Why Replicate a Forward?
A forward contract lets its holder agree today to buy an asset on a future date for a specified delivery price. A newly initiated forward normally requires no payment at the start, but it commits the parties to an exchange at delivery. Its future payoff can therefore be compared with the payoff from a portfolio assembled today using the asset and borrowing.
That comparison is useful because two portfolios with the same future cash flows should have the same value today in a market with no arbitrage. If a quoted forward delivery price makes the forward more or less valuable than its replicating portfolio, traders can combine the two positions to seek a risk-free profit. The argument below isolates this basic relationship for an asset that provides no income before delivery.
Replicating the Long Forward
A long forward with delivery price \(K\) requires the buyer to pay \(K\) at time \(n\) and receive the asset. If the asset’s spot price then is \(S_n\), the forward’s terminal payoff is \(S_n-K\): the asset received is worth \(S_n\), and the required payment is \(K\).
To create the same terminal payoff, buy one unit of the asset today and finance its purchase entirely by borrowing. The asset costs \(S_0\). At time \(n\), the loan balance is \(S_0(1+i)^n\), while the asset is worth \(S_n\). The financed spot position therefore has terminal net value \(S_n-S_0(1+i)^n\). The future asset appears in both payoffs, so matching the cash flows requires \(K=S_0(1+i)^n\).
Here \(K^*\) is the no-arbitrage delivery price for a new forward under the stated assumptions. It makes the forward’s terminal payoff identical to the payoff of the financed spot position. The forward itself has zero value at initiation when its delivery price is \(K^*\): there is no initial payment, and the matching strategy begins with a spot purchase funded by an equal loan.
The same result can be written using a discount factor. Let \(D(0,n)\) be the time-0 price of a risk-free payment of one dollar at time \(n\). A loan of \(S_0\) today requires repayment of \(S_0/D(0,n)\) at time \(n\). Thus, when that discount factor represents the available borrowing and lending terms, the matching delivery price is \(S_0/D(0,n)\). With a flat annual effective rate, \(D(0,n)=v^n\), giving the formula above.
Worked Examples
Worked Example: Matching a Forward to a Financed Purchase
A hypothetical machine component has a spot price of $120. A new two-year forward has a delivery price of $132.30. Assume a flat effective annual borrowing and lending rate of 5%, no income or carrying costs for the component, and no transaction costs. Determine whether the forward’s delivery price matches the financed spot position, and compare their payoffs if the component’s price at delivery is $150.
Method: replicate the long forward. Buying one component today costs $120. Borrowing that amount at 5% effective annual interest produces a repayment after two years of
The financed spot position is therefore worth the delivery-date component price minus $132.30. The stated forward requires a payment of $132.30 for the component, so its payoff is the delivery-date price minus $132.30 as well. In particular, the no-arbitrage delivery price is
If the component is worth $150 at delivery, the long forward payoff is \(150-132.30=\$17.70\). The financed spot position is worth \(150-132.30=\$17.70\). The two payoffs agree. This comparison avoids the common mistake of treating the delivery price as the asset’s current spot price: the forward payment is made in two years, so the spot cost must be accumulated to that date.
Check: In both positions, the investor ends with an asset worth $150 and owes $132.30, leaving $17.70.
Worked Example: Arbitrage When the Forward Is Overpriced
A hypothetical specialty metal has a spot price of $120. A one-year forward is quoted at $128. Assume a 5% effective annual borrowing and lending rate, no income or carrying costs for the metal, and no transaction costs. Show whether the quote is consistent with replication and calculate the risk-free gain if it is not.
Method: buy the asset and sell the forward. First find the delivery-date cost of carrying a spot purchase financed by borrowing:
The quoted delivery price, $128, exceeds the replicated cost by \(128-126=\$2\) per unit. To lock in that difference, buy one unit of metal today using a $120 loan, and enter a short forward agreeing to deliver that unit in one year for $128.
At initiation, the loan proceeds pay the $120 spot price, so the net cash flow is $0. At delivery, deliver the metal into the forward and receive $128. Repay the loan balance of $126. The remaining cash is
The gain does not depend on the metal’s delivery-date market price: the owned unit is delivered for the contract’s $128 payment. This arbitrage direction avoids the mistake of buying an overpriced forward. Instead, the trader sells the expensive forward and holds the asset that will be delivered.
Check: The asset purchase is financed at the outset, and the only remaining delivery-date cash flow is the $2 excess of the forward receipt over the loan repayment.
Worked Example: Arbitrage When the Forward Is Underpriced
A hypothetical instrument has a spot price of $200. A one-year forward is quoted at $205. Assume a 4% effective annual borrowing and lending rate, no income or carrying costs for the instrument, and that it can be short sold without cost. Determine whether an arbitrage is available and calculate its delivery-date profit per unit.
Method: short the asset and buy the forward. The financed spot position would require a delivery-date repayment of
The forward’s $205 delivery price is $3 below that amount. To capture the difference, short sell one unit today for $200, invest the $200 for one year, and enter a long forward to buy one unit at delivery for $205.
At initiation, the short-sale proceeds are invested, and the forward requires no initial payment. At delivery, the investment has grown to
Use $205 of that amount to pay for the asset under the long forward. Return the asset to close the short sale, leaving
The position’s delivery-date profit is $3 per unit, independent of the market price at delivery. The key is that the forward is bought because it is cheap relative to the cost of acquiring the asset through a financed spot purchase.
Check: The forward supplies the unit needed to close the short sale, while the investment covers the forward payment and leaves $3.
How Inconsistent Quotes Create Arbitrage
The replication relationship also tells us which side of the forward to trade. Under the model assumptions, compare the quoted delivery price \(K\) with the cost \(S_0(1+i)^n\) of financing a spot purchase through delivery.
| Forward quote | Relative price | Replicating trade | Delivery-date result |
|---|---|---|---|
| Overpriced | \(K>S_0(1+i)^n\) | Buy the spot asset with borrowed funds; sell the forward | Receive \(K\), repay \(S_0(1+i)^n\) |
| Underpriced | \(K<S_0(1+i)^n\) | Short sell the spot asset and invest the proceeds; buy the forward | Investment grows to \(S_0(1+i)^n\), and the forward costs \(K\) |
| Consistent | \(K=S_0(1+i)^n\) | The forward and financed spot position match | No certain gain from combining the positions |
In each arbitrage strategy, the position established at initiation has no net cash cost under the idealized assumptions. At delivery, the asset obtained from one leg closes the obligation created by the other. The remaining cash is the difference between the quoted forward price and the financed spot cost. This is why the comparison is about the delivery-date cost, not simply the spot price today.
If borrowing and lending rates differ, short selling is restricted, or transactions have costs, a trader may not be able to execute these exact positions. The equality derived here then need not describe an attainable market price; the relevant comparison must account for the actual available financing and trading terms. Likewise, an asset that pays income or incurs carrying costs does not satisfy the assumptions used in this derivation. Those cash flows change the replication and must be included rather than ignored.
Common Mistake: Discounting in the Wrong Direction
The forward delivery price is the spot purchase cost accumulated to the delivery date. It is not the spot price discounted back to time 0. For example, with \(S_0=\$100\), \(i=5\%\), and \(n=2\), the correct delivery-date amount is \(100(1.05)^2=\$110.25\). The calculation \(100/(1.05)^2\approx \$90.70\) discounts a future amount; it does not accumulate today’s spot purchase cost.
A useful check is to ask when the payment is made. A spot purchase costs money today, so financing that purchase requires repayment with interest at delivery. The repayment must be greater than the original spot price when \(i>0\). If a proposed forward price is below the spot price in this simple positive-rate model, recheck the time direction and assumptions before drawing a conclusion.
Check Your Understanding
For each problem, use the stated assumptions and compare the quoted price with the financed spot cost.
- A no-income asset has a spot price of $85. The effective annual borrowing rate is 4%, and delivery is in three years. What is the no-arbitrage delivery price? (Answer: $95.61)
- A no-income asset costs $240 today. The two-year discount factor is 0.9025, and borrowing and lending are available on those terms. What delivery price matches a financed spot purchase? (Answer: $265.93)
- A no-income asset costs $500 today. The effective annual rate is 6%, and a two-year forward is quoted at $570. Is it overpriced or underpriced, and by how much per unit relative to the financed spot cost? (Answer: Overpriced by $8.20 per unit)
- A no-income asset costs $1,250 today. The effective annual rate is 7%, and a one-year forward is quoted at $1,300. Assume short selling is permitted. State the arbitrage positions and the delivery-date profit per unit. (Answer: Short the asset, invest $1,250, and buy the forward; profit is $37.50 per unit)
- A no-income asset costs $600 today. The two-year discount factor is 0.94, and a two-year forward is quoted at $650. State the arbitrage direction and the delivery-date profit per unit. (Answer: Buy the asset with borrowed funds and sell the forward; profit is \(650-600/0.94=\)$11.70 per unit, rounded to the nearest cent)