|
Action
Initial Cash Flow
Cash Flow in 1 Year
1. Borrow S
0
dollars
S
0
2
S
0
(1 1 r
f
)
2. Buy stock for S
0
2
S
0
S
T
1 D
3. Enter short futures position
0
F
0
2 S
T
TOTAL
0
F
0
2 S
0
(1 1 r
f
) 1 D
The initial cash flow is zero by construction: The money necessary to purchase the stock
in step 2 is borrowed in step 1, and the futures position in step 3, which is used to hedge
the value of the stock position, does not require an initial outlay. Moreover, the total cash
flow at year-end is riskless because it involves only terms that are already known when the
contract is entered. If the final cash flow were not zero, all investors would try to cash in
on the arbitrage opportunity. Ultimately prices would change until the year-end cash flow
is reduced to zero, at which point F
0
would equal S
0
(1 1 r
f
) 2 D.
The parity relationship also is called the cost-of-carry relationship because it asserts
that the futures price is determined by the relative costs of buying a stock with deferred
delivery in the futures market versus buying it in the spot market with immediate delivery
and “carrying” it in inventory. If you buy stock now, you tie up your funds and incur a
time-value-of-money cost of r
f
per period. On the other hand, you receive dividend pay-
ments with a current yield of d. The net carrying cost advantage of deferring delivery of the
stock is therefore r
f
2 d per period. This advantage must be offset by a differential between
the futures price and the spot price. The price differential just offsets the cost-of-carry
advantage when F
0
5 S
0
(1 1 r
f
2 d ).
The parity relationship is easily generalized to multiperiod applications. We simply rec-
ognize that the difference between the futures and spot price will be larger as the maturity
of the contract is longer. This reflects the longer period to which we apply the net cost of
carry. For contract maturity of T periods, the parity relationship is
F
0
5 S
0
(1 1 r
f
2 d)
T
(22.2)
Return to the arbitrage strategy laid out in Example 22.8. What would be the three steps of the strategy
if F
0
were too low, say, $980? Work out the cash flows of the strategy now and in 1 year in a table like the
one in the example. Confirm that your profits equal the mispricing of the contract.
CONCEPT CHECK
22.5
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P A R T V I
Options, Futures, and Other Derivatives
Notice that when the dividend yield is less than the risk-free rate, Equation 22.2 implies
that futures prices will exceed spot prices, and by greater amounts for longer times to contract
maturity. But when d > r
f
, as is the case today, the income yield on the stock exceeds the for-
gone (risk-free) interest that could be earned on the money invested; in this event, the futures
price will be less than the current stock price, again by greater amounts for longer maturities.
You can confirm that this is so by examining the S&P 500 contract listings in Figure 22.1 .
Although dividends of individual securities may fluctuate unpredictably, the annualized
dividend yield of a broad-based index such as the S&P 500 is fairly stable, recently in the
neighborhood of a bit more than 2% per year. The yield is seasonal, however, with regular
peaks and troughs, so the dividend yield for the relevant months must be the one used.
Figure 22.5 illustrates the yield pattern for the S&P 500. Some months, such as January or
April, have consistently low yields, while others, such as May, have consistently high ones.
6
We have described parity in terms of stocks and stock index futures, but it should be
clear that the logic applies as well to any financial futures contract. For gold futures, for
example, we would simply set the dividend yield to zero. For bond contracts, we would
let the coupon income on the bond play the role of dividend payments. In both cases, the
parity relationship would be essentially the same as Equation 22.2.
The arbitrage strategy described above should convince you that these parity relation-
ships are more than just theoretical results. Any violations of the parity relationship give
rise to arbitrage opportunities that can provide large profits to traders. We will see in the
next chapter that index arbitrage in the stock market is a tool to exploit violations of the
parity relationship for stock index futures contracts.
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