No, you enter the sold date as the acquisition date. When you sold the option to open (STO) you established the short position, or you acquired it. The premium you received at that time is your proceeds. When you buy to close (BTC) the position, you disposed of it. You would enter that as the date sold. The premium you pay to close a position is your cost. If you the short option expire worthless, your cost basis is zero.
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