Put options give the right, but not the requirement to sell an underlying security at a specific price during a specific period.
The buyer of a put is someone who is looking to protect a position they think may lose value, essentially an insurance they can get a certain price for their position if the price on the open market is lower than the put option.
The seller of a put option is looking to gain value from the actual sale of the option in hopes that the underlying position will go up in value and they will not have to buy the underlying position at a higher value than is available on the open market.
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