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Are underlying assets supposed to be sold/bought immediately after being bought/sold in call/put option?
When you can exercie your option depends on your trading style. In the american options trading style (the most popular) you're allowed to exercice your options and make profit (if any) whenever you want before the expiration date. Thus, the decision of  exercising your option and make a profit out of it does not rely only on the asset price. The reason is, you already paid for the premium to get the option. So, if taken into account the underlying price AND your premium, your investment is profitable then you can exercice your contract anytime.