Hi LeeSang Yes, (b) is possible as a response, but (a) is not. "I haven't worn my watch today" is not a sentence I would expect to hear in any conversation. It suggests that although you did not put your watch on earlier today (and thus are not wearing it now) you might possibly put your watch on later today and then wear it for a while.
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LeeSangI understand (b) is the correct answer. but I think (a) may be possible.The present perfect is used for an action that started at some time in the past, and continues to the present.