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The Spot-Futures Parity Theorem
We have seen that a futures contract can be used to hedge changes in the value of the
underlying asset. If the hedge is perfect, meaning that the asset-plus-futures portfolio has
no risk, then the hedged position must provide a rate of return equal to the rate on other
risk-free investments. Otherwise, there will be arbitrage opportunities that investors will
exploit until prices are brought back into line. This insight can be used to derive the theo-
retical relationship between a futures price and the price of its underlying asset.
Suppose for simplicity that the S&P 500 index currently is at 1,000 and an investor
who holds $1,000 in a mutual fund indexed to the S&P 500 wishes to temporarily hedge
her exposure to market risk. Assume that the indexed portfolio pays dividends totaling
$20 over the course of the year, and for simplicity, that all dividends are paid at year-end.
Finally, assume that the futures price for year-end delivery of the S&P 500 contract is
1,010.
5
Let’s examine the end-of-year proceeds for various values of the stock index if the
investor hedges her portfolio by entering the short side of the futures contract.
22.4
Futures Prices
Final value of stock portfolio, S
T
$ 970
$ 990
$1,010
$1,030
$1,050
$1,070
Payoff from short futures position
(equals F
0
2 F
T
5 $1,010 2 S
T
)
40
20
0
2
20
2
40
2
60
Dividend income
20
20
20
20
20
20
TOTAL
$1,030
$1,030
$1,030
$1,030
$1,030
$1,030
5
Actually, the futures contract calls for delivery of $250 times the value of the S&P 500 index, so that each
contract would be settled for $250 times the index. With the index at 1,000, each contract would hedge about
$250 3 1,000 5 $250,000 worth of stock. Of course, institutional investors would consider a stock portfolio of
this size to be quite small. We will simplify by assuming that you can buy a contract for one unit rather than 250
units of the index.
The payoff from the short futures position equals the difference between the original
futures price, $1,010, and the year-end stock price. This is because of convergence: The
futures price at contract maturity will equal the stock price at that time.
Notice that the overall position is perfectly hedged. Any increase in the value of the
indexed stock portfolio is offset by an equal decrease in the payoff of the short futures
position, resulting in a final value independent of the stock price. The $1,030 total payoff is
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P A R T V I
Options, Futures, and Other Derivatives
the sum of the current futures price, F
0
5 $1,010, and the $20 dividend. It is as though the
investor arranged to sell the stock at year-end for the current futures price, thereby elimi-
nating price risk and locking in total proceeds equal to the futures price plus dividends paid
before the sale.
What rate of return is earned on this riskless position? The stock investment requires an
initial outlay of $1,000, whereas the futures position is established without an initial cash
outflow. Therefore, the $1,000 portfolio grows to a year-end value of $1,030, providing a
rate of return of 3%. More generally, a total investment of S
0
, the current stock price, grows
to a final value of F
0
1 D, where D is the dividend payout on the portfolio. The rate of
return is therefore
Rate of return on hedged stock portfolio 5
(F
0
1 D) 2 S
0
S
0
This return is essentially riskless. We observe F
0
at the beginning of the period when
we enter the futures contract. While dividend payouts are not perfectly riskless, they are
highly predictable over short periods, especially for diversified portfolios. Any uncertainty
is extremely small compared to the uncertainty in stock prices.
Presumably, 3% must be the rate of return available on other riskless investments. If
not, then investors would face two competing risk-free strategies with different rates of
return, a situation that could not last. Therefore, we conclude that
(F
0
1 D) 2 S
0
S
0
5 r
f
Rearranging, we find that the futures price must be
F
0
5 S
0
(1 1 r
f
) 2 D 5 S
0
(1 1 r
f
2 d)
(22.1)
where d is the dividend yield on the stock portfolio, defined as D / S
0
. This result is called
the spot-futures parity theorem. It gives the normal or theoretically correct relationship
between spot and futures prices. Any deviation from parity would give rise to risk-free
arbitrage opportunities.
Suppose that parity were violated. For example, suppose the risk-free interest rate
were only 1% so that according to Equation 22.1, the futures price should be
$1,000(1.01) 2 $20 5 $990. The actual futures price, F
0
5 $1,010, is $20 higher than
its “appropriate” value. This implies that an investor can make arbitrage profits by short-
ing the relatively overpriced futures contract and buying the relatively underpriced stock
portfolio using money borrowed at the 1% market interest rate. The proceeds from this
strategy would be as follows:
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