Options, Futures, and Other Derivatives, 9th Edition Solution Manual

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CHAPTER 1
Introduction
Practice Questions
Problem 1.8.
Suppose you own 5,000 shares that are worth $25 each. How can put options be used to
provide you with insurance against a decline in the value of your holding over the next four
months?
You should buy 50 put option contracts (each on 100 shares) with a strike price of $25 and an
expiration date in four months. If at the end of four months the stock price proves to be less
than $25, you can exercise the options and sell the shares for $25 each.
Problem 1.9.
A stock when it is first issued provides funds for a company. Is the same true of an exchange-
traded stock option? Discuss.
An exchange-traded stock option provides no funds for the company. It is a security sold by
one investor to another. The company is not involved. By contrast, a stock when it is first
issued is sold by the company to investors and does provide funds for the company.
Problem 1.10.
Explain why a futures contract can be used for either speculation or hedging.
If an investor has an exposure to the price of an asset, he or she can hedge with futures
contracts. If the investor will gain when the price decreases and lose when the price increases,
a long futures position will hedge the risk. If the investor will lose when the price decreases
and gain when the price increases, a short futures position will hedge the risk. Thus either a
long or a short futures position can be entered into for hedging purposes.
If the investor has no exposure to the price of the underlying asset, entering into a futures
contract is speculation. If the investor takes a long position, he or she gains when the asset’s
price increases and loses when it decreases. If the investor takes a short position, he or she
loses when the asset’s price increases and gains when it decreases.
Problem 1.11.
A cattle farmer expects to have 120,000 pounds of live cattle to sell in three months. The live-
cattle futures contract on the Chicago Mercantile Exchange is for the delivery of 40,000
pounds of cattle. How can the farmer use the contract for hedging? From the farmer’s
viewpoint, what are the pros and cons of hedging?
The farmer can short 3 contracts that have 3 months to maturity. If the price of cattle falls, the
gain on the futures contract will offset the loss on the sale of the cattle. If the price of cattle
rises, the gain on the sale of the cattle will be offset by the loss on the futures contract. Using
futures contracts to hedge has the advantage that it can at no cost reduce risk to almost zero.

Its disadvantage is that the farmer no longer gains from favorable movements in cattle prices.
Problem 1.12.
It is July 2010. A mining company has just discovered a small deposit of gold. It will take six
months to construct the mine. The gold will then be extracted on a more or less continuous
basis for one year. Futures contracts on gold are available on the New York Mercantile
Exchange. There are delivery months every two months from August 2010 to December 2011.
Each contract is for the delivery of 100 ounces. Discuss how the mining company might use
futures markets for hedging.
The mining company can estimate its production on a month by month basis. It can then short
futures contracts to lock in the price received for the gold. For example, if a total of 3,000
ounces are expected to be produced in September 2010 and October 2010, the price received
for this production can be hedged by shorting a total of 30 October 2010 contracts.
Problem 1.13.
Suppose that a March call option on a stock with a strike price of $50 costs $2.50 and is held
until March. Under what circumstances will the holder of the option make a gain? Under
what circumstances will the option be exercised? Draw a diagram showing how the profit on
a long position in the option depends on the stock price at the maturity of the option.
The holder of the option will gain if the price of the stock is above $52.50 in March. (This
ignores the time value of money.) The option will be exercised if the price of the stock is
above $50.00 in March. The profit as a function of the stock price is shown in Figure S1.1.
Figure S1.1 Profit from long position in Problem 1.13
Problem 1.14.
Suppose that a June put option on a stock with a strike price of $60 costs $4 and is held until
June. Under what circumstances will the holder of the option make a gain? Under what
circumstances will the option be exercised? Draw a diagram showing how the profit on a
short position in the option depends on the stock price at the maturity of the option.

The seller of the option will lose if the price of the stock is below $56.00 in June. (This
ignores the time value of money.) The option will be exercised if the price of the stock is
below $60.00 in June. The profit as a function of the stock price is shown in Figure S1.2.
Figure S1.2 Profit from short position In Problem 1.1
Problem 1.15.
It is May and a trader writes a September call option with a strike price of $20. The stock
price is $18, and the option price is $2. Describe the investor’s cash flows if the option is
held until September and the stock price is $25 at this time.
The trader has an inflow of $2 in May and an outflow of $5 in September. The $2 is the cash
received from the sale of the option. The $5 is the result of the option being exercised. The
investor has to buy the stock for $25 in September and sell it to the purchaser of the option
for $20.
Problem 1.16.
An investor writes a December put option with a strike price of $30. The price of the option is
$4. Under what circumstances does the investor make a gain?
The investor makes a gain if the price of the stock is above $26 at the time of exercise. (This
ignores the time value of money.)
Problem 1.17.
The Chicago Board of Trade offers a futures contract on long-term Treasury bonds.
Characterize the investors likely to use this contract.
Most investors will use the contract because they want to do one of the following:
a) Hedge an exposure to long-term interest rates.

b) Speculate on the future direction of long-term interest rates.
c) Arbitrage between the spot and futures markets for Treasury bonds.
Problem 1.18.
An airline executive has argued: “There is no point in our using oil futures. There is just as
much chance that the price of oil in the future will be less than the futures price as there is
that it will be greater than this price.” Discuss the executive’s viewpoint.
It may well be true that there is just as much chance that the price of oil in the future will be
above the futures price as that it will be below the futures price. This means that the use of a
futures contract for speculation would be like betting on whether a coin comes up heads or
tails. But it might make sense for the airline to use futures for hedging rather than
speculation. The futures contract then has the effect of reducing risks. It can be argued that an
airline should not expose its shareholders to risks associated with the future price of oil when
there are contracts available to hedge the risks.
Problem 1.19.
“Options and futures are zero-sum games.” What do you think is meant by this statement?
The statement means that the gain (loss) to the party with the short position is equal to the
loss (gain) to the party with the long position. In total, the gain to all parties is zero.
Problem 1.20.
A trader enters into a short forward contract on 100 million yen. The forward exchange rate
is $0.0080 per yen. How much does the trader gain or lose if the exchange rate at the end of
the contract is (a) $0.0074 per yen; (b) $0.0091 per yen?
a) The trader sells 100 million yen for $0.0080 per yen when the exchange rate is $0.0074
per yen. The gain is100 0 0006  millions of dollars or $60,000.
b) The trader sells 100 million yen for $0.0080 per yen when the exchange rate is $0.0091
per yen. The loss is100 0 0011  millions of dollars or $110,000.
Problem 1.21.
A trader enters into a short cotton futures contract when the futures price is 50 cents per
pound. The contract is for the delivery of 50,000 pounds. How much does the trader gain or
lose if the cotton price at the end of the contract is (a) 48.20 cents per pound; (b) 51.30 cents
per pound?
a) The trader sells for 50 cents per pound something that is worth 48.20 cents per pound.
Gain( 0 5000 0 4820) 50 000 900$ $ $=  −    = .
b) The trader sells for 50 cents per pound something that is worth 51.30 cents per pound.
Loss( 0 5130 0 5000) 50 000 650$ $ $=  −    = .
Problem 1.22.
A company knows that it is due to receive a certain amount of a foreign currency in four
months. What type of option contract is appropriate for hedging?

A long position in a four-month put option can provide insurance against the exchange rate
falling below the strike price. It ensures that the foreign currency can be sold for at least the
strike price.
Problem 1.23.
A United States company expects to have to pay 1 million Canadian dollars in six months.
Explain how the exchange rate risk can be hedged using (a) a forward contract; (b) an
option.
The company could enter into a long forward contract to buy 1 million Canadian dollars in
six months. This would have the effect of locking in an exchange rate equal to the current
forward exchange rate. Alternatively the company could buy a call option giving it the right
(but not the obligation) to purchase 1 million Canadian dollar at a certain exchange rate in six
months. This would provide insurance against a strong Canadian dollar in six months while
still allowing the company to benefit from a weak Canadian dollar at that time.
Further Questions
Problem 1.24 (Excel file)
Trader A enters into a forward contract to buy gold for $1000 an ounce in one year. Trader
B buys a call option to buy gold for $1000 an ounce in one year. The cost of the option is
$100 an ounce. What is the difference between the positions of the traders? Show the profit
per ounce as a function of the price of gold in one year for the two traders.
Trader A makes a profit of ST̶ 1000 and Trader B makes a profit of max(ST̶ 1000, 0) –100
where ST is the price of gold in one month. Trader A does better if ST is above $900 as
indicated in Figure S1.3.
Figure S1.3: Profit to Trader A and Trader B in Problem 1.24

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Problem 1.25
In March, a US investor instructs a broker to sell one July put option contract on a stock. The
stock price is $42 and the strike price is $40. The option price is $3. Explain what the
investor has agreed to. Under what circumstances will the trade prove to be profitable? What
are the risks?
The investor has agreed to buy 100 shares of the stock for $40 in July (or earlier) if the party
on the other side of the transaction chooses to sell. The trade will prove profitable if the
option is not exercised or if the stock price is above $37 at the time of exercise. The risk to
the investor is that the stock price plunges to a low level. For example, if the stock price
drops to $1 by July (unlikely but possible), the investor loses $3,600. This is because the put
options are exercised and $40 is paid for 100 shares when the value per share is $1. This
leads to a loss of $3,900 which is offset by the premium of $300 received for the options.
Problem 1.26
A US company knows it will have to pay 3 million euros in three months. The current
exchange rate is 1.4500 dollars per euro. Discuss how forward and options contracts can be
used by the company to hedge its exposure.
The company could enter into a forward contract obligating it to buy 3 million euros in three
months for a fixed price (the forward price). The forward price will be close to but not
exactly the same as the current spot price of 1.4500. An alternative would be to buy a call
option giving the company the right but not the obligation to buy 3 million euros for a a
particular exchange rate (the strike price) in three months. The use of a forward contract locks
in, at no cost, the exchange rate that will apply in three months. The use of a call option
provides, at a cost, insurance against the exchange rate being higher than the strike price.
Problem 1.27 (Excel file)
A stock price is $29. An investor buys one call option contract on the stock with a strike price
of $30 and sells a call option contract on the stock with a strike price of $32.50. The market
prices of the options are $2.75 and $1.50, respectively. The options have the same maturity
date. Describe the investor's position.
This is known as a bull spread and will be discussed in Chapter 11. The profit is shown in
Figure S1.4.

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Figure S1.4: Profit in Problem 1.27
Problem 1.28
The price of gold is currently $600 per ounce. Forward contracts are available to buy or sell
gold at $800 for delivery in one year. An arbitrageur can borrow money at 10% per annum.
What should the arbitrageur do? Assume that the cost of storing gold is zero and that gold
provides no income.
The arbitrageur should borrow money to buy a certain number of ounces of gold today and
short forward contracts on the same number of ounces of gold for delivery in one year. This
means that gold is purchased for $600 per ounce and sold for $800 per ounce. Assuming the
cost of borrowed funds is less than 33% per annum this generates a riskless profit.
Problem 1.29.
Discuss how foreign currency options can be used for hedging in the situation described in
Example 1.1 so that (a) ImportCo is guaranteed that its exchange rate will be less than
1.6600, and (b) ExportCo is guaranteed that its exchange rate will be at least 1.6200.
ImportCo can buy call options on £10,000,000 with a strike price of 1.6600. This will ensure
that it never pays more than $16,600,000 for the sterling it requires. ExportCo can buy put
options on £30,000,000 with a strike price of 1.6200. This will ensure that the price received
for the sterling will be above00,600,48$000,000,3062.1 = .
Problem 1.30.
The current price of a stock is $94, and three-month call options with a strike price of $95
currently sell for $4.70. An investor who feels that the price of the stock will increase is
trying to decide between buying 100 shares and buying 2,000 call options (20 contracts).
Both strategies involve an investment of $9,400. What advice would you give? How high does
the stock price have to rise for the option strategy to be more profitable?

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The investment in call options entails higher risks but can lead to higher returns. If the stock
price stays at $94, an investor who buys call options loses $9,400 whereas an investor who
buys shares neither gains nor loses anything. If the stock price rises to $120, the investor who
buys call options gains2000 (120 95) 9400 40 600$ − − = 
An investor who buys shares gains100 (120 94) 2 600$ − = 
The strategies are equally profitable if the stock price rises to a level, S, where100 ( 94) 2000( 95) 9400S S − = − −
or100S =
The option strategy is therefore more profitable if the stock price rises above $100.
Problem 1.31.
On July 17, 2009, an investor owns 100 Google shares. As indicated in Table 1.2, the share
price is $430.25 and a December put option with a strike price $400 costs $21.15. The
investor is comparing two alternatives to limit downside risk. The first involves buying one
December put option contract with a strike price of $400. The second involves instructing a
broker to sell the 100 shares as soon as Google’s price reaches $400. Discuss the advantages
and disadvantages of the two strategies.
The second alternative involves what is known as a stop or stop-loss order. It costs nothing
and ensures that $40,000, or close to $40,000, is realized for the holding in the event the
stock price ever falls to $40. The put option costs $2,115 and guarantees that the holding can
be sold for $4,000 any time up to December. If the stock price falls marginally below $400
and then rises the option will not be exercised, but the stop-loss order will lead to the holding
being liquidated. There are some circumstances where the put option alternative leads to a
better outcome and some circumstances where the stop-loss order leads to a better outcome.
If the stock price ends up below $400, the stop-loss order alternative leads to a better
outcome because the cost of the option is avoided. If the stock price falls to $380 in
November and then rises to $450 by December, the put option alternative leads to a better
outcome. The investor is paying $2,115 for the chance to benefit from this second type of
outcome.
Problem 1.32.
A trader buys a European call option and sells a European put option. The options have the
same underlying asset, strike price and maturity. Describe the trader’s position. Under what
circumstances does the price of the call equal the price of the put?
The trader has a long European call option with strike priceK and a short European put
option with strike priceK . Suppose the price of the underlying asset at the maturity of the
option isTS . IfTS K , the call option is exercised by the investor and the put option expires
worthless. The payoff from the portfolio isTS K− . IfTS K , the call option expires
worthless and the put option is exercised against the investor. The cost to the investor isTK S−
. Alternatively we can say that the payoff to the investor isTS K−

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amount). In all cases, the payoff isTS K− , the same as the payoff from the forward contract.
The trader’s position is equivalent to a forward contract with delivery priceK .
Suppose thatF is the forward price. IfK F= , the forward contract that is created has zero
value. Because the forward contract is equivalent to a long call and a short put, this shows
that the price of a call equals the price of a put when the strike price is F.

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CHAPTER 2
Mechanics of Futures Markets
Practice Questions
Problem 2.8.
The party with a short position in a futures contract sometimes has options as to the precise
asset that will be delivered, where delivery will take place, when delivery will take place, and
so on. Do these options increase or decrease the futures price? Explain your reasoning.
These options make the contract less attractive to the party with the long position and more
attractive to the party with the short position. They therefore tend to reduce the futures price.
Problem 2.9.
What are the most important aspects of the design of a new futures contract?
The most important aspects of the design of a new futures contract are the specification of the
underlying asset, the size of the contract, the delivery arrangements, and the delivery months.
Problem 2.10.
Explain how margins protect investors against the possibility of default.
A margin is a sum of money deposited by an investor with his or her broker. It acts as a
guarantee that the investor can cover any losses on the futures contract. The balance in the
margin account is adjusted daily to reflect gains and losses on the futures contract. If losses
are above a certain level, the investor is required to deposit a further margin. This system
makes it unlikely that the investor will default. A similar system of margins makes it unlikely
that the investor’s broker will default on the contract it has with the clearinghouse member
and unlikely that the clearinghouse member will default with the clearinghouse.
Problem 2.11.
A trader buys two July futures contracts on frozen orange juice. Each contract is for the
delivery of 15,000 pounds. The current futures price is 160 cents per pound, the initial
margin is $6,000 per contract, and the maintenance margin is $4,500 per contract. What
price change would lead to a margin call? Under what circumstances could $2,000 be
withdrawn from the margin account?
There is a margin call if more than $1,500 is lost on one contract. This happens if the futures
price of frozen orange juice falls by more than 10 cents to below 150 cents per lb. $2,000 can
be withdrawn from the margin account if there is a gain on one contract of $1,000. This will
happen if the futures price rises by 6.67 cents to 166.67 cents per lb.
Problem 2.12.
Show that, if the futures price of a commodity is greater than the spot price during the
delivery period, then there is an arbitrage opportunity. Does an arbitrage opportunity exist if
the futures price is less than the spot price? Explain your answer.
If the futures price is greater than the spot price during the delivery period, an arbitrageur

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buys the asset, shorts a futures contract, and makes delivery for an immediate profit. If the
futures price is less than the spot price during the delivery period, there is no similar perfect
arbitrage strategy. An arbitrageur can take a long futures position but cannot force immediate
delivery of the asset. The decision on when delivery will be made is made by the party with
the short position. Nevertheless companies interested in acquiring the asset will find it
attractive to enter into a long futures contract and wait for delivery to be made.
Problem 2.13.
Explain the difference between a market-if-touched order and a stop order.
A market-if-touched order is executed at the best available price after a trade occurs at a
specified price or at a price more favorable than the specified price. A stop order is executed
at the best available price after there is a bid or offer at the specified price or at a price less
favorable than the specified price.
Problem 2.14.
Explain what a stop-limit order to sell at 20.30 with a limit of 20.10 means.
A stop-limit order to sell at 20.30 with a limit of 20.10 means that as soon as there is a bid at
20.30 the contract should be sold providing this can be done at 20.10 or a higher price.
Problem 2.15.
At the end of one day a clearinghouse member is long 100 contracts, and the settlement price
is $50,000 per contract. The original margin is $2,000 per contract. On the following day the
member becomes responsible for clearing an additional 20 long contracts, entered into at a
price of $51,000 per contract. The settlement price at the end of this day is $50,200. How
much does the member have to add to its margin account with the exchange clearinghouse?
The clearinghouse member is required to provide20 2 000 40 000$ $  =  as initial margin for
the new contracts. There is a gain of (50,200− 50,000) 100= $20,000 on the existing
contracts. There is also a loss of(51 000 50 200) 20 16 000$ −   =  on the new contracts. The
member must therefore add40 000 20 000 16 000 36 000$ −  +  = 
to the margin account.
Problem 2.16.
On July 1, 2010, a Japanese company enters into a forward contract to buy $1 million with
yen on January 1, 2011. On September 1, 2010, it enters into a forward contract to sell $1
million on January 1, 2011. Describe the profit or loss the company will make in dollars as a
function of the forward exchange rates on July 1, 2010 and September 1, 2010.
Suppose1F and2F

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Problem 2.17.
The forward price on the Swiss franc for delivery in 45 days is quoted as 1.1000. The futures
price for a contract that will be delivered in 45 days is 0.9000. Explain these two quotes.
Which is more favorable for an investor wanting to sell Swiss francs?
The 1.1000 forward quote is the number of Swiss francs per dollar. The 0.9000 futures quote
is the number of dollars per Swiss franc. When quoted in the same way as the futures price
the forward price is1 1 1000 0 9091  =  . The Swiss franc is therefore more valuable in the
forward market than in the futures market. The forward market is therefore more attractive
for an investor wanting to sell Swiss francs.
Problem 2.18.
Suppose you call your broker and issue instructions to sell one July hogs contract. Describe
what happens.
Hog futures are traded on the Chicago Mercantile Exchange. (See Table 2.2). The broker will
request some initial margin. The order will be relayed by telephone to your broker’s trading
desk on the floor of the exchange (or to the trading desk of another broker).
It will be sent by messenger to a commission broker who will execute the trade according to
your instructions. Confirmation of the trade eventually reaches you. If there are adverse
movements in the futures price your broker may contact you to request additional margin.
Problem 2.19.
“Speculation in futures markets is pure gambling. It is not in the public interest to allow
speculators to trade on a futures exchange.” Discuss this viewpoint.
Speculators are important market participants because they add liquidity to the market.
However, contracts must be useful for hedging as well as speculation. This is because
regulators generally only approve contracts when they are likely to be of interest to hedgers
as well as speculators.
Problem 2.20.
Identify the three commodities whose futures contracts in Table 2.2 have the highest open
interest.
Based on the contract months listed, the answer is crude oil, corn, and sugar (world).
Problem 2.21.
What do you think would happen if an exchange started trading a contract in which the
quality of the underlying asset was incompletely specified?
The contract would not be a success. Parties with short positions would hold their contracts
until delivery and then deliver the cheapest form of the asset. This might well be viewed by
the party with the long position as garbage! Once news of the quality problem became widely
known no one would be prepared to buy the contract. This shows that futures contracts are
feasible only when there are rigorous standards within an industry for defining the quality of
the asset. Many futures contracts have in practice failed because of the problem of defining
quality.

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Problem 2.22.
“When a futures contract is traded on the floor of the exchange, it may be the case that the
open interest increases by one, stays the same, or decreases by one.” Explain this statement.
If both sides of the transaction are entering into a new contract, the open interest increases by
one. If both sides of the transaction are closing out existing positions, the open interest
decreases by one. If one party is entering into a new contract while the other party is closing
out an existing position, the open interest stays the same.
Problem 2.23.
Suppose that on October 24, 2010, you take a short position in an April 2011 live-cattle
futures contract. You close out your position on January 21, 2011. The futures price (per
pound) is 91.20 cents when you enter into the contract, 88.30 cents when you close out your
position, and 88.80 cents at the end of December 2010. One contract is for the delivery of
40,000 pounds of cattle. What is your total profit? How is it taxed if you are (a) a hedger and
(b) a speculator? Assume that you have a December 31 year end.
The total profit is40 000 (0 9120 0 8830) 1 160$   −  = 
If you are a hedger this is all taxed in 2011. If you are a speculator40 000 (0 9120 0 8880) 960$   −  =
is taxed in 2010 and40 000 (0 8880 0 8830) 200$   −  =
is taxed in 2011.
Further Questions
Problem 2.24
Trader A enters into futures contracts to buy 1 million euros for 1.4 million dollars in three
months. Trader B enters in a forward contract to do the same thing. The exchange (dollars
per euro) declines sharply during the first two months and then increases for the third month
to close at 1.4300. Ignoring daily settlement, what is the total profit of each trader? When the
impact of daily settlement is taken into account, which trader does better?
The total profit of each trader in dollars is 0.03×1,000,000 = 30,000. Trader B’s profit is
realized at the end of the three months. Trader A’s profit is realized day-by-day during the
three months. Substantial losses are made during the first two months and profits are made
during the final month. It is likely that Trader B has done better because Trader A had to
finance its losses during the first two months.
Problem 2.25
Explain what is meant by open interest. Why does the open interest usually decline during the
month preceding the delivery month? On a particular day there are 2,000 trades in a
particular futures contract. Of the 2,000 traders on the long side of the market, 1,400 were
closing out position and 600 were entering into new positions. Of the 2,000 traders on the
short side of the market, 1,200 were closing out position and 800 were entering into new
positions. What is the impact of the day's trading on open interest?

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Open interest is the number of contract outstanding. Many traders close out their positions
just before the delivery month is reached. This is why the open interest declines during the
month preceding the delivery month. The open interest went down by 600. We can see this in
two ways. First, 1,400 shorts closed out and there were 800 new shorts. Second, 1,200 longs
closed out and there were 600 new longs.
Problem 2.26
One orange juice future contract is on 15,000 pounds of frozen concentrate. Suppose that in
September 2009 a company sells a March 2011 orange juice futures contract for 120 cents
per pound. In December 2009 the futures price is 140 cents. In December 2010 the futures
price is 110 cents. In February 2011 the futures price is 125 cents. The company has a
December year end. What is the company's profit or loss on the contract? How is it realized?
What is the accounting and tax treatment of the transaction is the company is classified as a)
a hedger and b) a speculator?
The price goes up during the time the company holds the contract from 120 to 125 cents per
pound. Overall the company therefore takes a loss of 15,000×0.05 = $750. If the company is
classified as a hedger this loss is realized in 2011, If it is classified as a speculator it realizes a
loss of 15,000×0.20 = $3000 in 2009, a gain of 15,000×0.30 = $4,500 in 2010 and a loss of
15,000×0.15 = $2,250 in 2011.
Problem 2.27.
A company enters into a short futures contract to sell 5,000 bushels of wheat for 250 cents
per bushel. The initial margin is $3,000 and the maintenance margin is $2,000. What price
change would lead to a margin call? Under what circumstances could $1,500 be withdrawn
from the margin account?
There is a margin call if $1000 is lost on the contract. This will happen if the price of wheat
futures rises by 20 cents from 250 cents to 270 cents per bushel. $1500 can be withdrawn if
the futures price falls by 30 cents to 220 cents per bushel.
Problem 2.28.
Suppose that there are no storage costs for crude oil and the interest rate for borrowing or
lending is 5% per annum. How could you make money on August 4, 2009 by trading
December 2009 and June 2010 contracts on crude oil? Use Table 2.2.
The December 2009 settlement price for oil is $75.62 per barrel. The June 2010 settlement
price for oil is $79.41 per barrel. You could go long one December 2009 oil contract and
short one June 2010 contract. In December 2009 you take delivery of the oil borrowing
$75.62 per barrel at 5% to meet cash outflows. The interest accumulated in six months is
about 75.62×0.05×0.5 or $1.89. In December the oil is sold for $79.41 per barrel which is
more than the amount that has to be repaid on the loan. The strategy therefore leads to a
profit. Note that this profit is independent of the actual price of oil in June 2010 or December
2009. It will be slightly affected by the daily settlement procedures.
Problem 2.29.
What position is equivalent to a long forward contract to buy an asset atK on a certain date
and a put option to sell it forK on that date?

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The equivalent position is a long position in a call with strike priceK .
Problem 2.30. (Excel file)
The author’s Web page (www.rotman.utoronto.ca/~hull/data) contains daily closing prices
for the December 2001 crude oil futures contract and the December 2001 gold futures
contract. (Both contracts are traded on NYMEX.) You are required to download the data and
answer the following:
a) How high do the maintenance margin levels for oil and gold have to be set so that
there is a 1% chance that an investor with a balance slightly above the maintenance
margin level on a particular day has a negative balance two days later (i.e. one day
after a margin call). How high do they have to be for a 0.1% chance. Assume daily
price changes are normally distributed with mean zero.
b) Imagine an investor who starts with a long position in the oil contract at the
beginning of the period covered by the data and keeps the contract for the whole of
the period of time covered by the data. Margin balances in excess of the initial margin
are withdrawn. Use the maintenance margin you calculated in part (a) for a 1% risk
level and assume that the maintenance margin is 75% of the initial margin. Calculate
the number of margin calls and the number of times the investor has a negative
margin balance and therefore an incentive to walk away. Assume that all margin calls
are met in your calculations. Repeat the calculations for an investor who starts with a
short position in the gold contract.
The data for this problem in the 7th edition is different from that in the 6th edition.
a) For gold the standard deviation of daily changes is $15.184 per ounce or $1518.4 per
contract. For a 1% risk this means that the maintenance margin should be set at3263.224.1518 
or 4996 when rounded. For a 0.1% risk the maintenance
margin should be set at0902.324.1518  or 6636 when rounded.
For crude oil the standard deviation of daily changes is $1.5777 per barrel or $1577.7
per contract. For a 1% risk, this means that the maintenance margin should be set at3263.227.1577 
or 5191 when rounded. For a 0.1% chance the maintenance
margin should be set at0902.327.1577  or 6895 when rounded. NYMEX
might be interested in these calculations because they indicate the chance of a trader
who is just above the maintenance margin level at the beginning of the period having
a negative margin level before funds have to be submitted to the broker.
b) For a 1% risk the initial margin is set at 6,921 for on crude oil. (This is the
maintenance margin of 5,191 divided by 0.75.) As the spreadsheet shows, for a long

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CHAPTER 3
Hedging Strategies Using Futures
Practice Questions
Problem 3.8.
In the Chicago Board of Trade’s corn futures contract, the following delivery months are
available: March, May, July, September, and December. State the contract that should be
used for hedging when the expiration of the hedge is in
a) June
b) July
c) January
A good rule of thumb is to choose a futures contract that has a delivery month as close as
possible to, but later than, the month containing the expiration of the hedge. The contracts
that should be used are therefore
(a) July
(b) September
(c) March
Problem 3.9.
Does a perfect hedge always succeed in locking in the current spot price of an asset for a
future transaction? Explain your answer.
No. Consider, for example, the use of a forward contract to hedge a known cash inflow in a
foreign currency. The forward contract locks in the forward exchange rate — which is in
general different from the spot exchange rate.
Problem 3.10.
Explain why a short hedger’s position improves when the basis strengthens unexpectedly and
worsens when the basis weakens unexpectedly.
The basis is the amount by which the spot price exceeds the futures price. A short hedger is
long the asset and short futures contracts. The value of his or her position therefore improves
as the basis increases. Similarly it worsens as the basis decreases.
Problem 3.11.
Imagine you are the treasurer of a Japanese company exporting electronic equipment to the
United States. Discuss how you would design a foreign exchange hedging strategy and the
arguments you would use to sell the strategy to your fellow executives.
The simple answer to this question is that the treasurer should
1. Estimate the company’s future cash flows in Japanese yen and U.S. dollars
2. Enter into forward and futures contracts to lock in the exchange rate for the
U.S. dollar cash flows.
However, this is not the whole story. As the gold jewelry example in Table 3.1 shows, the
company should examine whether the magnitudes of the foreign cash flows depend on the
exchange rate. For example, will the company be able to raise the price of its product in U.S.

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dollars if the yen appreciates? If the company can do so, its foreign exchange exposure may
be quite low. The key estimates required are those showing the overall effect on the
company’s profitability of changes in the exchange rate at various times in the future. Once
these estimates have been produced the company can choose between using futures and
options to hedge its risk. The results of the analysis should be presented carefully to other
executives. It should be explained that a hedge does not ensure that profits will be higher. It
means that profit will be more certain. When futures/forwards are used both the downside
and upside are eliminated. With options a premium is paid to eliminate only the downside.
Problem 3.12.
Suppose that in Example 3.4 the company decides to use a hedge ratio of 0.8. How does the
decision affect the way in which the hedge is implemented and the result?
If the hedge ratio is 0.8, the company takes a long position in 16 NYM December oil futures
contracts on June 8 when the futures price is $68.00. It closes out its position on November
10. The spot price and futures price at this time are $75.00 and $72. The gain on the futures
position is(72 68 00) 16 000 64 000−    = 
The effective cost of the oil is therefore20 000 75 64 000 1 436 000  −  =  
or $71.80 per barrel. (This compares with $71.00 per barrel when the company is fully
hedged.)
Problem 3.13.
“If the minimum-variance hedge ratio is calculated as 1.0, the hedge must be perfect." Is this
statement true? Explain your answer.
The statement is not true. The minimum variance hedge ratio isS
F

 
It is 1.0 when0 5= 
 and2S F=
  . Since1 0 
 the hedge is clearly not perfect.
Problem 3.14.
“If there is no basis risk, the minimum variance hedge ratio is always 1.0." Is this statement
true? Explain your answer.
The statement is true. Using the notation in the text, if the hedge ratio is 1.0, the hedger locks
in a price of1 2F b+ . Since both1F and2b are known this has a variance of zero and must be
the best hedge.

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Problem 3.16.
The standard deviation of monthly changes in the spot price of live cattle is (in cents per
pound) 1.2. The standard deviation of monthly changes in the futures price of live cattle for
the closest contract is 1.4. The correlation between the futures price changes and the spot
price changes is 0.7. It is now October 15. A beef producer is committed to purchasing
200,000 pounds of live cattle on November 15. The producer wants to use the December live-
cattle futures contracts to hedge its risk. Each contract is for the delivery of 40,000 pounds of
cattle. What strategy should the beef producer follow?
The optimal hedge ratio is1 2
0 7 0 6
1 4

  = 

The beef producer requires a long position in200000 0 6 120 000  =  lbs of cattle. The beef
producer should therefore take a long position in 3 December contracts closing out the
position on November 15.
Problem 3.17.
A corn farmer argues “I do not use futures contracts for hedging. My real risk is not the
price of corn. It is that my whole crop gets wiped out by the weather.”Discuss this viewpoint.
Should the farmer estimate his or her expected production of corn and hedge to try to lock in
a price for expected production?
If weather creates a significant uncertainty about the volume of corn that will be harvested,
the farmer should not enter into short forward contracts to hedge the price risk on his or her
expected production. The reason is as follows. Suppose that the weather is bad and the
farmer’s production is lower than expected. Other farmers are likely to have been affected
similarly. Corn production overall will be low and as a consequence the price of corn will be
relatively high. The farmer’s problems arising from the bad harvest will be made worse by
losses on the short futures position. This problem emphasizes the importance of looking at
the big picture when hedging. The farmer is correct to question whether hedging price risk
while ignoring other risks is a good strategy.
Problem 3.18.
On July 1, an investor holds 50,000 shares of a certain stock. The market price is $30 per
share. The investor is interested in hedging against movements in the market over the next
month and decides to use the September Mini S&P 500 futures contract. The index is
currently 1,500 and one contract is for delivery of $50 times the index. The beta of the stock
is 1.3. What strategy should the investor follow? Under what circumstances will it be
profitable?
A short position in50 000 30
1 3 26
50 1 500
 
  =
 
contracts is required. It will be profitable if the stock outperforms the market in the sense that
its return is greater than that predicted by the capital asset pricing model.
Problem 3.19.
Suppose that in Table 3.5 the company decides to use a hedge ratio of 1.5. How does the

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decision affect the way the hedge is implemented and the result?
If the company uses a hedge ratio of 1.5 in Table 3.5 it would at each stage short 150
contracts. The gain from the futures contracts would be55.2$70.150.1 =
per barrel and the company would be $0.85 per barrel better off.
Problem 3.20.
A futures contract is used for hedging. Explain why the daily settlement of the contract can
give rise to cash flow problems.
Suppose that you enter into a short futures contract to hedge the sale of a asset in six months.
If the price of the asset rises sharply during the six months, the futures price will also rise and
you may get margin calls. The margin calls will lead to cash outflows. Eventually the cash
outflows will be offset by the extra amount you get when you sell the asset, but there is a
mismatch in the timing of the cash outflows and inflows. Your cash outflows occur earlier
than your cash inflows. A similar situation could arise if you used a long position in a futures
contract to hedge the purchase of an asset and the asset’s price fell sharply. An extreme
example of what we are talking about here is provided by Metallgesellschaft (see Business
Snapshot 3.2).
Problem 3.21.
The expected return on the S&P 500 is 12% and the risk-free rate is 5%. What is the expected
return on the investment with a beta of (a) 0.2, (b) 0.5, and (c) 1.4?
a)0 05 0 2 (0 12 0 05) 0 064 +    −  =  or 6.4%
b)0 05 0 5 (0 12 0 05) 0 085 +    −  =  or 8.5%
c)0 05 1 4 (0 12 0 05) 0 148 +    −  =  or 14.8%
Further Questions
Problem 3.22
A company wishes to hedge its exposure to a new fuel whose price changes have a 0.6
correlation with gasoline futures price changes. The company will lose $1 million for each 1
cent increase in the price per gallon of the new fuel over the next three months. The new

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