful about this. The argument is that a model permitting free money is a bad model, which is a modelling principle rather than a fact about markets. Real markets have frictions, minimum tick sizes, latency, and hard capacity limits that could in principle sustain small violations.
• It assumes impacts simply add up. The framework treats each trade as contributing an independent, decaying push, with the pushes summing linearly. That is a strong assumption, and it sits awkwardly with the concavity it is trying to accommodate.
• The results are about a specific family of models. The clean conclusions concern power-law and exponential forms. The general problem of characterising every arbitrage-free impact-and-decay pair is harder, and the paper does not fully close it.
• Nobody is trading in the model. The strategies considered are deterministic schedules in a market that reacts mechanically. There are no other participants who might notice you executing a "manipulation" strategy and adjust.
• The exponents are still contested. The claim that reality sits near the arbitrage boundary depends on empirical exponents that different researchers measure differently.

The one-line takeaway

Gatheral showed that you cannot pick your market impact function and your impact decay function independently, because most combinations imply a strategy that makes free money by trading in circles, and in particular the convenient assumption of exponentially decaying impact forces you into linear impact, which the data flatly rejects.