Finance 6324
FINN 6210 / BPHD 8240
Spring Semester 2021
Problem Set 2: Forwards and Futures
Assignment Due: March 9 by 11:59 pm. You can work in a group with no more than two students.
Please hand in one copy per group and list the names of the students in the group. Email your write
up to me ( XXXXXXXXXX) as an email attachment. Send only Word or PDF documents.
1. (Lecture Note 1) Answer the following questions.
a. Describe the profit from the following portfolio: a short forward contract on an asset and a
long European call option on the asset with the same maturity as the forward contract and
a strike price that is equal to the forward price of the asset at the time the portfolio is set
up.
. Describe the profit from the following portfolio: a long forward contract on an asset and a
long European put option on the asset with the same maturity as the forward contract and
a strike price that is equal to the forward price of the asset at the time the portfolio is set
up.
2. (Lecture Note 2) The cu
ent spot price of a ba
el of oil, ?0, is $70.63. The per year
continuously compounded risk-free rate of interest, ?, is 3%, storage cost, ?, is 2%, and
convenience yield, ?, is 8%. The expected return of oil, ??, is 12%. Answer the following
questions for a futures contract with a maturity, ?, of 6 months.
a. What is the expected spot price on the maturity date of the contract?
. What is the no a
itrage futures price?
c. Is the futures price in part b in contango or normal backwardation?
d. Give your answer in part c, would a speculator go long or short in the future contract?
Explain your answer.
3. (Lecture Note 2) In the Spring of 1999, the U.S. Dollar-Deutschemark (DM) exchange rate
was $0.5405 per DM. The U.S. and German interest rates (annualized, continuously
compounded) were r = 6% and fr = 7.5%, respectively. Answer the following questions.
a. What was the no-a
itrage forward price of DM for a 3 month forward contract?
. Suppose the actual quoted price for a 3 month forward contract was $0.5632 per DM.
Explain whether or not there was an a
itrage opportunity. If one did exist, use an a
itrage
table to demonstrate how you could have profited. The a
itrage table should have the
following column titles: “Transaction”, “Payoff (now)”, and “Payoff (at T)”.
mailto: XXXXXXXXXX
2
c. Suppose the actual quoted price for a 3 month forward contract was $0.5102 per DM.
Explain whether or not there was an a
itrage opportunity. If one did exist, use an a
itrage
table to demonstrate how you could have profited. The a
itrage table should have the
following column titles: “Transaction”, “Payoff (now)”, and “Payoff (at T)”.
d. In March of 1999, suppose that one month earlier (i.e., in Fe
uary) a foreign cu
ency
trader entered into a 3-month long forward contract at the forward price that you calculated
in part a. The contract was for the delivery of 1 million Marks. The March spot Dollar-DM
exchange rate was $ XXXXXXXXXXWhat was the value in March of this long forward contract?
4. (Lecture Note 2) Consider a 9-month forward contract on Amazon.com Inc. (AMZN). The
cu
ent price of one share is $1,670, and the annual continuously compounded risk-free interest
ate is 6%. Answer the following questions.
a. Compute the no-a
itrage forward price of the stock for a 9-month contract.
. Suppose a small private investor can bo
ow money at 8% per year with quarterly
compounding and can lend money at 5.5% per year with semiannual compounding.
Suppose the actual quoted forward price for a 9-month contract is 1,850.25 per share of
AMZN. Explain whether there is an a
itrage opportunity. In your explanation, use an
a
itrage table with the following column titles: “Transaction”, “Payoff (now)”, and
“Payoff (9 months)”.
c. What would the bo
owing or lending rate have to be to eliminate the a
itrage opportunity
in part b of this question? Your answer should report the rate in the compounding interval
of the bo
owing (quarterly) or lending (semiannual) rate.
5. (Lecture Note 3) Fidelity Investments has hired you to help them develop enhanced equity
index funds using index futures contracts. Fidelity’s initial idea is to offer an S&P 500 index
fund that promises twice (200%) the S&P 500 index. Consider the following information.
The company plans to start the portfolio with $250 million invested to replicate the S&P 500
index. Of course, before using derivatives the portfolio will have a beta coefficient of 1.0.
(Assuming the beta is computed relative to the S&P 500 index.) For the computations below,
assume the cu
ent level of the S&P 500 index is 1,288 and the dividend yield of the index is
1.2% per year. The risk-free rate of interest is 5% per year. Both the dividend yield and the
isk-free rate are continuously compounded.
Answer the following questions.
a. Cu
ently the December S&P 500 stock index futures price is 1, XXXXXXXXXXAssuming a
emaining maturity of exactly four months, compute the no a
itrage December S&P 500
futures price.
3
. Given the no a
itrage index futures price that you calculated in a, is there an a
itrage
opportunity? If there is, use an a
itrage table to illustrate the transactions and the payoffs
now and in four months. The a
itrage table should have the following column titles:
“Transaction”, “Payoff (now)”, and “Payoff (at T)”.
c. What would the risk-free rate of interest have to be to eliminate the a
itrage opportunity?
d. How many December S&P 500 futures contracts would you have to establish a position in
to make the portfolio twice as volatile as the S&P 500 index? Should you be short or long?
Use the cu
ent December futures price of 1,310.50.
e. Suppose you establish the futures position in part d. If the S&P 500 index turns out to be
1,416.80 in three months, compute the value of the portfolio plus the index futures position.
Show that the combined position has a beta coefficient of about 2.0. Assume in three
months the December futures contract has exactly 1 month to maturity.
f. Suppose you establish the futures position in part d. If the S&P index turns out to be
XXXXXXXXXXin three months, compute the value of the portfolio plus index futures position.
Show that the combined position has a beta coefficient of about 2.0. Assume in three
months the December futures contract has exactly 1 month to maturity.
6. (Lecture Note 2) The accounting department is preparing the firm’s quarterly financial
statements. As part of that process, they would like you to value some forward contracts.
Answer the following questions.
a. Compute the value of a long forward contract on the Chinese Yuan with a remaining
maturity of 8 months. The cu
ent spot exchange rate is 0.15 U.S. dollars per Yuan, the
U.S. domestic risk-free rate of interest (with continuous compounding) is 3 percent per
year, and the Chinese risk-free rate of interest (with continuous compounding) is 4.25
percent per year. The delivery price in this previously-negotiated forward contract is 0.12
U.S. dollars per Yuan and the size of the contract is 25,000,000 Yuan.
. What is the value of a short forward contract on Chinese Yuan with exactly the same
contract terms as the long forward contract in part a of this question?
c. Compute the value of a long forward contract on rhodium with a remaining maturity of 9
months. The cu
ent spot price of rhodium is $2,500 per ounce and the risk-free rate of
interest (with continuous compounding) is 3.5% percent per year. The delivery price in the
previously-negotiated forward contract is $2,475 per ounce. Storage costs are $0.15 per
ounce every 3 months and are payable in advance. The size of the contract is 100,000
ounces.
d. What is the value of a short forward contract on rhodium with exactly the same contract
terms as the long contract in part c of this question?
4
7. (Lecture Note 2) Consider a 6-month forward contract on gold. The cu
ent spot price is $1,000
per ounce and the annual continuously compounded risk-free interest rate is 5%. Assume
storage costs and the convenience yield are zero. Answer the following questions.
a. Compute the no-a
itrage forward price of gold for a 6-month forward contract. Round
your answer to two places after the decimal place, e.g XXXXXXXXXXrounds to 1.02.
. Suppose you can bo
ow money at 8 percent per year with monthly compounding and can
lend money at 4.5 percent per year with monthly compounding. Compute the continuously
compounded bo
owing and lending rates. Round your answer to four places after the
decimal place, e.g., XXXXXXXXXXrounds to XXXXXXXXXXor 7.66%.
c. Suppose the actual quoted forward price for a 6-month contract is $1,027.50 per ounce of
gold. Explain whether there is an a
itrage opportunity. If one does exist, use an a
itrage
table to demonstrate how you can make a riskless a
itrage profit. The a
itrage table
should have the following column titles: “Transaction”, “Payoff (now)”, and “Payoff (6
months)”.
8. (Lecture Note 2) Suppose the spot price of winter wheat is $5.98 per bushel. Storage costs are
$0.15 per bushel every 3 months and are payable in advance. The interest rate is 5 percent per
year with continuous compounding. Assume that winter wheat is a pure consumption
commodity. Answer the following questions.
a. Suppose the actual quoted price for a nine-month forward contract is $6.55 per bushel.
Explain whether there is an a
itrage opportunity. If one does exist, use an a
itrage table
to demonstrate how you can make a riskless a
itrage profit. The a
itrage table should
have the following column titles: “Transaction”, “Payoff (now)”, and “Payoff (9 months)”.
. Suppose the actual quoted price for a nine-month forward contract is $6.80 per bushel.
Explain whether there is an a
itrage opportunity. If one does exist, use an a
itrage table
to