My question regarding the answer of the below problem, should not the short experience loss when the value of the contract increase (i.e interest decrease) because the contract will be exercised?
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You do not exercise forward contracts because that’s an committed agreement. You would exercise an option because that’s a contingent claim.
the interest rate increase would increase the price if new forwards. The mark to market formula would result in a loss in value for the short when the interest increases, not when the interest decreases.
It depends on the type of option. You can’t exercise a European early, but you can exercise an American option early (although you usually wouldn’t want to do that).