The confusing terminology mentioned in the question deals with entering and exiting option orders. In review, there are two main ways in which you can participate in options, you can either buy an option or write an option. When you buy an option you are buying the right to either buy (call option) or sell (put option) the underlying asset at a set price before a set date. If you write an option, you are selling this right for a premium.

When you enter a trade, you are essentially opening a position, hence the orders "sell to open" and "buy to open." If you are buying an option, either a put or a call, you must enter a "buy to open" order. If you are writing an option, also referred to as selling an option, you must enter a "sell to open" order.

Now, to exit an order, you need to close your options position. If you bought an option, you need to use a "sell to close" order, which is essentially like owning a stock where you sell it back into the market to close the position. If you wrote an option, you will need to use the "buy to close" order. While it may seem odd that you would buy to close a position, by taking a long position in the option, you neutralize the rights you sold when you wrote the option with the rights gained when you buy the new options, which closes your position.

In summary, a person holding a short position (contract writer) can sell to open (enter a contract) or buy to close (close a position). A person holding a long position (contract purchaser) can buy to open (enter a position) or sell to close (close a position).

For more information, check out Options Basics Tutorial and Stock Basics Tutorial.

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