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Are there empirical estimates for how trades routed through Citadel and other intermediaries differ from trading directly on the exchange?

I've read elsewhere that a large volume of small trades lets them sell to large entities the ability to avoid all the HFT players. It's less about making money on the small trades.



I'm not sure what kind of differences you're asking about, but they're required to give at least as good of a price as the exchange is offering, and in many cases they give a better price




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