When you buy a listed option, you do not end up owing anything to the person on the other side. The Options Clearing Corporation steps into the middle and becomes buyer to every seller and seller to every buyer. It issues the contracts, holds the margin collected from clearing members, and guarantees performance if a member fails.
The OCC also runs the machinery around assignment and exercise: it processes exercise notices after the close and allocates assignments to clearing members, who then allocate to individual accounts. It publishes contract adjustment memos when a corporate action changes the deliverable.
Do not confuse it with the Office of the Comptroller of the Currency, which shares the initials. See occ-comptroller-of-the-currency.
Related: occ-comptroller-of-the-currency, assignment, exercise, call-option