Offsetting the option is a way of exiting an option position. To offset a position, a person who had bought a call would have to sell a call with the same strike price and expiration, or a person who sold a call would have to buy a call with the same strike price and expiration. A put would work the same way. Allowing the option to expire on the expiration date is a way of exiting an option position. If an option is not exercised by its expiration date, it simply becomes worthless. Exercising the option is a way of exiting an option position. The option may be exercised by choosing to purchase the underlying security (if a call), or choosing to sell the underlying security (if a put). There are three ways to exit an option position, and returning the option to the party it was purchased from is not one of them. The option may be exercised by choosing to purchase the underlying security (if a call), or choosing to sell the underlying security (if a put). Or the position may be offset. To offset a position, a person who had bought a call would have to sell a call with the same strike price and expiration, or a person who sold a call would have to buy a call with the same strike price and expiration. A put would work the same way. The third way to exit an option position is to simply allow the option to expire on its expiration date, at which time it becomes worthless.
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