UNIVERSITY OF TORONTO

Joseph L. Rotman School of Management

RSM332 Practice problems - Introduction to derivatives

1 Multiple Choice Questions

1. You buy one August call option with a strike price of 50 and one August
put option with the same strike price. The call price is 4.25 and the put
price is 4.50. You strategy is useful if you believe that the stock price ...
A will be lower than 41.25 in August
B will be between 41.25 and 58.75 in August
C will be higher than 58.75 in August
D both A) and C) are correct
E none of the above
2. With the stock at 15 per share, an investor desires the following payo↵
structure

Assuming all options are European, this can be achieved by

A Long the stock, long a put (X=20), short a call (X=10).
B Buy a bond with face value=10, short a call(X=10), long a call(X=20)
C Buy a bond with face value=20, long a put(X=10), short a put(X=20).
D Buy a Bond with face value=20, short 2 puts (X=10), short 2 puts
(X=20)
E None of the above
3. The S&P index is currently at 1000. The one-year interest rate is 6%
(annually compounded) and the S&P index is assumed to pay no dividend
in the coming year. The price on an index forward contract with a 1-year
maturity is 1070. An arbitrageur can

1
 A realize a profit at maturity of 10 by borrowing 1000, buying the
index, and entering into the futures contract short.
B realize a profit at maturity of 10 by shorting the index, investing
1000 and entering into the futures contract long.
C realize a profit at maturity of 10 by shorting the index, investing
1000 and entering into the futures contract short.
D realize a profit at maturity of 10 by borrowing 1000, buying the
index, and entering into the futures contract long.
E Realize a profit at maturity of 10 by buying the index for 1000 and
selling it in one year.
4. You are working as a risk manager for a taxi company. You are wor-
ried that a sudden change in fuel price hurts your profit. You will need
1,000,000 liters of fuel next year. Which of the following strategy bests
protects you against fuel price movements.
A take of a short position in a forward contract for 1,000,000 liters of
fuel
B take a long position in a forward contract for 3,000,000 liters of fuel
C short put options on a total of 1,000,000 liters of fuel
D none of the above

2 Numerical Questions
1. Suppose your company intends to purchase 1 million pounds of copper
one year later. The company wishes to hedge 80% of its exposure and will
use forward contracts. One forward contract is for the delivery of 25,000
pounds of copper. Today’s spot price of copper is 372 per pound. The
risk-free rate is 3%
(a) What should be the forward price?
(b) How should you achieve the hedging objective of the company?
2. You are working for a gold mine. You run a regression of your firm’s profit
on the price of gold per ounce (PAU ) and obtain the following estimates:

P rof it = 0.95M + 2.15M PAU

If you would like to completely immunize you firm against gold price risk
next year and assuming that your profit will keep following the same
dynamics, what should you do? Assume that each forward contract is
for the delivery of 1,000 ounces of gold.
3. You are working as a risk manager for Bombardier Inc. The firm has just
signed a 1 billion USD contract for the sale of 12 airplanes to a customer
in California. Each airplane costs 80 millions CAD to produce. Both the

2
 revenues from the sale and the total cost of production are assumed to be
paid in two years, when the planes are delivered.
The management team is concerned that a devaluation in the USD a↵ects
the firm’s operating profit. Your task is to use options to ensure that the
operating profit on the contract be at least 10 millions CAD in two years.
Using option contracts (each option represents 10,000 USD), show how
you can implement this objective. Describe your strategy carefully, i.e.,
what type(s) of options do you buy/sell and in what quantities.
4. You now work as an option trader and want to invest in a portfolio of
call options on AAPL stock. All options have a maturity of 1 year and
we denote by C(X) a call option with a strike price of X. You consider
the following strategy: buy 1 C(X = 50), sell 3 C(X = 150), buy 1
C(X = 200), buy 1 C(X = 250).
(a) Draw the gross payo↵ diagram of the portofolio at maturity. What
is the gross payo↵ if the stock of AAPL is 180 in a year?
(b) Implement the same payo↵ diagram using put options and one-year
zero coupon bonds.
5. You would like to create a security with the following payo↵ as a function
of the price (ST ) of an underlying stock at time T .
15

10

0
0 5 10 15 20 25 30 35 40 45 50

(a) Suppose you can only trade put options on the underlying stock, show
how you can use put options to create the above security.
(b) Suppose you can only trade the stock, call options on the underlying
stock as well as risk-free bonds, show how you can use call options and
bonds to create the above security.

3
3 Solutions
MCQs: D, C, A, D

Numerical questions:
1. (a) F = 383.16
(b) Your total exposure is 1M pounds. You wish to hedge 0.8M pounds.
You need a total of 0.8M/25,000=32 contracts. Since you intend to buy
copper in a year and wish you lock the future price, you enter a long
position in 32 contracts.
2. Each long position in the forward contract pays out 1000PAU 1000F .
You profit function with N forwards become

P rof itnew = 0.95M + 2.15M PAU + N 1000(PAU F)

In order to be immunized we need @P rof itnew /@PAU = 2.15M +1000N =

0.
Solving for N: N = 2, 150. In other words, you need to take a short
position in 2,150 contracts of gold for next year.
3. Denoting the USDCAD exchange rate by F , Bombardier’s profit (in CAD
millions) in two years is

⇧ = 1000F 80 ⇥ 12 = 1000F 960

The objective of the management is to generate an operating profit of at

least 10 millions CAD. Therefore, the desired payo↵ is:

max(⇧, 10) = max(1000F 960, 10)

= max(0, 10 1000F + 960) + 1000F 960
= max(0, 970 1000F ) + 1000F 960
= 1000 max(0, 0.97 F ) +⇧
| {z }
Put option payo↵

Therefore Bombardier should buy a total of 100,000 (= 110,000

billion
) USDCAD
put options with a maturity of two years and a strike price of 0.97.

4. (a) The gross payo↵ diagram of the portfolio at maturity is:

4
 Gross Payo↵

100

50

ST
0
50 100 150 200 250

50

If S = 180, then all the Call options with X 180 are not exercised, so
the gross payo↵ is given by:

Payo↵ = max(0, 180 50) 3 max(0, 180 150) = 130 90 = 40.

(b) Sell a 1-year zero coupon bond with a face value of 50, buy 1 P(250),
buy 1 P(200), sell 3 P(150), buy 1 P(50).
5. (a) In order to obtain the desired payo↵ using put options, we can buy 1
put option with exercise price 10, buy one put option with exercise price
20, sell two put options with exercise price 30, and buy a put option
with exercise price 40.
(b) Using the put-call parity, we can convert the above strategy into posi-
tions that involve risk-free bond, stock, and calls. Specifically, we can buy
a risk-free bond with face value of 10, sell a stock, buy a call option with
exercise price 10, buy a call option with exercise price 20, sell two call
options with exercise price 30, and buy a call option with exercise price
40.

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 UNIVERSITY OF TORONTO
Joseph L. Rotman School of Management

RSM332 Practice problems - Option Pricing II

1 Multiple Choice Questions

1. Portfolio X consists of 300 shares of stock and 280 puts on that stock.
Portfolio Y consists of 150 shares of stock. The put delta is = 0.5.
Which portfolio has a higher dollar exposure to a change in stock price?
A Portfolio Y
B Portfolio X
C The two portfolios have the same exposure.
D X if the stock price increases and Y if it decreases.
E None of the above
2. Suppose ABC’s stock price is currently 50. In the next six months it will
either fall to 30 or rise to 80. What is the option delta of a call option
with an exercise price of 50?
A 0.375
B 0.500
C 0.600
D 0.75
3. If a call option is far ’out of the money’ the value of the option will be:

A Less than the value of a put option with the same exercise price
B Zero
C Greater than the value of a put option with the same exercise price
D Equal to the value of a put option with the same exercise price
E None of the above is correct

4. Suppose XYS’s stock price is currently 20. A one year call option on the
stock with an exercise price of 12 has a value of 9.43. Calculate the
price of an equivalent put option if the one year risk-free interest rate is
5% (annual rate).

A 0.86
B 9.43
C 8.00
D 12.00

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 5. Suppose that the stock price of Bank IOU is currently 25. In the next
year it will either fall to 15 or rise to 40. What is the current value of a
one-year call option with an exercise price of 20? The one-year risk-free
interest rate is 5% (annual rate).
A 20.00
B 8.57
C 9.52
D 13.10
6. An increase in a stock’s volatility:
A Increases the associated call option value
B Decreases the associated put option value
C Increases or decreases the option value, depending on the level of
interest rates
D Does not change either the put or call option value because put-call
parity holds
E Both (A) and (B) are correct

2 Numerical Questions
1. The current price of a non-dividend paying stock is 100. You also collect
additional information on a series of European options written on the
stock. Options with the same characteristics are grouped into three sets:
A, B, and C:

Set Maturity (in years) Call Put Strike

A 1 7.5 15.35 XA
B 1 CB 49.5 150
C TC 12.12 4.5 100

The term structure of interest rates is flat with r = 2%/year.

(a) Assuming no-arbitrage, find the missing information in the table above,
XA , CB , and TC .
(b) Consider the following trading strategy: buy one call option A and
sell one call option B. You would like to generate a holding period return
of 25% when the options expire in a year. What price of the underlying
asset in a year, S1 , would give you such a return.
(c) Suppose that the European call option in set B, CB , is currently
trading for 2. Is there an arbitrage opportunity? If so, explain how can
you take advantage of it.
(d) Which of the following statement is true? Circle the right answer.

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 If the traded call option in part (c), CB , were an American call option
instead of a European call option and assuming that all the other options
stay the same, then
i there would be no arbitrage opportunity.
ii you could perform the exact same arbitrage strategy as in part (c) but
only if you could find an American put with the same characteristics
as CB .
iii you would do the same arbitrage as in part (c).
iv there is not enough information to conclude anything.

2. Consider a Put and a Call option, written on a stock with the following
characteristics (both options have the same characteristics): strike price
X = 100, maturity T = 2 years. The risk-free interest rate is constant
and equal to 2%, annually compounded. The stock price is expected to
take the following values in the future (each period is one year):

144

120

100 108

90

81

(a) Using a two-period binomial tree, compute the price of a European

Call option written on the stock using a dynamic replication strategy.
(b) Using a two-period binomial tree, compute the price of a European
Put option written on the stock using a dynamic replication strategy.
(c) Using the price obtained in part (a), verify that the Put-Call Parity
relation holds.

3. Consider an option on a stock ABC that pays no dividend. The option has
the following characteristics: strike price X = 22, maturity T = 2 years.
Assume that the term structure of interest rate is flat at a rate (annually
compounded) of 1% and that the stock prices can take the following path
over the next two years (each period is one year):

3
 28.8

24

20 21.6

18

16.2

(a) Using a dynamic replicating porfolio, compute the price P of an Eu-

ropean Put option on stock ABC.
(b) Using a dynamic replicating porfolio, compute the price P of an Amer-
ican Put option on stock ABC.
(c) Which option is more valuable and why?

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3 Solutions
MCQs: (1) B, (2) C, (3) A, (4) A, (5) B, (6) A.

Numerical Questions:
1. (a) The missing information can be obtained using the Put-Call parity.
The missing information solves:
XA
= PA + S CA ) XA = (15.35 + 100 7.5)(1 + 2%) = 110
1 + 2%
XB
CB = PB + S = 2.44
(1 + 2%)
✓ ◆
XC XC
= PC + S CC ) TC = ln / ln(1 + 2%) = 4
(1 + 2%)TC PC + S C C

(b) The Bull Spread strategy in a year has the following payo↵:

⇧1 = max(S1 110, 0) max(S1 150, 0)

The cost of that portfolio is CA CB .

We want the holding period return in a year to be equal to 25%, that is:
⇧1
R1 = 1 = 25%
CA CB
max(S1 110, 0) max(S1 150, 0)
() 1.25 =
5.06
() 6.325 = max(S1 110, 0) max(S1 150, 0)

Clearly, for S1 < 110 and S1 > 150, this equation does not hold, therefore
it must be that 110  S1  150. It follows that

6.325 = S1 110 ) S1 = 116.32

(c) The market price is di↵erent from the no-abritrage price, so there is
an arbitrage. Comparing the market price (2) to the no-arbitrage price
(2.44), we notice that the call is underpriced, so we’d like to buy it and
use the Put-Call Parity to build a replicating portfolio for the call, that
is:

Actions 0 S1  150 150 < S1

Buy CB 2.0 0 S1 150
Sell PB +49.5 (150 S1 ) 0
Sell S +100 S1 S1
150 150
Deposit (1+2%) (1+2%) 150 150
Net Cash-Flows 0.44 0 0

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 This strategy generates a strictly positive cash-flow today with no obli-
gations in the future, so this is a risk-free lunch. The profit per trade is
0.44.
(d) iii. It is never optimal to exercise an American Call option on a non-
dividend paying stock prior to maturity. Therefore the American call is
essentially the same as the European call and the same arbitrage strategy
can be done.
2. (a) The European Call option price tree is:

44

21.9608

10.45752 8

3.1372

To obtain these, we computed the following values for the hedging porftolio
at each node (assuming that the risk-free bond has a face value of 100):
u = 1, Bu = 100, d = 0.296, Bd = 24, = 0.62745, and B =
53.33. The final price for the option is obtained using:
1 1
C= ⇥S +B ⇥ = 0.62745 ⇥ 100 + 53.33 ⇥ = 10.45752.
1 + 2% 1 + 2%
(b) Doing the same for an European Put option:

6.57439 0

11.1764

19

To obtain these, we computed the following values for the hedging porftolio
at each node (assuming that the risk-free bond has a face value of 100):
u = 0, Bu = 0, d = 0.7037, Bd = 76, = 0.37254, and B = 44.706.
The final price for the option is obtained using:
1 1
P = ⇥S +B ⇥ = 0.37254 ⇥ 100 + 44.706 ⇥ = 6.57439.
1 + 2% 1 + 2%

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 (c) The Put Call parity implies that the price of a put is given by:

X 100
P =C S+ = 10.45752 100 + = 6.5744,
(1 + r)T (1 + r)T

which is the same price as that obtained in (b).

3. (a) The European Put option price path is:

0.2508

2.4627 0.4

3.7821

5.8

To obtain these, we computed the following values for the hedging porftolio
at each node (assuming that the risk-free bond has a face value of 100):
u = 0.05556, Bu = 1.6, d = 1, Bd = 22, = 0.5885, and
B = 14.376. The final price for the option was obtained using:
1 1
PE = ⇥S+B⇥ = 0.5885 ⇥ 20 + 14.376 ⇥ = 2.4627.
1 + 1% 1 + 1%

(b) To obtain the American Put option value, we just need to check if it
is optimal to exercise early at any nodes. We then only have to change
those nodes and all the preceding ones. Those not a↵ected can stay the
same. We notice that at node d, it is optimal to exercise the put since
the intrisic value = 22 18 = 4 > 3.7821. Recomputing the Put price, we
obtain:

0.2508

2.5993 0.4

5.8

7
The replicating strategy components are: = 0.6249, and B = 15.25.
And the put price is:
1
PA = 0.6249 ⇥ 20 + 15.25 ⇥ = 2.5993.
1 + 1%

(c) PA > PE because the expected future cash-flow of the American option
in node d is larger than that of the European option because early exercise
was optimal.