NASDAQ

The Nasdaq is a virtual (listed) exchange, where all of the trading is done by computers. The process is similar to the above, in that the seller provides an asking price and the buyer provides a bidding price. However, buyers and sellers are electronically matched. One or more Nasdaq market makers will always provide a bid and ask price at which they will always purchase or sell 'their' stock.

NYSE:

The New York Stock Exchange is a physicalexchange, where much of the trading is done face-to-face on a trading floor. This is also referred to as a "listed" exchange (because only stocks listed with the exchange may be traded). Orders enter by way of brokerage firms that are members of the exchange and flow down to floor brokers who go to a specific spot on the floor where the stock trades. At this location, known as the trading post, there is a specific person known as the specialist whose job is to match buy orders and sell orders. Prices are determined using an auction method known as "open outcry": the current bid price is the highest amount any buyer is willing to pay and the current ask price is the lowest price at which someone is willing to sell; if there is a spread, no trade takes place. For a trade to take place, there must be a matching bid and ask price. (If a spread exists, the specialist is supposed to use his own resources of money or stock to close the difference, after some time.) Once a trade has been made, the details are sent back to the brokerage firm, who then notifies the investor who placed the order. Although there is human contact in this process, don't think that the NYSE is still in the Stone Age; computers do play a huge role in the process, especially for so-called "program trading".

(printed under the GNU Free Documentation License from  Wikipedia)

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