Now suppose that F < 352.635, say F = 351. Now the forward is undervalued relative
to spot by $1.635, so we buy forward and sell spot to take advantage of the mispricing. Specifically:
1. Enter into a long forward position to purchase 1 oz of gold in three months at the delivery price of $351.
2. Short 1 oz of gold in the spot market and hold the short position for three months. 3. Invest the proceeds of $350 from the short sale for three months at the interest rate of
3%.
The investment in this strategy plays the same role as the borrowing in the earlier strategy. The cash flows are summarized in Table 3.2. Once again, there are no net cash flows at inception since the cash inflow of $350 from the short sale is matched by the cash outflow of $350 for the investment. There are no net interim cash flows since gold has no holding costs.
At maturity, we pay $351 and receive 1 oz of gold from the forward contract that we use to cover our short position. We also receive a cash inflow of e(0.03)(1/4)× 350 = $352.635
from the investment. Thus, there is a net cash inflow of $1.635, representing our arbitrage
profits. ■
The assumption that there are no holding costs or benefits is often not a reasonable one. Holding financial assets such as bonds or equities may result in holding benefits in the form of coupons or dividends. Holding commodities may involve substantial storage and insurance costs; the costs of storing oil, for instance, amount, on an annualized basis, to about 20% of the cost of the oil itself. Such interim costs or benefits affect the cost of the replication strategy and should be taken into account in calculating the forward price. The following example deals with such a situation.
Example 3.2
Consider a six-month forward contract on a bond. Suppose the current spot price S of the bond is $95 and that the bond will pay a coupon of $5 in three months’ time. Finally, suppose the rate of interest is 10% for all maturities. What is the arbitrage-free forward price of the bond?In terms of our notation, we are given S= 95, T = 6 months = 1/2 year, and r = 10%. Since holding the bond involves a cash inflow, M is negative and is given by minus the present value of $5 receivable in 3 months:
M = −e−(0.10)(0.25)× 5 = −4.877
Therefore, from the forward-pricing formula (3.4), we have
F = e(0.10)(1/2)× (95 − 4.877) = e(0.10)(1/2)× (90.123) = 94.74
Any other delivery price leads to an arbitrage opportunity.
(A) Arbitrage from an Overvalued Forward
Suppose F > 94.74, for example, F = 95.25. Then, the forward is overvalued relative to spot by $0.51, so we should buy spot, sell forward, and borrow. There are many ways to set up the precise strategy. One is to split the initial borrowing of $95 into two parts with one part repaid in three months with the $5 coupon and the other part repaid after six months with the delivery price received from the forward contract. More precisely:
1. Enter into a short forward position to sell the bond in six months’ time for $95.25. 2. Buy 1 unit of the spot asset for $95 and hold it up to T .
3. Borrow P V (5)= e−(0.10)(1/4)× 5 = $4.877 for repayment in three months and $90.123 for repayment in six months.
The cash flows from this strategy are summarized in Table 3.3. There are no net cash flows at inception since the cash outflow of $95 required to purchase the bond is matched by the total inflows from the borrowings (4.877 + 90.123 = 95). The only interim cash flows occur in three months. At that point, an amount of $5 is due to repay the three-month borrowing, but we receive $5 as coupon from the bond we hold. Thus, there are no net cash flows at this point either.
At maturity of the forward contract, there is a cash inflow of $95.25 from the forward position when the bond is delivered, and a cash outflow of
e(0.10)(0.50)× 90.123 = 94.74
towards repaying the six-month borrowing. Thus, there is a net cash inflow of $95.25 − 94.74 = 0.51, representing the arbitrage profits.
(B) Arbitrage from an Undervalued Forward
Now suppose F < 94.74, say F = 94.25. Then the forward is undervalued relative to spot by $0.49, so we buy forward, sell spot, and invest. In greater detail:
1. Enter into a long forward position to sell the bond in six months’ time for $94.25.
TABLE 3.3 Cash Flows in Example 3.2 from Arbitraging an Overvalued Forward
Source of Cash Flow Initial Cash Flow Interim Cash Flow Final Cash Flow
Short forward – – +95.25
Long spot −95.000 +5.000 –
3-month borrowing +4.877 −5.000 –
6-month borrowing +90.123 – −94.74
TABLE 3.4 Cash Flows in Example 3.2 from Arbitraging an Undervalued Forward
Source of Cash Flow Initial Cash Flow Interim Cash Flow Final Cash Flow
Long forward – – −94.25
Short spot +95.000 −5.000 –
3-month investment −4.877 +5.000 –
6-month investment −90.123 – +94.74
Net cash flows – – +0.49
2. Short 1 unit of the bond for $95 and hold the short position up to T .
3. Invest P V (5)= e−(0.10)(1/4)× 5 = $4.877 for three months and $90.123 for six months. Table 3.4 summarizes the resulting cash flows. There are no net initial cash flows. There is a cash inflow of $5 after three months from the three-month investment, but there is also a coupon of $5 due on the short bond. Thus, there is no net cash flow at this point either. After six months, the contract is at maturity. At this point, we receive
e(0.10)(0.50)× 90.123 = $94.74
from the six-month investment. We pay $94.25 on the forward contract and receive the bond, which we use to close out the short position. This leaves us with a net cash inflow of
$0.49 representing arbitrage profits. ■
3.4
Forward Pricing on Currencies and Related Assets
An important difference between a currency and other underlyings such as wheat is that when we buy and store one bushel of wheat, it remains one bushel of wheat at maturity (assuming, of course, that the rats don’t get at it!). In contrast, when we buy and store currency, the currency earns interest at the appropriate rate, so one unit of the currency grows to more than one unit over time. This means that the fundamental forward pricing formula (3.3) must be modified for such cases.
As a specific motivation, consider a currency forward contract (say, on British pound sterling denoted £) maturing in T years. An investor taking a long position in this contract pays the delivery price $F at time T and receives £1 at that point. To replicate this outcome using the spot asset, the investor cannot simply buy £1 today and hold it to T . Why not? The pound sterling the investor holds earns interest at the rate applicable to T -year sterling deposits, so the £1 would grow to more than £1 at T . For example, if T = 3 months and the three-month interest rate on sterling is 8%, then the initial £1 will grow to
e(0.08)(1/4) = £1.02
in three months, so the investor will end up overreplicating the outcome of the forward contract.
To correct for this, we must take interest yield into account in constructing the replicating strategy. We do this by adjusting the number of units of the spot currency we buy at the outset so that we are left with exactly one unit at maturity. In this example, this may be accomplished by buying only £(1/1.02) = £0.98 initially. When this amount is invested at the 8% rate for three months, we will receive £1 at maturity.
We describe the forward pricing formula that results when the replicating strategy is modified in this way. Then we provide an example to illustrate the arguments.