Impact Must Be Linear, or Else: Huberman and Stanzl on Price Manipulation

Every model of trading needs an assumption about how your own trades move the price. Trade a lot, the price moves a lot. Fine. But exactly how much? Is the relationship a straight line, a curve that flattens out, a curve that steepens?

You might think this is a detail to be settled empirically, with no deep theory involved. Gur Huberman and Werner Stanzl showed that is wrong. There is a theoretical constraint on the shape, and it is a severe one. Get the shape wrong and your model implies that a trader can manufacture profit out of thin air by simply trading in a clever pattern, no information required.

The problem: the shape of impact is not a free choice

Consider a round trip that ends where it started. You buy some shares, you sell some shares, and by the end your position is zero and the fundamental value of the stock has not changed. You have created no value. You have taken no view. You have simply shuffled shares back and forth.

Your profit from this should be negative. You consumed liquidity, you paid the spread, you moved prices against yourself. Trading costs money. Any model that says otherwise is describing a universe where you can get rich by pointlessly churning, which is not a universe anyone believes in.

Huberman and Stanzl call a strategy that generates positive expected profit from such a round trip a price manipulation strategy, and they call the ability to build strategies with unboundedly good risk-reward ratios out of them a quasi-arbitrage. Their question: which impact functions rule these out?

The key idea via analogy: the crooked exchange rate booth

Imagine a currency booth whose exchange rate depends on how much you exchange at once. Suppose swapping a small amount gets you a proportionally better rate than swapping a large amount.

Now here is your money machine. Change a huge amount of pounds into euros in one big transaction, accepting the poor bulk rate. Then change your euros back into pounds in many small transactions, each of which gets the good small-lot rate. If the rate schedule is shaped the wrong way, the many small trips back can be worth more than the single big trip out, and you have made money by doing nothing at all.

The problem is entirely about the curvature of the rate schedule. If the rate is a straight line in size, then splitting up a trade gains you nothing and combining trades costs you nothing. Every route from A to B has the same total cost. There is no shape to exploit.

That is Huberman and Stanzl's theorem, and it is remarkably clean. If price impact is permanent and does not depend on time, then the only impact function that rules out price manipulation is a linear one. Any curvature at all, in either direction, opens up a strategy for extracting free money by splitting, combining, or reordering trades in a round trip.

That is a startling result, because it says a purely theoretical requirement, no free lunches, pins down the functional form of something you would have expected to be a purely empirical matter.

But there is a crucial second half to the paper, and it is the half that saves the empirical literature.

Real price impact has two components. Permanent impact is the part that never goes away: the market genuinely revises its view of the stock because you traded, so your buying moved the fair value up for good. Temporary impact is the part that fades: you paid up to get filled quickly, and once you stop pushing, the price relaxes back.

Huberman and Stanzl show that the linearity requirement bites on the permanent part only. The temporary part, the transient cost of demanding immediacy, can take a much more general shape. It can be concave, it can be a square root, it can be whatever the data says, because it does not persist and therefore cannot be harvested by round-tripping.

This is the reconciliation the field needed. All the empirical evidence for concave, square-root-shaped impact can happily live in the temporary component. The permanent component, the part that shifts the price forever, must be a straight line.

Why it mattered

• It made no-arbitrage a design constraint on impact models. After this paper, "does my impact model permit manipulation?" became a question every serious modeller had to answer. Gatheral's later work on the compatibility of impact shape and decay shape is the direct sequel.
• It rescued the linear-impact assumption from being an embarrassing convenience. Almgren and Chriss, and the models built on them, assume linear permanent impact. Critics complained this was chosen purely for tractability. Huberman and Stanzl showed there is a principled reason it has to be that way.
• It gave the field a clean division of labour. Permanent impact: linear, informational, unavoidable. Temporary impact: whatever shape the data says, transient, a cost of immediacy. That decomposition organises a great deal of subsequent work.
• It explained why bad models produce insane optimal strategies. If you build an execution optimiser on a manipulable impact model, the optimiser will hunt down the manipulation and hand you a "strategy" that is really just a bug.

The honest limitations

• The theorem is about a model, not about reality. It says which models are internally coherent. It does not prove that real-world permanent impact is exactly linear. It proves that if you assume permanent, time-independent impact and you want no free lunches, linearity follows. Reality may violate the premises.
• Real markets do have manipulation. Cornering, ramping, banging the close: these are real strategies that real people go to prison for. The paper's framework says they should not be possible, which tells you the framework is missing something, probably strategic interaction between participants and asymmetric information.
• Frictions can sustain small violations. Tick sizes, fees, latency, and capacity limits mean a theoretically available free lunch may be uneatable in practice. Slight violations of linearity might be perfectly stable in a real market.
• It says nothing about how you should actually trade. This is a consistency result, not an execution algorithm. It constrains the models, it does not solve them.

The one-line takeaway

Huberman and Stanzl proved that the permanent part of price impact must be a straight line, because any curvature at all lets a trader manufacture free money by splitting and recombining a round-trip trade, while the temporary, fading part of impact is free to be as curved as the data demands.