Forward put call parity

I am absolutely spinning my wheels here, I cannot understand why the FC bond instrument is adjusted so the FV is strike minus future price…the FC should leave me with either value of S(T) or X and if the call strike is equal to the future price established initially the call intrinsic value equals the contracts value at expiration. Assuming the option is in the money why isn’t the bonds FV still equal to X???

S(T) = F/(1+rf)^T

use that in your put call parity

P + S = X/(1+Rr)^T + C

C = P + S - X/(1+rf)^T

= P + (F-X)/(1+Rf)^T

It’s not the algebra I’m stuck on really

not sure what you are asking…

I am showing it algebraically - but the fact is that the Futures Price F = S * (1+rf)^T hence F/(1+rf)^t = S (spot price of the Stock (bond here) is substituted into the Put-Call Parity for a Forward contract. So Futures Price is substituted in the PCP instead of Stock Price.

Okay I think it’s a little clearer now, can you comment tho on the situation where strike is less that FT, implying that that bond may be shorted rather than invested in. That part I still don’t get. Let’s say strike is 8, FT is 10, and at exp the underlying is at 20. The FC combination of debt and long call should either 0 or ST - x, which 20 - 8 seems to so why am I shorting a bond with a FV of -2? I figure this has something to do with arbitrage on the put side but I don’t see it.

the ST at expiry has NO ROLE to play in the PCP for forwards. You will always get the Ft only.

So here Ft - X = 2 (depending on Put or Call, it would be +2 or -2.