Forward Price, Delivery Price, and Forward Contract Value

# Forward price confusion


# Forward price confusion












What exactly is the "forward price"?

I have read that it is the price the long position in the forward contract will pay the short position for delivery at maturity (this is agreed upon at the time of conception of the forward). This seems to indicate to me that it is equal to the 'delivery price'. Which seems to suggest that it is a constant over time: it does not change. In Hull I read however things like "the delivery price does not change (...) but the forward price is likely to do so." How can the forward price change? Puzzling. I do not comprehend. As always, the problem is, of course, in the definitions: I do not know apparently what are the precise definitions of these terms.

I also read somewhere: "Note that the forward price is not the price of the forward contract." So, what does the formula $F(t) = S(t)e^{-q(T-t)} - K e^{-r(T-t)}$ represent then?

In general, I am confused by the following terms:

- Forward price

- Delivery price

- Price of forward contract

Can someone provide a precise, unambiguous and mathematically sound explanation of these terms?

[PS: Please refrain from generic links to other questions on this site--I can assure you I have seen them all, and I would not ask this question if any of those questions/answers would've helped me!]

## Answer by Rylan (score 5)

https://quant.stackexchange.com/a/79184

Note that forwards can behave differently in different markets; let's assume we're talking about an equity market and specifically on some stock $S(t)$ paying a continuous dividend of $q$. Also, let's fix $s < t < T$, and make the typical assumptions that we use when doing no-arbitrage pricing such as being able to borrow/lend at a single rate $r$, no transactions costs, etc. (Probably a few others that I always forget to list.)

$F(t, T)$ is the forward price observed at time $t$ for delivery of one unit of $S(T)$ at time $T$. Note that $F(t, T)$ is $\mathcal F(t)$ measurable while $S(T)$ isn't.

An interpretation of $F(t, T)$ is that you could agree with a counterparty at time $t$ that, at time $T$, you pay them $F(t, T)$ (the amount of money you agreed at time $t$) and they give you $S(T)$.

We indeed have $F(t, T) = S(t)e^{(r-q)(T-t)}$. This comes from replication. Specifically, suppose at time $t$ you borrow $K = S(t)e^{-q(T-t)}$ from the bank, and you buy $e^{-q(T-t)}$ units of $S$. You reinvest your dividends in $S$ throughout your holding period, giving you $S(T)$ at time $T$, and your loan has accrued interest, meaning you will have $S(t)e^{-q(T-t)}e^{r(T-t)} =S(t)e^{(r-q)(T-t)}$. If $F(t, T)$ was any other price, there would be an arbitrage opportunity.

You can also agree to enter a forward contract with a counterparty at time $s$. This is an agreement to, at time $T$, pay the delivery price $d(s, T)$ and receive $S(T)$. In principle, $d(s, T)$ can be whatever two counterparties agree on, but if they want to structure it in such a way that the contract has zero value for both parties at time $s$, then $d(s, T) = F(s, T)$. The price of this forward contract should be the profit (or loss) you'd get from unwinding the trade or taking an opposite position. The price of the forward contract at time $t$ would be $F(t, T) - d(s, T)$ (for the party agreeing to pay $d(s, T)$; the other party would have the negative of this as their price.) Depending on conventions, this value could also be discounted making the price $e^{-r(T-t)}(F(t, T) - d(s, T))$, as the money only changes hands at time $T$.

When someone says that the forward price changes, what they likely mean is that $F(s, T)$ is not necessarily equal to $F(t, T)$ -- we can see this from the formula I supplied.

In summary, one way we can tie it all together: A forward contract (in this context) is an agreement to, at time $T$, exchange a stock for a fixed amount of money. The delivery price is the "fixed amount of money" I referred to earlier. The price of the forward contract is how much money you could have if you unwound (or hedged) the forward contract today, and the forward price is the amount of money you'd need to borrow/invest to replicate a forward contract with a price of zero.

## Answer by D Stanley (score 5)

https://quant.stackexchange.com/a/79189

Less quantitative answer:

Suppose I want to enter into a contract to buy corn in 6 months. The seller will look at the current price of corn, among other factors, and make an offer for which they are willing to sell their corn in 6 months. Suppose that price is \$4.50 per bushel. $4.50 is the forward price.

If I decline and come back the next day, the seller will again look at the current price and other factors, and may offer a different price to sell in 6 months. Suppose that price is \$4.60 per bushel. $4.60 is now the new forward price.

If I agree to that price, then we write a contract to exchange corn at that price. I am guaranteed to buy the corn at \$4.60 per bushel in 6 months. $4.60 is the delivery price of my contract.

Now suppose tomorrow someone else comes to the seller and wants to buy corn and they offer \$4.75 per bushel. \$4.75 is now the current forward price. My delivery price has not changed, but the forward price has.

Bottom line -

The agreed-upon delivery price for an existing contract does not change, but the price at which you can enter into those contracts certainly changes over time.

More precisely:

Forward price = current price at which I can enter into a forward contract (changes over time)

Delivery price = agreed-upon price of an executed contract (does not change)

"Price of forward contract" is less clear, and can be used differently depending on the context. It may refer to the value of a forward (delivery price minus current forward price), or one of the other two prices mentioned above.

> So, what does the formula $F(t)=S(t)e^{−q(T−t)}−Ke^{−r(T−t)}$ represent then?

That represents the present value of an existing forward contract with delivery price $K$ and current forward price $S$. Over time $T-t$, the forward price is expected to decline as either the stock pays dividends or the commodity has costs to hold it. In addition, you won't receive the delivery price until the contract matures in $T-t$ years, so the present value is that amount discounted by the interest rate.

"Price of a Forward Contract" is a bit misleading since you can't buy a forward contract at an arbitrary delivery price. You can only enter into new forward contracts at the current forward price, so the "price" of a contract at origination is zero.

That's a lot of concepts baked into one formula (carry costs, time value of money, etc.) that must be understood before the formula as a whole can be understood.