Selling an option before maturity
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?
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.