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Does the option carry with it some identifying information about who wrote it, or is the person who gets assigned determined some other way?

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For listed options, the exchange uses a random process to assign the exercised option to the clearing members of the exchange. The assigned firm then must assign the option to their accounts that wrote the options by some equitable process. The process is required to be approved by the exchange and is usually a random process or FIFO (first in-first out).

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