BUSS386, Spring 2023
Problem Set 1
Topics 1 through 4 (Slide 24)
Due: 5pm, Monday, April 3
Turn in your answers to questions with an asterisk (*).
1. (FFOM, Part of 4.32) The following table gives the prices of bonds:
Bond principal
($)
100
100
100
100
Time to maturity
(years)
0.50
1.00
1.50
2.00
Annual coupon†
($)
0.0
0.0
6.2
8.0
Bond price
($)
98
95
101
104
† Half the stated coupon is assumed to be paid every six months.
(a) Calculate zero rates for maturities of 6 months, 12 months, 18 months, and
24 months.
(b) What are the forward rates for the periods: 6 months to 12 months, 12 months
to 18 months, 18 months to 24 months?
(c) Estimate the price of a two-year bond providing a semiannual coupon of 7%
per annum.
2. (OFOD, 1.20) On July 1, 2021, a company enters into a forward contract to buy
10 million Japanese yen on January 1, 2022. On September 1, 2021, it enters into
a forward contract to sell 10 million Japanese yen on January 1, 2022. Describe
the payoff from this strategy.
3. (OFOD, 1.10) Suppose that a June put option to sell a share for $60 costs $4 and
is held until June. Under what circumstances will the seller of the option (i.e.,
the party with the short position) make a profit? Under what circumstances will
the option be exercised? Draw a diagram illustrating how the profit from a short
position in the option depends on the stock price at maturity of the option. Ignore
the time value of money.
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4. (FFOM, 1.35) The current price of a stock is $94, and three-month European 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?
5. (FFOM, 2.28) A company enters into a short futures contract to sell 5,000 bushels
of wheat for 450 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circumstances could $1,500 be withdrawn from the margin account?
6. (FFOM, 2.29) 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 if
the June and December futures contracts for a particular year trade at $80 and
$86, respectively. (Note: For this question, you may assume that the interest
accumulated on $100 for six months is approximately 100 × 5% × 6/12 or $2.5.)
7. (FFOM, 9.26) The price of a stock is $40. The price of a one-year European put
option on the stock with a strike price of $30 is quoted as $7 and the price of a
one-year European call option on the stock with a strike price of $50 is quoted as
$5. Suppose that an investor buys 100 shares, shorts 100 call options, and buys
100 put options. Draw a diagram illustrating how the investor’s profit or loss
varies with the stock price over the next year. How does your answer change if
the investor buys 100 shares, shorts 200 call options, and buys 200 put options?
Ignore the time value of money.
8. (FFOM, 9.28) Use DerivaGem to calculate the value of an American put option
on a non-dividend-paying stock when the stock price is $30, the strike price is $32,
the risk-free rate is 5%, the volatility is 30%, and the time to maturity is 1.5 years.
(Choose “Binomial American” for the Option Type and 50 time steps.)
(a) What is the option’s intrinsic value?
(b) What is the option’s time value?
(c) What would a time value of zero indicate? What is the value of an option
with zero time value?
(d) Using a trial and error approach, calculate how low the stock price would
have to be for the time value of the option to be zero.
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9. (FFOM, 3.22) It is now June. A company knows that it will sell 5,000 barrels of
crude oil in September. It uses the October CME Group futures contract to hedge
the price it will receive. Each contract is on 1,000 barrels of “light sweet crude.”
What position should it take? What price risk is it still exposed to after taking
the position?
10. Assume the interest rate is 4% per annum with semiannual compounding, the 6month forward price for company XYZ is $1020, and use these premiums for XYZ
options with 6 months to expiration:
Call
Put
Strike
$950 $120.405 $51.777
1000
93.809 74.201
84.470 84.470
1020
1050
71.802 101.214
1107
51.873 137.167
Be sure to account for the interest cost of each position when you compute profit.
(a) Suppose you buy the XYZ stock for $1000 and buy a 950-strike put. Construct
payoff and profit diagrams for this position. Verify that you obtain the same
payoff and profit diagram by investing $931.37 in zero-coupon bonds and
buying a 950-strike call.
(b) Suppose you invest in the XYZ stock for $1000, buy a 950-strike put, and sell
a 1050-strike call. Draw a profit diagram for this position. What is the net
option premium? If you wanted to construct a zero-cost collar keeping the
put strike equal to $950, in what direction would you have to change the call
strike?
11. (OFOD, 1.19) In the 1980s, Bankers Trust developed index currency option notes
(ICONs). These were bonds in which the amount received by the holder at maturity varied with a foreign exchange rate. One example was its trade with the Long
Term Credit Bank of Japan. The ICON specified that if the yen/USD exchange
rate, ST , is greater than 169 yen per dollar at maturity (in 1995), the holder of
the bond receives $1,000. If it is less than 169 yen per dollar, the amount received
by the holder of the bond is
169
−1 .
1, 000 − max 0, 1, 000
ST
When the exchange rate is below 84.5, nothing is received by the holder at maturity.
Show that this ICON is a combination of a regular bond and two options.
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12. (*) Natick, MA and Indianapolis, IN (January 25, 2006) - Boston Scientific Corporation (NYSE: BSX) and Guidant Corporation (NYSE: GDT) today announced
that the Board of Directors of Guidant has unanimously approved and entered
into the merger agreement provided to Guidant by Boston Scientific on January
17, 2006. Under that agreement, Boston Scientific will acquire all the outstanding
shares of Guidant for a combination of cash and stock worth $80 per Guidant share,
or approximately $27 billion in aggregate. Prior to entering into this agreement
with Boston Scientific, Guidant terminated its merger agreement with Johnson &
Johnson. Under the terms of the merger agreement between Boston Scientific and
Guidant, each share of Guidant common stock will be exchanged for $42.00 in cash
and $38.00 in Boston Scientific common stock, based on the average closing price
of Boston Scientific common stock during the 20 consecutive trading day period
ending three days prior to the closing date. If the average closing price of Boston
Scientific common stock during this period is less than $22.62, Guidant shareholders will receive 1.6799 Boston Scientific shares for each share of Guidant common
stock, and if the average closing price of Boston Scientific common stock during
this period is greater than $28.86, Guidant shareholders will receive 1.3167 Boston
Scientific shares for each share of Guidant common stock. Guidant shareholders
will own approximately 36 percent of the combined company.
(a) Assuming this deal was completed as originally announced, draw the payoff
to shareholders of Guidant at closing as a function of the Boston Scientific
stock market value.
(b) Show how the payoffs to Guidant shareholders can be replicated using combinations of zero coupon bonds (borrowing/lending), Boston Scientific stock,
and options on Boston Scientific stock. Be precise about relevant quantities
and strike prices.
13. (*) Consider the following prices for coffee options which were taken from WSJ.
The units are dollars per pound of coffee.
Strike July Calls
1.60
0.1125
1.65
0.1000
1.70
0.0900
1.75
0.0760
July Puts
0.1635
0.2000
0.2410
0.2770
The futures price for July delivery is $1.549/lb. Ignore the time value of the money.
Suppose I am a coffee producer (i.e., I sell coffee). Assume my break-even price
for coffee is $1.55/lb.
(a) If I want to lock in a selling price for coffee, what futures position should
I take? What profit have I locked in? (Note: In this context, profit is the
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difference between the “all-in” price and the break-even price ($1.55/lb). For
example, if you sell coffee at $1.70 after considering all involved costs, you
make a profit of 1.70 − 1.55 or $0.15.)
(b) I would like to protect myself against drops in the July price of coffee but still
benefit if the July price of coffee rises. What option position should I take if
I want a $1.65/lb floor—i.e., the option provides a hedge against price drops
to $1.65 or lower—on my selling price? What is the lowest “all-in” price (i.e.
the net cost or revenue from the option plus the revenue from the coffee) that
I will receive for the coffee? Draw the profit diagram with and without the
floor. At what July spot price for coffee does the hedged position start to do
better than the unhedged position?
(c) How would I get a collar with a $1.60 floor and a $1.70 cap—i.e., my profit is
capped when the price of coffee is $1.70 or higher? Draw the profit diagram
for the collar. At what July spot price for coffee does the collar start to do
better than the unhedged position?
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