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Is it true that approved market makers are simply taking the other side of the orders from retail or maybe institutions whatever the price is without actually being on the order book, and they also won’t be on the footprint chart even after the execution?

I think there’re differences between market makers that’s officially approved by the exchange and is not basically HFTs and doesn’t manipulate the price, and market maker who has enough liquidity and basically is HFT but isn’t approved and just using market-making-like strategy and actually manipulates the price.. and what everyone calls market maker nowadays seems pretty much like the latter one… is it true…?

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Being recognised market maker allow to have some advantages: on some trading venues they can see the flows of incoming orders with more transparency, on others they pay less fees.

Market makers do not "blinding" buy and sell, they implement strategies that are based on two principles

  • understand the flows: are they informed? informed flows are buying before the price goes permanently up or selling before it goes permanently down. If you want more details, see for instance Kyle's model.
    Having access to retail flows (that are mostly uninformed) helps.
  • control of the inventory list: adjust you quotes to unwind your inventory when it is too much imbalanced. See Dealing with the Inventory Risk. A solution to the market making problem, By Guéant, L and Fernandez Tapia.

I think that your point about "HFT manipulating prices" is not true. Market authorities are monitoring markets and punish bad practices; no market participant can durably manipulate prices. At worst one can consider that HFT are creating negative externalities by pushing other participants to invest in data processing. If you want more details you can have a look at Market Microstructure in Practice, by L and Laruelle.

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An official market maker at an exchange is supposed to continuously provide double sided quotes for the security they register for (they aren't generally market makers for all securities). These market makers are definitely "in the foot print" all the time because they are required to be in the order book continuously. They are almost all the time required to be within a certain spread in the order book (i.e. price between their own bid and offer).

In return for this, like @lehalle mentioned, they get preferential fees. The "early view" of orders that he states is part of what is called order book priority. Given the same price, orders from market makers may match early before other orders, subjected to some restrictions.

In order for a market maker to fulfill their obligation, they may employ an HFT algorithm (to quote in a quickly changing market), but registered market makers are always part of the footprint of the executions that happen in an exchange.

Some resources: Obligations of market makers:

Incentives for market makers (fees):

Incentives for market markers (order book priority/execution priority):

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