Non arbitrage condition
from section Yield Curve Movement and the Forward Curve in chapter YTM in Relation to Spot and Forward Rates in LES
the claim is that, if the future spot rates evolve as current forward curve, then forward contract price should remain constant no matter when we calculate it
according to forward pricing model, the forward price now is F_{A,B-A} = DF_B/DF_A
suppose t periods elapsed (t<A), the forward price at time t is F_{A-t,B-A} = DF_{B-t}/DF_{A-t}
so the claim is essentially saying these two are equal
now, according to the LES, this is correct because DF_{B-t}=DF_B/DF_t, and DF_{A-t}=DF_A/DF_t
this is using the assumption that future spot rates evolve as current forward curve
but how? it seems to me the assumption is talking about rates whereas in the proof it is using DFs. I know DFs are related to spot rates, but how are DFs related to forward curve? Can someone show me what exactly does the assumption say mathematically?