The arithmetic at expiration
The intrinsic value of one share is what the right is worth the moment it expires: for a call, the amount the stock finishes above the strike, and for a put, the amount it finishes below it. Neither can be negative, because the owner of an option simply does not exercise a right that is worth nothing.
A long position pays the premium and collects that payoff, so the profit is the payoff on the whole position minus the premium and both commissions. A short position collects the premium and pays the payoff, which is the same expression with the sign reversed. The break-even price is the strike plus, for a call, the cost per share of the premium and the commissions - the price at which the payoff exactly repays what the position cost.