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Is it possible for a trade to have positive market impact? We are accustomed to a trade having negative market impact and essentially a cost associated to it, either immediately through price impact or through a change in the order book. I have had heard polarizing opinion on this thus far; one trader tells me "no its impossible market impact is always negative" and the other "of course its possible your trade can provide liquidity to the market if there's matching flow". Thoughts?

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Personally I believe that the market impact can only be negative from a new trade completion.

If you want to address the second trader's view about providing liquidity to the market consider this hypothetical:

An illiquid and infrequently traded market has three actors A, B, C, and an observer. Each actor maintains a public view of the mid-market and a private view as follows:

Actor Public Price Private Info
A 100 Assumes the price is 100 no other information.
B 100 Is short wanting to cover will pay upto 105 but hides this interest.
C 100 Assumes the price is 100 no other information.

The observer can see the public price is 100, but the latent (hidden) information is that the price is at least 105 since if any offers in reasonable size were to be shown less than 105 they would be bought by B.

Suppose that C now acquires an interest to sell for whatever reason and offers the market at 102. B immediately trades this and with all parties satiated, the state reverts to:

Actor Public Price Private Info
A 102 Assumes the price is 102 no other information.
B 102 Assumes the price is 102 no other information.
C 102 Assumes the price is 102 no other information.

From the perspective of the Observer and that of C, C's trade has had a positive market impact, from 100 to 102.

But from the point of view of knowing the latent market the trade has had a negative market impact since B would have paid upto 105 but managed to buy at 102 instead and has nothing to do after that trade.

Thus, this example depends heavily upon your definitions which become particularly unclear in illiquid markets. For frequently traded and liquid markets I suspect that this will all go away, i.e. the latent market will converge to the observed market in which case the impact can again only be negative.

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