In the financial engineering course I am taking we are studying how to use the binomial model to price derivatives, one of which is the forward. For this question it is related to a forward contract on a stock.
At time $t = T$ the forward contract is equal to the value of the underlying at time $T$. From there we can work backwards using the following formula:
$F_t = \frac{1}{R}[q * F_u + (1 - q) * F_d]$
Where $R$ is the interest rate, $Q = {q, 1-q}$ are the risk-neutral probabilities, and $F_u$ and $F_d$ are the next period's up and down price for the forward, respectively.
If we work this back in the lattice to $T = 0$ we get a price of some kind. See the example image below:
This is the pricing for a futures contract on an underlying stock that is priced using the binomial model.
Where I am confused is that in order to derive the price of a forward or a future we know that at $t = 0$ the contract must be worth $0$. Intuitively this is because you don't "purchase" a forward. You just agree to be beholden to it. This is unlike an option, where the counterparty will get some kind of premium for taking on the risk.
But this binomial model clearly shows at $t = 0$ it's worth something! What does the number at $t = n$ represent here in the binomial pricing model?