Options on Forwards someone, forward, contract

lunarfollies

CFAI Vol 6 Reading 62 Page 216

Could someone please clarify what the strike price of options on forwards relates to i.e is it the strike price to buy the forward contract or the underlying asset?

The text shows the payoff on call options on forwards as (St -X) if St>X and 0 if St <= X. This is the same payoff as for calls on the underlying asset.

My interpretation of calls on a forward contract is that X is the strike price to buy the underlying forward contract and not the underlying asset per se. Hence if the underlying asset price is St, a forward contract with price F(0,T) has value [St - F(0,T)], with a resultant payoff of X - [St-F(0,T)] on the call. Obviously I am getting it wrong.

Thanks