Assignment 1
Question 1. Margin Account and Settlement
Suppose that you bought two one-year gold futures contracts when the one-year futures
price of gold was US$1,340.30 per troy ounce. You then closed the position at the end of
the sixth trading day. The initial margin requirement is US$5,940 per contract, and the
maintenance margin requirement is US$5,400 per contract. One contract is for 100 troy
ounces of gold. The daily prices on the intervening trading days are shown in the following
table.
Day
Settlement Price
0
1340.30
1
1345.50
2
1339.20
3
1330.60
4
1327.70
5
1337.70
6
1340.60
Assume that you deposit the initial margin and do not withdraw the excess on any given
day. Whenever a margin call occurs on Day t, you would make a deposit to bring the
balance up to meet the initial margin requirement at the start of trading on Day t+1, i.e.,
the next day.
a.
What are the initial margin and maintenance margin on your margin account?
Ans. $11,880 & $10,800
b.
Fill the appropriate numbers in the blank cells in the following table. (Hint: See
solution to Q19 in Lesson 2 Learning Activity.)
Day
Settlement
price per
troy ounce
0
$1340.30
1
$1345.50
2
$1339.20
3
$1330.60
4
$1327.70
5
$1337.70
6
$1340.60
Mark-toMarket
Other
Entries
1
Account
Balance
Explanation
Margin
Call?
Y/N
c.
What is your total profit after you closed out your position?
Question 2. Binomial Model and Option Pricing
The shares of XYZ Inc. are currently selling for $120 per share. The shares are expected to
go up by 10 percent or down by 5 percent in each of the following two months (Month 1 and
Month 2). XYZ Inc. is also expected to pay a dividend yield of 2 percent at the end of Month
1. The risk-free rate is 0.5 percent per month.
a.
b.
What is the value of an American call option on XYZ shares, with an exercise price of
$125 and two months to expiration? Use the binomial model to obtain the answer.
Draw a binomial tree diagram for this American call option, showing the share price,
call price, and whether the call should be exercised at each state during the next two
months.
Question 3. Currency Option Pricing with Binomial Model
On January 11, the spot exchange rate for the U.S. dollar is $0.70 per Canadian dollar. In
one year’s time, the Canadian dollar is expected to appreciate by 20 percent or depreciate
by 15 percent. We have a European put option on U.S. dollars expiring in one year, with an
exercise price of 1.39 CND$/US$, that is currently selling for a price of $2.93. Each put
option gives the holder the right to sell 10,000 U.S. dollars. The current one-year Canadian
Treasury Bill rate is 2 percent, while the one-year U.S. Treasury Bill rate is 3 percent, both
compounded annually. Treat the Canadian dollar as the domestic currency.
a.
What is the estimated value of this put option by using the binomial model?
b.
Calculate the estimated value of this put option for U.S. T-Bill rates of 0%, 1%, 2%,
4%, 5%, and 6%. Plot these values in a graph (by hand or using Excel), with put
option values on the y-axis and U.S. T-bill rates on the x-axis. What can we conclude
about the relationship between foreign interest rates and foreign currency put option
values?
c.
Calculate the estimated value of this put option for Canadian T-Bill rates of 0%, 1%,
2%, 4%, 5%, and 6%. Plot these values in a graph (by hand or using Excel), with
put option values on the y-axis and Canadian T-bill rates on the x-axis. What can we
conclude about the relationship between domestic interest rates and foreign currency
put option values?
Question 4. Option Pricing with Black-Scholes-Merton Model
Today is January 12, 2017. The shares of XYZ Inc. are currently selling for $120 per share.
The shares have an estimated volatility of 25%. XYZ Inc. is also expected to pay a dividend
of $1.50 with an ex-dividend date of January 25, 2017. The risk-free rate is 6.17 percent
2
per year with continuous compounding. Assume that one call option gives the holder the
right to purchase one share.
a.
Use the Black-Scholes-Merton model to estimate the fair value of a European call
option on XYZ shares, with exercise price of $125 and expiration date of March 21,
2017. (Note that 2017 is not a leap year.)
b.
This European call option has a market price of $3.00. Is it correctly priced? If not,
how can an investor use the put-call parity to take advantage of this arbitrage
opportunity?
Question 5. Volatility and Option Hedging
Today, is January 4, 2016. IBM common stock is selling at $135.95 per share. The stock has
a dividend yield of 4% per year. The following table contains the monthly stock prices for
IBM shares during the last 12 months.
Month (2015)
IBM Share Price
January
148.46
February
157.92
March
156.51
April
167.04
May
166.69
June
159.82
July
159.16
August
146.52
September
143.62
October
138.78
November
139.42
December
137.62
A call option with a March 18, 2016 expiration date and an exercise price of $130 is
currently trading at $6.50. Each option entitles the holder to purchase 100 IBM shares. The
risk-free rate is 0.58%, compounded continuously. Shares and options can only be bought
and sold in whole numbers. Note that 2016 is a leap year.
a.
Compute the historical volatility in terms of annualized standard deviation on the IBM
shares, using the 12-month price data in the table above. Note that the volatility
should be calculated on the stock returns and not on the stock prices. Obtain your
answer to four decimal places (or two decimal places in percentage).
b.
Based on the market price of $6.50, derive the implied volatility on the IBM shares.
You may use the BlackScholesMertonImpliedVolatility10e.xlsm file provided by the
textbook’s authors to derive the implied volatility. Take a screen shot of the answer
provided in this Excel spreadsheet, and copy and paste it into your answer for this
3
question. Obtain your answer to four decimal places (or two decimal places in
percentage).
c.
Construct a delta-hedge position on January 4, 2016 involving the sale of 1,000
calls. Then rebalance the portfolio at the end of the next day, when the share price
goes down to $135 per share. Assume the market call price is correct. That is, use
the implied volatility as the correct volatility for the IBM shares. (You may calculate
the deltas using the formula or the BlackScholesMertonBinomial10e.xlsm file
provided by the textbook’s authors. If you use the latter, include a screen shot of the
Excel spreadsheet in your answer.)
Obtain the value of this delta-hedge portfolio after it has been rebalanced. Compare
this value to the target value of the portfolio should its initial value be invested at
the risk-free rate. Explain the difference.
d.
There is another call option on IBM shares with an exercise price of $125 and the
same expiration date (March 18, 2016). Construct a delta- and gamma-hedge
portfolio on January 4, 2016 involving the sale of 1,000 of the 130-call option. Then
rebalance the portfolio at the end of the next day, when the share price goes down
to $135 per share. Again, use the implied volatility as the correct volatility for the
IBM shares. (You may calculate the deltas and gammas using the formula or the
BlackScholesMertonBinomial10e.xlsm file provided by the textbook’s authors. If you
use the latter, include a screen shot of the Excel spreadsheet in your answer.)
Obtain the value of this delta-and-gamma-hedged portfolio after it has been
rebalanced. Compare this value to the target value of the portfolio should its initial
value be invested at the risk-free rate. Explain the difference.
e.
Explain the difference between the delta-hedged portfolio value in part (c) and the
delta-and-gamma-hedged portfolio value in part (d).
Question 6. Protective Put
Suncor Energy Inc. (SU) shares are listed on the New York Stock Exchange. At 9:30 a.m. on
January 14, 2016, these shares sold for $21.85 per share. The volatility on the returns of
Suncor shares is approximately 24%. The following call and put option contracts were
available for the months of January, February, and March:
CALLS
Strike/Expiry
January 22, 2016
February 19, 2016
March 18, 2016
23
0.34
0.72
0.96
24
0.13
0.41
0.69
25
0.25
0.26
0.40
PUTS
Strike/Expiry
January 22, 2016
February 19,
2016
March 18, 2016
23
1.28
2.01
2.14
4
24
2.63
2.80
2.92
25
3.60
3.70
3.95
Each option contract involves 100 shares. The risk-free rates for these three expiration
dates are 0.6%, 1%, and 1.2%. All three rates are continuously compounded.
Given the information on Suncor shares and options above, construct a protective put using
the 23-put with February expiration. Hold the protective put position until expiration.
a.
Write out the payoff and profit function.
b.
Use a table to show the payoffs and profits when the put option expires in-themoney and out-of-the-money.
c.
Calculate the potential profits for this protective put, using share prices ranging from 0
to 26. Plot a graph of these potential profits, with share prices on the x-axis, and profits
on the y-axis. (Hint: It may be easier to do this in an Excel spreadsheet.)
d.
What is the breakeven share price at expiration for this protective put?
e.
What is the maximum profit and maximum loss on this protective put?
Question 7. Box Spread
Use the data on Suncor Inc. presented in Question 6 above to answer this question.
a.
Construct a box-spread using the March option contracts with exercise prices of 24
and 25.
b.
Construct a profitable riskless arbitrage opportunity using this box-spread, with the
requirement of $0 investment today. Calculate the NPV of the riskless profit.
5