Quotes are current pricing information about individual securities on an exchange.
A potential investor will refer to a current quote to see what price a security traded at most recently. A quote will also show the bid and ask prices, which indicates the price other buyers are attempting to buy the security for (bid), and the price sellers are trying to sell it for (ask).
If you are selling, you're going to get the bid price, and if you're buying, you're going to pay the ask price. The difference between the two is called the spread, and will basically be pocketed by the broker or specialist that handles the transaction. A security with a spread of zero indicates high liquidity and is referred to as a frictionless asset or trade.
Quotes used to be viewed on ticker tape, which was constantly printed out during trading hours with the most recent price information, and this is where the shorthand ticker symbols for companies originated. This information was originally transmitted via telegraph.
Now of course quotes can be viewed on demand with price information updated instantaneously. Bid and ask prices are recorded in an electronic order book and a matching engine finds the closest bid and ask prices.
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