Chapter 1
Introduction to Derivatives
Question 1.1
This problem offers different scenarios in which some companies may have an interest to hedge
their exposure to temperatures that are detrimental to their business. In answering the problem, it
is useful to ask the question: Which scenario hurts the company, and how can it protect itself?
a) A soft drink manufacturer probably sells more drinks when it is abnormally hot. She dislikes
days at which it is abnormally cold because people are likely to drink less, and her business
suffers. She will be interested in a cooling-degree-day futures contract because it will make
payments when her usual business is slow. She hedges her business risk.
b) A ski-resort operator may fear large losses if it is warmer than usual. It is detrimental to her
business if it does not snow in the beginning of the season or if the snow is melting too fast at
the end of the season. She will be interested in a heating-degree-day futures contract because
it will make payments when her usual business suffers, thus compensating the losses.
c) During the summer months, an electric utility company, such as one in the south of the
United States, will sell a lot of energy during days of excessive heat because people will use
their air conditioners, refrigerators, and fans more often, thus consuming a lot of energy and
increasing profits for the utility company. In this scenario, the utility company will have less
business during relatively colder days, and the cooling-degree-day futures offers a possibility
to hedge such risk.
Alternatively, we may think of a utility provider in the northeast during the winter months, a
region where people use many additional electric heaters. This utility provider will make
more money during unusually cold days and may be interested in a heating-degree-day
contract because that contract pays off if the primary business suffers.
d) An amusement park operator fears bad weather and cold days because people will abstain
from going to the amusement park during cold days. She will buy a cooling-degree-day
future to offset her losses from ticket sales with gains from the futures contract.
Question 1.2
A variety of counterparties are imaginable. For one, we could think about speculators who have
differences in opinion and who do not believe that we will have excessive temperature variations
during the life of the futures contracts. Thus, they are willing to take the opposing side, receiving
a payoff if the weather is stable.
Alternatively, there may be opposing hedging needs: Compare the ski-resort operator and the
soft drink manufacturer. The cooling-degree-day futures contract will pay off if the weather is
relatively mild, and we saw that the resort operator will buy the futures contract. The buyer of

2 Chapter 1/Introduction to Derivatives
the cooling-degree-day futures will make a loss if the weather is cold (which means that the
seller of the contract will make a gain). Since the soft drink manufacturer wants additional
money if it is cold, she may be interested in taking the opposite side of the cooling-degree-day
futures.
Question 1.3
a) Remember that the terminology bid and ask is formulated from the market makers
perspective. Therefore, the price at which you can buy is called the ask price. Furthermore,
you will have to pay the commission to your broker for the transaction. You pay:
($41.05 × 100) + $20 = $4,125.00
b) Similarly, you can sell at the market maker’s bid price. You will again have to pay a
commission, and your broker will deduct the commission from the sales price of the shares.
You receive:
($40.95 × 100) − $20 = $4,075.00
c) Your round-trip transaction costs amount to:
$4,125.00 − $4,075.00 = $50
Question 1.4
In this problem, the brokerage fee is variable and depends on the actual dollar amount of the
sale/purchase of the shares. The concept of the transaction cost remains the same: If you buy the
shares, the commission is added to the amount you owe, and if you sell the shares, the
commission is deducted from the proceeds of the sale.
a) ($41.05 × 100) + ($41.05 × 100) × 0.003 = $4,117.315
= $4,117.32
b) ($40.95 × 100) − ($40.95 × 100) × 0.003 = $4,082.715
= $4,082.72
c) $4,117.32 − $4,082.72 = $34.6
The variable (or proportional) brokerage fee is advantageous to us. Our round-trip transaction
fees are reduced by $15.40.
Question 1.5
In answering this question, it is important to remember that the market maker provides a service
to the market. He stands ready to buy shares into his inventory and sell shares out of his
inventory thus providing immediacy to the market. He is remunerated for this service by earning
the bid-ask spread. The market maker buys the security at a price of $100, and he sells it at a

Chapter 1/Introduction to Derivatives 3
price of $100.10. If he buys 100 shares of the security and immediately sells them to another
party, he is earns a spread of:
100 × ($100.12 − $100) = 100 × $0.12 = $12.00
Question 1.6
A short sale of XYZ entails borrowing shares of XYZ and then selling them and receiving cash.
Therefore, initially, we will receive the proceeds from the sale of the asset less the proportional
commission charge:
300 × ($30.19) − 300 × ($30.19) × 0.005 = $9,057 × 0.995
= $9,011.72
When we close out the position, we will again incur the commission charge, which is added to
the purchasing cost:
300 × ($29.87) + 300 × ($29.87) × 0.005 = $8,961 × 1.005
= $9,005.81
Finally, we subtract the cost of covering the short position from our initial proceeds to receive
total profits: $9,011.72 − $9,005.81 = $5.91. We can see that paying the commission charge
twice significantly reduces the profits we can make.
Question 1.7
a) A short sale of JKI stock entails borrowing shares of JKI, then selling them and receiving
cash, and we learned that we sell assets at the bid price. Therefore, initially, we will receive
the proceeds from the sale of the asset at the bid (ignoring the commissions and interest).
After 180 days, we cover the short position by buying the JKI stock, and we saw that we will
always buy at the ask. Therefore, we earn the following profit:
400 × ($25.125) − 400 × ($23.0625) = $10,050 − $9,225.00
= $825.00
b) We have to pay the commission twice. The commission will reduce our profit:
400 × ($25.125) − 400 × ($25.125) × 0.003 − (400 × [$23.0625] + 400 × [$23.0625])
= $10,050 × 0.997 − $9,225 × 1.003
= $10,019.85 − $9,252.675
= $767.175.

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4 Chapter 1/Introduction to Derivatives
c) The proceeds from short sales minus the commission charge are $10,019.85 (or $10,050 if
you ignore the commission charge). Since the six-month interest rate is given and the period
of our short sale is exactly half a year, we can directly calculate the interest we could earn
(and that we now lose) on a deposit of $10,019.85:
$10,019.85 × (0.03) = $300.5955
= $300.60
or without taking into account the commission charge:
$10,050.00 × (0.03) = $301.50.
Question 1.8
We learned from the main text that short selling is equivalent to borrowing money and that a
short seller will often have to deposit the proceeds of the short sale with the lender as collateral.
A short seller is entitled to earn interest on his collateral, and the interest rate he earns is called
the short rebate in the stock market. Usually, the short rebate is close to the prevailing market
interest rate. Sometimes, though, a particular stock is scarce and difficult to borrow. In this case,
the short rebate is substantially less than the current market interest rate, and an equity lender can
earn a nice profit in the form of the difference between the current market interest rate and the
short rebate.
By signing an agreement as mentioned in the problem, you give your brokerage firm the
possibility to act as an equity lender, using the shares of your account. Brokers want you to sign
such an agreement because they can make additional profits.
Question 1.9
If we borrow an asset from a lender, we have the obligation to make any payments to the lender
that she is entitled to as a stockholder. Because the lender is entitled to the dividend on the day
the stock goes ex dividend but does not receive it from the company because we have sold her
stock, we must provide the dividend. This payment is tax-deductible for us.
In a perfect capital market, we would expect that the stock price falls exactly by the amount of
the dividend on the ex-date. Therefore, we should not care.
However, two complications may arise. First, we may have borrowed a large amount of shares,
and the increased dividend forces us to pay more to the lender, and we may not have the
additional required money.
An unexpected increase of the dividend could be a sign that the company is doing well.
Empirically, we observe sharp price increases after such announcements. As we have a short
position in the stock, we make money if the stock price falls. Therefore, an unexpected increase
in the dividend is very bad for our position, and we should care!

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Chapter 1/Introduction to Derivatives 5
Question 1.10
The main textbook explains short selling as follows. “The sale of a stock you do not already own
is called a short-sale.”Short positions are those resulting from short sales. At NASDAQ for
example, each financial firm is required to report its overall short positions in all accounts in
NASDAQ-listed securities twice a month. The short positions from these reports are aggregated
and the resulting number is called short interest in NASDAQ stocks. The short interest statistics
are widely observed, and some financial market participants believe that short interest is a good
measure of market sentiment.
You can easily access statistics on short interest on individual stocks on Nasdaq
(www.nasdaq.com). Once you are on the home page, go to the pull-down menu “Quotes and
Research,” then go to “Short Interest” in the folder “Company Profile.” Once you are on that
page, you can enter up to 25 tickers and retrieve the short interest in the stocks of your choice.
In general, stocks that lend themselves to some speculation and stocks around corporate events
(mergers and acquisition, dividend dates, etc.) with uncertain outcomes will have a particularly
high short interest.
It is theoretically possible to have short interest of more than 100 percent because some market
participants (e.g., market makers) have the ability to short sell a stock without having a locate
(i.e., having someone who actually owns the stock and has agreed to lend it).
Question 1.11
We are interested in borrowing the asset “money.” Therefore, we go to an owner (or if you
prefer, to a collector) of the asset, called Bank. The Bank provides the $100 of the asset money
in digital form by increasing our bank account. We sell the digital money by going to the ATM
and withdrawing cash. After 90 days, we buy back the digital money for $102 by depositing cash
into our bank account. The lender is repaid, and we have covered our short position.
Question 1.12
We are interested in borrowing the asset “money” to buy a house. Therefore, we go to an owner
of the asset, called Bank. The Bank provides the dollar amount, say $250,000, in digital form in
our mortgage account. As $250,000 is a large amount of money, the bank is subject to substantial
credit risk (e.g., we may lose our job) and demands a collateral. Although the money itself is
not subject to large variations in price (besides inflation risk, it is difficult to imagine a reason for
money to vary in value), the Bank knows that we want to buy a house, and real estate prices vary
substantially. Therefore, the Bank wants more collateral than the $250,000 they are lending.

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6 Chapter 1/Introduction to Derivatives
In fact, as the Bank is only lending up to 80 percent of the value of the house, we could get a
mortgage of $250,000 for a house that is worth $250,000 ÷ 0.8 = $312,500. We see that the bank
factored in a haircut of $312,500 − $250,000 = $62,500 to protect itself from credit risk and
adverse fluctuations in property prices.
We buy back the asset money over a long horizon of time by reducing our mortgage through
annuity payments.
Question 1.13
For example, go to the website of the CME group (www.cmegroup.com). Under the tab
“Products & Trading,” you can choose the asset class you are interested in. If you click, for
example, on “Equity Index,” you will come to a product listing of all equity products CME
offers. Once you click on any particular product, say E-mini S&P 500 futures, you get to a
separate page with the latest quotes of this product. If you click on the link “Contract
Specifications,” you will see all details regarding ticker symbol, contract size, tick size, traded
months, settlement, etc. of the product.
CME group has a Daily Volume and Open Interest Report that summarizes exchange-wide
volume, including futures and options volume. You can access it by clicking on the “Market
Data” tab and choosing the volume report.
If the notional values were cut in half, one would expect trading volume and open interest to
double but the notional value traded to stay the same.
Question 1.14
We have the following for the different trades and different dealers:
a)
A B C
Trade 1 +5 –5
Trade 2 +15 –15
Trade 3 +10 –10
Trade 4 –20 +20
Total 0 +5 –5
Therefore, trader A’s net position is zero, trader B is long five contracts, and trader C is short
five contracts.

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Chapter 1/Introduction to Derivatives 7
b) Trading volume is equal to 5 + 15 + 10 + 20 = 50.
Open interest is equal to 5.
Notional value of trading volume is equal to 50 contracts × $100/contract = $5,000
Notional value of open interest is equal to 5 contracts × $ 100/contract = $500.
c) Now all traders have a net position of zero. Trading volume has increased by five contracts to
55. The notional value of trading volume is equal to 55 × $100 = $5,500. Open interest and
the notional value of open interest are equal to zero.
d) Trading volume and open interest would be identical; the notional values of trading volume
and open interest would be lower.

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Chapter 2
An Introduction to Forwards and Options
Question 2.1
The payoff diagram of the stock is just a graph of the stock price as a function of the
stock price:
In order to obtain the profit diagram at expiration, we have to finance the initial
investment. We do so by selling a bond for $50. After one year, we have to pay back: $50
× (1 + 0.1) = $55. The second figure (on the next page) shows the graph of the stock, of
the bond to be repaid, and of the sum of the two positions, which is the profit graph. The
arrows show that at a stock price of $55, the profit at expiration is indeed zero.

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10 Part One/Insurance, Hedging, and Simple Strategies
In order to obtain the profit diagram at expiration, we have to lend out the money we
received from the short sale of the stock. We do so by buying a bond for $50. After one
year, we receive from the investment in the bond: $50 × (1 + 0.1) = $55. The second
figure shows the graph of the sold stock, of the money we receive from the investment in
the bond, and of the sum of the two positions, which is the profit graph. The arrows show
that at a stock price of $55, the profit at expiration is indeed zero.
Question 2.3
The position that is the opposite of a purchased call is a written call. A seller of a call
option is said to be the option writer or to have a short position. The call option writer is
the counterparty to the option buyer, and his payoffs and profits are just the opposite of
those of the call option buyer.
Similarly, the position that is the opposite of a purchased put option is a written put
option. Again, the payoff and profit for a written put are just the opposite of those of the
purchased put.
It is important to note that the opposite of a purchased call is NOT the purchased put. If
you do not see why, please draw a payoff diagram with a purchased call and a purchased
put.

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Chapter 2/An Introduction to Forwards and Options 11
Question 2.4
a) The payoff to a long forward at expiration is equal to:
Payoff to long forward = Spot price at expiration – forward price
Therefore, we can construct the following table:
Price of asset in six months Agreed forward price Payoff to the long forward
40 50 −10
45 50 −5
50 50 0
55 50 5
60 50 10
b) The payoff to a purchased call option at expiration is:
Payoff to call option = max[0, spot price at expiration – strike price]
The strike is given: It is $50. Therefore, we can construct the following table:
Price of asset in six months Strike price Payoff to the call option
40 50 0
45 50 0
50 50 0
55 50 5
60 50 10
c) If we compare the two contracts, we immediately see that the call option has a
protection for adverse movements in the price of the asset: If the spot price is below
$50, the buyer of the call option can walk away and need not incur a loss. The buyer
of the long forward incurs a loss, but he has the same payoff as the buyer of the call
option if the spot price is above $50. Therefore, the call option should be more
expensive. It is this attractive option to walk away that we have to pay for.

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12 Part One/Insurance, Hedging, and Simple Strategies
Question 2.5
a) The payoff to a short forward at expiration is equal to:
Payoff to short forward = forward price – spot price at expiration
Therefore, we can construct the following table:
Price of asset in six months Agreed forward price Payoff to the short forward
40 50 10
45 50 5
50 50 0
55 50 −5
60 50 −10
b) The payoff to a purchased put option at expiration is:
Payoff to put option = max[0, strike price – spot price at expiration]
The strike is given: It is $50. Therefore, we can construct the following table:
Price of asset in six months Strike price Payoff to the call option
40 50 10
45 50 5
50 50 0
55 50 0
60 50 0
c) The same logic as in question 2.4 (c) applies. If we compare the two contracts, we see
that the put option has a protection for increases in the price of the asset: If the spot
price is above $50, the buyer of the put option can walk away and need not incur a
loss. The buyer of the short forward incurs a loss and must meet her obligations.
However, she has the same payoff as the buyer of the put option if the spot price is
below $50. Therefore, the put option should be more expensive. It is this attractive
option to walk away if things are not as we want that we have to pay for.
Question 2.6
We need to solve the following equation to determine the effective annual interest rate:
$91 × (1 + r) = $100. We obtain r = 0.0989, which means that the effective annual
interest rate is approximately 9.9 percent.

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Chapter 2/An Introduction to Forwards and Options 13
Remember that when we drew profit diagrams for the forward or call option, we drew the
payoff on the vertical axis, and the index price at the expiration of the contract on the
horizontal axis. In this case, the particularity is that the default-free, zero-coupon bond
will pay exactly $100, no matter what the stock price is. Therefore, the payoff diagram is
just a horizontal line, intersecting the y-axis at $100.
The textbook provides the answer to the question concerning the profit diagram in the
section “Zero-Coupon Bonds in Payoff and Profit Diagrams.” When we were calculating
profits, we saw that we had to find the future value of the initial investment. In this case,
our initial investment is $91. How do we find the future value? We use the current risk-
free interest rate and multiply the initial investment by it. However, as our bond is
default-free and does not bear coupons, the effective annual interest rate is exactly the 9.9
percent we have calculated before. Therefore, the future value of $91 is $91 × (1 +
0.0989) = $100, and our profit in six months is zero!
Question 2.7
a) It does not cost anything to enter into a forward contract—we do not pay a premium.
Therefore, the payoff diagram of a forward contract coincides with the profit diagram.
The graphs have the following shape:
b) We have seen in question 2.1 that in order to obtain the profit diagram at expiration of
a purchase of XYZ stock, we have to finance the initial investment. We did so by
selling a bond for $50. After one year, we had to pay back: $50 × (1 + 0.1) = $55.
Therefore, our total profit at expiration from the purchase of a stock that was financed
by a loan was: $ST − $55, where ST is the value of one share of XYZ at expiration.
But this profit from buying the stock, and financing it is the same as the profit from

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14 Part One/Insurance, Hedging, and Simple Strategies
our long forward contract, and both positions do not require any initial cash—but
then, there is no advantage in investing in either instrument.
c) The dividend is only paid to the owner of the stock. The owner of the long forward
contract is not entitled to receive the dividend because she only has a claim to buy the
stock in the future for a given price, but she does not own it yet. Therefore, it does
matter now whether we own the stock or the long forward contract. Because
everything else is the same as in part a) and b), it is now beneficial to own the share:
We can receive an additional payment in the form of the dividend if we own the stock
at the ex-dividend date. This question hints at the very important fact that we have to
be careful to take into account all the benefits and costs of an asset when we try to
compare prices. We will encounter similar problems in later chapters.
Question 2.8
We saw in question 2.7 (b) that there is no advantage in buying either the stock or the
forward contract if we can borrow to buy a stock today (so both strategies do not require
any initial cash) and if the profit from this strategy is the same as the profit of a long
forward contract. The profit of a long forward contract with a price for delivery of $53 is
equal to: $ST − $53, where ST is the (unknown) value of one share of XYZ at expiration
of the forward contract in one year. If we borrow $50 today to buy one share of XYZ
stock (that costs $50), we have to repay in one year: $50 × (1 + r). Our total profit in one
year from borrowing to buy one share of XYZ is therefore: $ST − $50 × (1 + r). Now we
can equate the two profit equations and solve for the interest rate r:
$ST − $53 = $ST − $50 × (1 + r)
⇔ $53 = $50 × (1 + r)
⇔ $53 1
$50  = r
⇔ r = 0.06
Therefore, the one-year effective interest rate that is consistent with no advantage to
either buying the stock or forward contract is 6 percent.
Question 2.9
a) If the forward price is $1,100, then the buyer of the one-year forward contract
receives at expiration after one year a profit of: $ST − $1,100, where ST is the
(unknown) value of the S&R index at expiration of the forward contract in one year.
Remember that it costs nothing to enter the forward contract.
Let us again follow our strategy of borrowing money to finance the purchase of the
index today, so that we do not need any initial cash. If we borrow $1,000 today to buy
the S&R index (that costs $1,000), we have to repay in one year: $1,000 × (1 + 0.10)
= $1,100. Our total profit in one year from borrowing to buy the S&R index is
therefore: $ST − $1,100. The profits from the two strategies are identical.

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Chapter 2/An Introduction to Forwards and Options 15
b) The forward price of $1,200 is worse for us if we want to buy a forward contract. To
understand this, suppose the index after one year is $1,150. While we have already
made money in part a) with a forward price of $1,100, we are still losing $50 with the
new price of $1,200. As there was no advantage in buying either the stock or forward
at a price of $1,100, we now need to be “bribed” to enter into the forward contract.
We somehow need to find an equation that makes the two strategies comparable
again. Suppose that we lend some money initially together with entering into the
forward contract so that we will receive $100 after one year. Then, the payoff from
our modified forward strategy is: $ST − $1,200 + $100 = $ST − $1,100, which equals
the payoff of the “borrow to buy index” strategy. We have found the future value of
the premium somebody needs us to pay. We still need to find out what the premium
we will receive in one year is worth today.
We need to discount it: $100/(1 + 0.10) = $90.91.
c) Similarly, the forward price of $1,000 is advantageous for us. As there was no
advantage in buying either stock or forward at a price of $1,100, we now need to
“bribe” someone to sell this advantageous forward contract to us. We somehow need
to find an equation that makes the two strategies comparable again. Suppose that we
borrow some money initially together with entering into the forward contract so that
we will have to pay back $100 after one year. Then, the payoff from our modified
forward strategy is: $ST − $1,000 − $100 = $ST − $1,100, which equals the payoff of
the “borrow to buy index” strategy. We have found the future value of the premium
we need to pay. We still need to find out what this premium we have to pay in one
year is worth today. We simply need to discount it: $100/(1 + 0.10) = $90.91. We
should be willing to pay $90.91 to enter into the one-year forward contract with a
forward price of $1,000.
Question 2.10
a) Figure 2.6 depicts the profit from a long call option on the S&R index with six
months to expiration and a strike price of $1,000 if the future price of the option
premium is $95.68. The profit of the long call option is:
max[0, ST − $1,000] − $95.68
⇔ max[−$95.68, ST − $1,095.68]
where ST is the (unknown) value of the S&R index at expiration of the call option in
six months. In order to find the S&R index price at which the call option diagram
intersects the x-axis, we have to set the above equation equal to zero. We get: ST −
$1,095.68 = 0 ⇔ ST = $1,095.68. This is the only solution, as the other part of the
maximum function, −$95.68, is always less than zero.
b) The profit of the six month forward contract with a forward price of $1,020 is: $ST −
$1,020. In order to find the S&R index price at which the call option and the forward
contract have the same profit, we need to set both parts of the maximum function of
the profit of the call option equal to the profit of the forward contract and see which
part permits a solution. First, we see immediately that $ST − $1,020 = $ST − $1,095.68

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16 Part One/Insurance, Hedging, and Simple Strategies
does not have a solution. But we can solve the other leg: $ST − $1,020 = −$95.68 ⇔
ST = $924.32, which is the value given in the exercise.
Question 2.11
a) Figure 2.8 depicts the profit from a long put option on the S&R index with six months
to expiration and a strike price of $1,000 if the future value of the put premium is
$75.68. The profit of the long put option is:
max[0, $1, 000 − ST ] − $75.68
⇔ max[−$75.68, $924.32 − ST ]
where ST is the (unknown) value of the S&R index at expiration of the put option in
six months. In order to find the S&R index price at which the put option diagram
intersects the x-axis, we have to set the above equation equal to zero. We get: $924.32
− ST = 0 ⇔ ST = $924.32. This is the only solution, as the other part of the maximum
function, −$75.68, is always less than zero.
b) The profit of the short six-month forward contract with a forward price of $1,020 is:
$1,020 − $ST. In order to find the S&R index price at which the put option and the
sold forward contract have the same profit, we need to set both parts of the maximum
function of the profit of the put option equal to the profit of the forward contract and
see which part permits a solution. First, we see immediately that $1,020 − $ST =
$924.32 − $ST does not have a solution. But we can solve the other leg: $1,020 − ST =
−$75.68 ⇔ ST = $1,095.68, which is the value given in the exercise.
Question 2.12
a) Long Forward
The maximum loss occurs if the stock price at expiration is zero (the stock price
cannot be less than zero because companies have limited liability). The forward then
pays 0 – Forward price. The maximum gain is unlimited. The stock price at expiration
could theoretically grow to infinity; there is no bound. We make a lot of money if the
stock price grows to infinity (or to a very large amount).
b) Short Forward
The profit for a short forward contract is forward price – stock price at expiration.
The maximum loss occurs if the stock price rises sharply; there is no bound to it, so it
could grow to infinity. The maximum gain occurs if the stock price is zero.
c) Long Call
We will not exercise the call option if the stock price at expiration is less than the
strike price. Consequently, the only thing we lose is the future value of the premium
we paid initially to buy the option. As the stock price can grow very large (and
without bound), and our payoff grows linearly in the terminal stock price once it is
higher than the strike, there is no limit to our gain.

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Chapter 2/An Introduction to Forwards and Options 17
d) Short Call
We have no control over the exercise decision when we write a call. The buyer of the
call option decides whether to exercise it or not, and he will only exercise the call if
he makes a profit. As we have the opposite side, we will never make any money at
the expiration of the call option. Our profit is restricted to the future value of the
premium, and we make this maximum profit whenever the stock price at expiration is
smaller than the strike price. However, the stock price at expiration can be very large
and has no bound, and as our loss grows linearly in the terminal stock price, there is
no limit to our loss.
e) Long Put
We will not exercise the put option if the stock price at expiration is larger than the
strike price. Consequently, the only thing we lose whenever the terminal stock price
is larger than the strike is the future value of the premium we paid initially to buy the
option. We will profit from a decline in the stock prices. However, stock prices
cannot be smaller than zero, so our maximum gain is restricted to strike price less the
future value of the premium, and it occurs at a terminal stock price of zero.
f) Short Put
We have no control over the exercise decision when we write a put. The buyer of the
put option decides whether to exercise or not, and he will only exercise if he makes a
profit. As we have the opposite side, we will never make any money at the expiration
of the put option. Our profit is restricted to the future value of the premium, and we
make this maximum profit whenever the stock price at expiration is greater than the
strike price. However, we lose money whenever the stock price is smaller than the
strike; hence, the largest loss occurs when the stock price attains its smallest possible
value, zero. We lose the strike price because somebody sells us an asset for the strike
that is worth nothing. We are only compensated by the future value of the premium
we received.
Question 2.13
a) In order to be able to draw profit diagrams, we need to find the future values of the
call premia. They are:
i) 35-strike call: $9.12 × (1 + 0.08) = $9.8496
ii) 40-strike call: $6.22 × (1 + 0.08) = $6.7176
iii) 45-strike call: $4.08 × (1 + 0.08) = $4.4064

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Chapter 2/An Introduction to Forwards and Options 19
b) Intuitively, whenever the 45-strike option pays off (i.e., has a payoff bigger than
zero), the 40-strike and the 35-strike options pay off. However, there are some
instances in which the 40-strike option pays off and the 45-strike option does not.
Similarly, there are some instances in which the 35-strike option pays off but neither
the 40-strike nor the 45-strike pay off. Therefore, the 35-strike offers more potential
than the 40- and 45-strike, and the 40-strike offers more potential than the 45-strike.
We pay for these additional payoff possibilities by initially paying a higher premium.
Question 2.14
In order to be able to draw profit diagrams, we need to find the future values of the put
premia. They are:
a) 35-strike put: $1.53 × (1 + 0.08) = $1.6524
b) 40-strike put: $3.26 × (1 + 0.08) = $3.5208
c) 45-strike put: $5.75 × (1 + 0.08) = $6.21
We get the following payoff diagrams:

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20 Part One/Insurance, Hedging, and Simple Strategies
We get the profit diagram by deducting the option premia from the payoff graphs. The
profit diagram looks as follows:
Intuitively, whenever the 35-strike put option pays off (i.e., has a payoff bigger than
zero), the 40-strike and the 35-strike options also pay off. However, there are some
instances in which the 40-strike option pays off and the 35-strike option does not.
Similarly, there are some instances in which the 45-strike option pays off but neither the
40-strike nor the 35-strike pay off. Therefore, the 45-strike offers more potential than the
40- and 35-strike, and the 40-strike offers more potential than the 35-strike. We pay for
these additional payoff possibilities by initially paying a higher premium. It makes sense
that the premium is increasing in the strike price.
Question 2.15
The nice thing that lead us to the notion of indifference between a forward contract and a
loan- financed stock index purchase whenever the forward price equaled the future price
of the loan was that we could already tell today what we had to pay back in the future. In
other words, the return on the loan, the risk-free interest rate r, was known today, and we
removed uncertainty about the payment to be made. If we were to finance the purchase of
the index by short selling IBM stock, we would introduce additional uncertainty because
the future value of the IBM stock is unknown. Therefore, we could not calculate today
the amount to be repaid, and it would be impossible to establish an equivalence between
the forward and loan-financed index purchase today. The calculation of a profit diagram
would only be possible if we assumed an arbitrary value for IBM at expiration of the
futures, and we would have to draw many profit diagrams with different values for IBM
to get an idea of the many possible profits we could make.

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Chapter 3
Insurance, Collars, and Other Strategies
Question 3.1
This question is a direct application of the Put-Call-Parity [equation (3.1)] of the textbook.
Mimicking Table 3.1., we have:
S&R Index S&R Put Loan Payoff −(Cost + Interest) Profit
900.00 100.00 −1,000.00 0.00 −95.68 −95.68
950.00 50.00 −1,000.00 0.00 −95.68 −95.68
1,000.00 0.00 −1,000.00 0.00 −95.68 −95.68
1,050.00 0.00 −1,000.00 50.00 −95.68 −45.68
1,100.00 0.00 −1,000.00 100.00 −95.68 4.32
1,150.00 0.00 −1,000.00 150.00 −95.68 54.32
1,200.00 0.00 −1,000.00 200.00 −95.68 104.32
The payoff diagram looks as follows:

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Chapter 3/Insurance, Collars, and Other Strategies 23
We can see from the table and from the payoff diagram that we have in fact reproduced a call
with the instruments given in the exercise. The profit diagram below confirms this hypothesis.
Question 3.2
This question constructs a position that is the opposite to the position of Table 3.1. Therefore, we
should get the exact opposite results from Table 3.1. and the associated figures. Mimicking Table
3.1., we indeed have:
S&R Index S&R Put Payoff −(Cost + Interest) Profit
−900.00 −100.00 −1,000.00 1,095.68 95.68
−950.00 −50.00 −1,000.00 1,095.68 95.68
−1,000.00 0.00 −1,000.00 1,095.68 95.68
−1,050.00 0.00 −1,050.00 1,095.68 45.68
−1,100.00 0.00 −1,100.00 1,095.68 −4.32
−1,150.00 0.00 −1,150.00 1,095.68 −54.32
−1,200.00 0.00 −1,200.00 1,095.68 −104.32
On the next page, we see the corresponding payoff and profit diagrams. Please note that they
match the combined payoff and profit diagrams of Figure 3.5. Only the axes have different
scales.

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Chapter 3/Insurance, Collars, and Other Strategies 25
Question 3.3
In order to be able to draw profit diagrams, we need to find the future value of the put premium,
the call premium, and the investment in zero-coupon bonds. We have for:
the put premium: $51.777 × (1 + 0.02) = $52.81,
the call premium: $120.405 × (1 + 0.02) = $122.81, and
the zero-coupon bond: $931.37 × (1 + 0.02) = $950.00
Now, we can construct the payoff and profit diagrams of the aggregate position:
Payoff diagram:
From this figure, we can already see that the combination of a long put and the long index looks
exactly like a certain payoff of $950, plus a call with a strike price of 950. But this is the
alternative given to us in the question. We have thus confirmed the equivalence of the two
combined positions for the payoff diagrams. The profit diagrams on the next page confirm the
equivalence of the two positions (which is again an application of the Put-Call-Parity).

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26 Part One/Insurance, Hedging, and Simple Strategies
Profit diagram for a long 950-strike put and a long index combined:
Question 3.4
This question is another application of Put-Call-Parity. Initially, we have the following cost to
enter into the combined position: We receive $1,000 from the short sale of the index, and we
have to pay the call premium. Therefore, the future value of our cost is: ($120.405 − $1,000) ×
(1 + 0.02) = −$897.19. Note that a negative cost means that we initially have an inflow of
money.
Now, we can directly proceed to draw the payoff diagram:

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Chapter 3/Insurance, Collars, and Other Strategies 27
We can clearly see from the figure that the payoff graph of the short index and the long call
looks like a fixed obligation of $950, which is alleviated by a long put position with a strike price
of 950. The following profit diagram, including the cost for the combined position, confirms this:
Question 3.5
This question is another application of Put-Call-Parity. Initially, we have the following cost to
enter into the combined position: We receive $1,000 from the short sale of the index, and we
have to pay the call premium. Therefore, the future value of our cost is: ($71.802 − $1,000) ×
(1 + 0.02) = −$946.76. Note that a negative cost means that we initially have an inflow of
money.

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28 Part One/Insurance, Hedging, and Simple Strategies
Now, we can directly proceed to draw the payoff diagram:
In order to be able to compare this position to the other suggested position of the exercise, we
need to find the future value of the borrowed $1,029.41. We have: $1,029.41 × (1 + 0.02) =
$1,050. We can now see from the figure that the payoff graph of the short index and the long call
looks like a fixed obligation of $1,050, which is exactly the future value of the borrowed amount,
and a long put position with a strike price of 1,050. The following profit diagram, including the
cost for the combined position we calculated above, confirms this. The profit diagram is the
same as the profit diagram for a loan and a long 1,050-strike put with an initial premium of
$101.214.

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Chapter 3/Insurance, Collars, and Other Strategies 29
Profit diagram of going short the index and buying a 1,050-strike call:
Question 3.6
We now move from a graphical representation and verification of the Put-Call-Parity to a
mathematical representation. Let us first consider the payoff of (a). If we buy the index (let us
name it S), we receive at the time of expiration T of the options simply ST.
The payoffs of part (b) are a little bit more complicated. If we deal with options and the
maximum function, it is convenient to split the future values of the index into two regions: one
where ST < K and another one where ST ≥ K . We then look at each region separately, and hope
to be able to draw a conclusion when we look at the aggregate position.
We have for the payoffs in (b):
Instrument ST < K = 950 ST ≥ K = 950
Get repayment of loan $931.37 × 1.02 = $950 $931.37 × 1.02 = $950
Long call option max (ST − 950, 0) = 0 ST − 950
Short put option − max ($950 − ST , 0) 0
= ST − $950
Total ST ST

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30 Part One/Insurance, Hedging, and Simple Strategies
We now see that the total aggregate position only gives us ST , no matter what the final index
value is—but this is the same payoff as in part (a). Our proof for the payoff equivalence is
complete.
Now let us turn to the profits. If we buy the index today, we need to finance it. Therefore, we
borrow $1,000 and have to repay $1,020 after one year. The profit for part (a) is thus: ST −
$1,020.
The profits of the aggregate position in part (b) are the payoffs, less the future value of the call
premium plus the future value of the put premium (because we have sold the put), and less the
future value of the loan we gave initially. Note that we already know that a riskless bond is
canceling out of the profit calculations. We can again tabulate:
Instrument ST < K ST ≥ K
Get repayment of loan $931.37 × 1.02 = $950 $931.37 × 1.02 = $950
Future value of given loan −$950 −$950
Long call option max (ST − 950, 0) = 0 ST − 950
Future value call premium −$120.405 × 1.02 = −$122.81 −$120.405 × 1.02 = −$122.81
Short put option − max ($950 − ST , 0) 0
= ST − $950
Future value put premium $51.777 × 1.02 = $52.81 $51.777 × 1.02 = $52.81
Total ST − 1020 ST − 1020
Indeed, we see that the profits for parts (a) and (b) are identical as well.
Question 3.7
Let us first consider the payoff of (a). If we short the index (let us name it S), we have to pay at
the time of expiration T of the options: −ST .
The payoffs of part (b) are more complicated. Let us look again at each region separately, and
hope to be able to draw a conclusion when we look at the aggregate position.
We have for the payoffs in (b):
Instrument ST < K ST ≥ K
Make repayment of loan −$1,029.41 × 1.02 = −$1,050 −$1,029.41 × 1.02 = −$1,050
Short call option − max (ST − 1,050, 0) = 0 − max (ST − 1,050, 0)
= 1,050 − ST
Long put option max ($1,050 − ST , 0) 0
= $1,050 − ST
Total −ST −ST

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Solution Manual For Derivatives Markets, 3rd Edition

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