# Confusion in forward contract pricing on a stock using the binomial model

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?

• That must be the forward price, no? – Alex C Feb 28 '18 at 20:34
• @AlexC It appears I'm confusing "forward price" and "forward valuation" here. To be honest, I didn't realize there is two different numbers. Could you explain more? – user20664 Feb 28 '18 at 20:35
• You agree now to a forward price, but you don't have to pay now. It will be paid at delivery, i.e. at maturity. – Alex C Feb 28 '18 at 20:37
• the "forward price" is the price written into the contract and agreed to by both parties (it is a bit like the strike price of an option). The value of the forward is the answer to the question what is contract worth today, if we want to transfer it to another person, or if we want to negotiate with the counterparty for cancellation (i.e. how much money does the counterparty want/need to receive/pay to break out of this agrrement they signed with us). As you said at the moment the forward is signed its value is zero, it could be cancelled without either party paying anything. But later on, no. – Alex C Feb 28 '18 at 21:13
• @AlexC Thank you for your reply. It's slightly more detailed than a current answer. I'd appreciate you turning this into a full answer. What you said makes sense, so TODAY, since the underlying is worth 100 at time zero, the futures contract is worth approximately 100, so you wouldn't make any money at t=0. As the future price evolves, the contract is worth strike - stock_price. If you sold it to a different party this is what you would. Am I understanding this correctly now? – user20664 Feb 28 '18 at 21:18