I know a stock price is determined by supply and demand, but what I am actually asking is: Who monitors the supply and demand then sets the price accordingly? Is it done automatically by a computer system at the stock exchange market, or is there a person who monitors the daily action and then modifies the stock price manually during the day according to that? And if the stock price is modified each day by a person, does this person work directly for the company who issued the stock, or for the stock exchange who trades it? Also what about the bid/ask prices? Are they set by an automated system or by a person?

The NYSE is a bit more ancient. I don't know if they still use this system or not, but they have specialists for a given stock. More popular traded stocks may have multiple specialists, and a specialist might take a bunch of thinly traded stocks. Usually, brokers are the ones placing orders to the specialist and the specialist will match up the buyers and the sellers and take a cut. They might have gone electronic by now. The specialist works for the stock exchange and pays a large fee for a seat on the exchange.
The NASDAQ is a fully automated computer system. The tons of brokers instead of talking to the specialist just input their orders. The array of buy and sell orders form the bid/ask. When there's a match, the trade executes by computer.




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