We are now in position to start introducing actual financial derivative contracts, including the pricing mechanics in the Black case. The simplest are the forward and futures contracts, which we already encountered in Section 2.1. Note, however, that what we introduced there were the forward and futures prices, which are not the same thing as the PVs of a forward or a futures contract. Some explanation is definitely in order.
Consider a forward contract, with payout function VT = FT, T − K = ST − K at time T shown in Figure 2.2. As introduced in Section 2.2.6, the choice of strike K for which the forward is costless to enter into is denoted F0, T, i.e. .
If the forward was not entered into on a costless basis, or if time has passed since the original strike was set and the commodity price has moved, then the strike price K today will not be equal to F0, T and we need to PV it (i.e. to obtain the present value). Quite trivially we obtain
In the event where the final settlement of the forward is at a later time Tstl than the fixing T, with T < Tstl, we have
Consider now the T-futures contract, with strike ...