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If a put seller closes early, what happens to the buyer?
The original option writer (seller) can close his short position in the contracts he wrote by purchasing back matching contracts (i.e. contracts with the same terms: underlying, option type, strike price, expiration date) from any others who hold long positions, or else who write new matching contract instances. Rather than buyer and seller settling directly, options are settled through a central options clearing house, being the Options Clearing Corporation for exchange-listed options in the U.S. See also Wikipedia - Clearing house (finance). So, the original buyer of the put maintains his position (insurance) and the clearing process ensures he is matched up with somebody else holding a matching obligation, if he chooses to exercise his put. I also answered a similar question but in more detail, here.