Forward Contracts

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).