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Having a long position in a stock means you own the security.
You have the right to collect the dividends or interest the security pays, the right to sell it or give it away when you wish, and the right to keep any profits if you do sell.
Similarly, you have a long position in an option when you hold the option, and you have the right to exercise it before expiration or sell it.
The term is also used to describe a position that's maintained by your brokerage firm or bank on your behalf. For example, if your firm holds stocks for you in street name, you are said to be long on their books.
Having a long position is the opposite of having a short position in a security. A short position means you have borrowed shares through your broker, sold them, and must return them, plus interest, at some point in the future.
Similarly, a short position in an option means that you have sold the option, giving the holder the right to exercise and committing yourself to fulfilling the terms should exercise occur and you're assigned to meet them.