The value of a forward contract
1. The idea
A forward contract costs nothing on the day it is agreed: its delivery price is that day’s forward price, so neither side pays to enter it. Later, prices move and the contract gains or loses value. To find that value, compare it with a new forward agreed today for the same delivery date, at today’s forward price \(F_0\). The new contract is worth nothing. The two differ only in their delivery prices, so at delivery the old contract gives its buyer \(F_0 - K\) more than the new one, where \(K\) is the old delivery price. The old contract’s value today, \(f\), is the present value of that difference:
\[ f = (F_0 - K)\,e^{-rT}, \]
where \(r\) is the risk-free rate with continuous compounding and \(T\) the time left until delivery, in years. The seller’s value is the opposite, \(-f\): one side’s gain is the other’s loss.
For a share without dividends, today’s forward price is \(F_0 = S_0 e^{rT}\), with \(S_0\) the spot price (the first lesson of this level), so the value is \(f = S_0 - Ke^{-rT}\). If the share pays dividends before delivery, with present value \(I\), then \(f = S_0 - I - Ke^{-rT}\).
An example. A trader agreed some time ago to buy a share for $50, and delivery is now six months away. The share trades at $54 and pays no dividends, and the risk-free rate is 4%. The contract is worth \(f = 54 - 50\,e^{-0.04 \times 0.5} = \$4.99\) to the buyer, and \(-\$4.99\) to the seller.
Turned around, a forward price also gives the spot price: \(S_0 = F_0 e^{-rT}\).
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The value on a chart
First, the value on the day a forward is agreed.
Now move the spot price. The buyer’s value rises one for one with the share, and the seller’s falls by the same amount.
4. One worked example, then your turn
In the worked example, we value a seller’s contract on a share that pays a dividend.
Now a contract valued from today’s forward price, with hints, one step at a time.
And one on your own: the spot price behind a forward price.
5. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer for a forward on a share, a forward valued from today’s forward price, and a share price from a forward price. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: