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There is one problem that bothers me:

Let’s say I buy a European put option with a certain maturity date with premium \$1.6 Suppose that the market price of the put option rises before maturity (\$3) and that I sell it to earn the difference in the market prices of the option (\$1.4),

will I become the writer/seller of the option? In other words, will my payoff at maturity be $-\max\{ K-S_T,0 \}$?

But if that is the case, and I sell the put option to buyer $B$, who later re-sells it to another buyer $C$ (who holds until maturity), will buyer $B$ be the new writer of the option, who bears responsibility of the purchase at maturity?

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    $\begingroup$ When you sell an option that you own you no longer have any rights/obligations at maturity. Those rights pass to the person who bought the option from you. At maturity nothing happens as far as you are concerned, it is a matter for someone else (the person who is currently long and the person who is currently short the option). $\endgroup$ – Alex C Oct 4 '19 at 4:46
  • $\begingroup$ I have a question, @Alex C. Your comment is a perfectly valid answer to this question. So is there a reason for writing it out as a comnent, as opposed to an answer? (I am just curious) $\endgroup$ – Dhruv Gupta Oct 4 '19 at 5:32
  • $\begingroup$ @AlexC However, my textbook says that the seller of the put option bears a responsibility to purchase at the strike price at maturity. Or is this "seller" the institution that creates the option? $\endgroup$ – Richard Oct 4 '19 at 8:58
  • $\begingroup$ Stricly speaking the textbook should say "the short seller of the option...". $\endgroup$ – Alex C Oct 4 '19 at 14:39
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The vast majority of options are traded on an exchange, which means that you actually have a contract with the exchange, not a third party. So if you buy an option, you initiate a contract with the exchange. When you sell it on the exchange, the original contract you have with the exchange is voided and a new contract between the exchange and the buyer is generated (the exchange takes care of this automatically).

my textbook says that the seller of the put option bears a responsibility to purchase at the strike price at maturity

Correct, if you sell to open a position. If you sell to close a position (meaning you previously bought a contract and are now selling it), your original option goes away.

will I become the writer/seller of the option?

Technically, yes, but that position offsets the put that you initially bought, so the exchange just cancels the initial position.

Financially it's the same effect. Suppose you bought a put, then sold another put at the same strike $K$ with the same maturity. If the puts are in the money at expiration, then you would be obligated to buy the stock for $K$ via your sold put, but then would exercise the option to sell it via the bought put at the same price $K$. So you have no net profit or loss in the transaction.

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  • $\begingroup$ Your statements related to the exchange's role are incorrect. The exchange is never the other side of the contract unless it's an OTC market. Much more commonly the exchange provides a market place, most typically in the form of a limit order book. Buyers and sellers bid for the given instrument on that exchange, when prices match a trade executes which establishes a contract between buyer and seller. Then, in the case of regulated futures/options markets, a clearing house steps in and novates the contract such that the buyer and seller positions now are contracted against the clearing house. $\endgroup$ – Ian Ash Oct 4 '19 at 15:28
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    $\begingroup$ Hmm, it’s the OTC markets where investors trade with investors, cutting out the exchange. You are semantically correct in the distinction between an “exchange” and its associated “clearing house”. However, the distinction is semantic to anyone more interested in the economics of trading than the legal plumbing of the system’s operations. $\endgroup$ – demully Oct 5 '19 at 7:40

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