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If I look at the trading hours announcement of the New Stock Exchange, I see that there are seven different entities listed (NYSE, American, Arca, National and 3 options-related entities).

How are the four equity entities (NYSE, American, Arca and National) related? Are they just four different collections of stocks that are all traded by the same people, or are there actually four completely separate exchanges in different rooms with different people and different rules, or is it somewhere between these two extremes?

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NYSE American Equities used to be known as the American Stock Exchange. It is a separate exchange from the NYSE.

NYSE Arca is owned by Intercontinental Exchange and is located in Chicago. It trades stocks and options.

NYSE National is an electronic exchange with a taker/maker schedule.

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Yes, these are different and separate exchanges that were originally separate from the New York Stock exchange (now all are owned by the Intercontinental Exchange group).

Most of them are electronic, meaning that trades are completed electronically through the computers in the exchange's data centers (in Mahwah, NJ) rather than on a physical trading floor.

From the regulator's point-of-view they are still treated as separate exchange with separate members, separate listings (though most stocks have one exchange that is the 'primary' listing, and other 'secondary' listings), separate rule-books, and different fee-schedules. Many members are members of more than one exchange, and most of the rules and pricing are similar. Most of the exchange servers now run on much of the same software as the other exchanges, with slight variations to account for the differences in rules.

One way to look at the relative importance of each is to see how much volume is trading on the exchange you are interested in. BATS (now owned by CBOE) offers a real-time report:

https://markets.cboe.com/us/equities/market_share/

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