UNIVERSITY OF TORONTO AT SCARBOROUGH
DEPARTMENT OF MANAGEMENT
MGTC71: Introduction to Derivatives Markets
Problem Set 1 (Supplemental Questions)
1. Explain the similarities and differences between forwards and futures contract on the same asset. You may
want to consider (but not limited to) the following points for the differences:
a) How trade is conducted
c) Counter-party risk
2. A trader enters into a short futures contract to sell 5,000 bushels of wheat for 200 cents per bushel. The
initial margin is $3,000 and the maintenance margin is $2,000. What price change would lead to a margin
3. The fictional country of Heaven has announced a floating exchange rate policy against a basket of other
currencies, including the dollar ($), euro () and yen () with weights of one-half for the dollar and one-
quarter each for the euro and yen (so that 1H = 0.5$ + 0.25 + 0.25). Suppose that there are spot and
futures markets for the Heaven currency against the other big three. A US company has just signed a
contract to sell equipments to this country, to be paid on the first of January 2012. Show how this company
could hedge this contract.
4. A Swiss company purchased a product from a US firm. The delivery date is in March (40 days from now),
and the purchase price of the machine tools is SFR 5million. The US firm is concerned about exchange rate
movements between now and the delivery date and considers hedging its exposure. The futures price for
March is $0.7021 USD/SFR. One futures contract is for SFR 125,000.
a) Construct the optimal hedge using the futures contract described above. How many contracts does the US
firm have to buy/sell in order to hedge the exchange rate risk?
b) Demonstrate that this hedge is perfect by looking at scenarios where the SFR is $0.65, $0.70 and $0.75.
5. You have a $100 million portfolio of U.S. stocks and you want to hedge your portfolio using 1-year futures
on the U.S. market stock index (S&P 500). Today, the S&P 500 index is 1100 points with a zero dividend
yield, and the annual interest rate is 6.1837% [6%, C.C.]. In this case, one futures contract has a value of
$250 1100 = $275,000.
a) What should be the 1-year S&P 500 futures price in order to preclude arbitrage opportunities? [Lecture-4]
b) Which position should you take in the 1-year S&P 500 futures if the beta of the portfolio relative to the
S&P 500 is 1.4.
c) Assume that the beta of your portfolio is 1.4 and that you apply the hedging strategy proposed in (b).
Compute the value of the hedged portfolio in one year if the S&P 500 is 1000, 1100, or 1200. In the three
cases, compare the return of the hedged portfolio to the risk-free interest rate. Briefly explain your result.
6. A company needs to buy 1 million gallons of heating oil in June 2012. The current futures price for this
expiration date is $0.8190 per gallon, while the current spot price is $0.9602. Each futures contract is for
42,000 gallons of oil.
1 a) Suppose that the correlation between changes in futures and spot prices over the life of the hedge is 0.98
and that both have equal variances. How many futures contracts should the company trade to set up a
hedge? Should it use a long or a short hedge?
b) Now suppose that, in fact, in June 2012 (6 months from now) the spot price of heating oil is $0.7800 per
gallon. How much did the company profit/lose on its futures position
c) (Lecture 4) If the storage cost is 1% for oil and the current interest rate is 6% per annum, what is the
convenience yield on oil between now and June 2012? What does it measure?
7. You are the owner of $100m worth of a 10yr bond with that pays an annual coupon of 8%. The yield curve
is currently flat at 5% annual rate. You wish to hedge your interest rate exposure, but the only instrument
available to you is the 6-month future on a 2-year treasury discount bond (face value of $100,000).
a) Find the continuous compounded equivalent of the annual rate of 5%.
b) Find the two bonds price and duration.
c) How will you hedge your exposure using the futures contract? Solve for the number of futures contracts
you wish to buy or sell. (Assume the future price of the treasury bond is 100,000*1.05 /1.05 = 2
d) Suppose the yield curve does not change in 3 months. Are you still perfectly hedged now? Illustrate that
by showing that you are not protected from a parallel shift in the yield curve now.
e) So what you need to do to protect yourself from future move of interest rates?
8. Consider a five-year bond paying $100 after 1, 2, 3, 4, and 5 years along with a principal payment of $5000
after five years. The discount rate is 2.5% (continuously compounded, constant over the entire 5 years).
a) What is the value of the bond? What is its duration? If the discount falls to 2.4%, how much does the value
of the bond change?
b) Suppose there were a four-year bond offering the same structure of payments ($100 per year until the
principal is repaid in 4 year) what is its value at the two interest rates and its duration?
c) Suppose you own the bond in (b). Design a strategy to hedge your position using the bond in (a).
9. Suppose a one-year-long forward contract on a non-dividend-paying stock is entered into when the stock
price is $50 and the risk-free interest rate is 5% per annum with continuous compounding.
a) What are the forward price and the initial value of the forward contract?
b) Six months after the signing of the forward contract, the price of the stock rises to $55 and the risk-free
interest rate is still 5%. What is the new market forward price for the same contract (maturing in 6 months)?
What is the value of the forward contract signed 6 months ago?
10. Suppose you can borrow or lend LIBOR and are given following rates:
Current Dollar/Mark Exchange Rate 1.3890 DM/$
90 day Forward Dollar/Mark Rate 1.3830 DM/$
90 day LIBOR-equivalent Deutschemark Rate 4.60%
180 day U.S. $ LIBOR 6.70%
All interest rates are quoted in annualized, continuously-compounded form, and are the same for
borrowing or lending.
a) What must the 90 day LIBOR rate be for there to be no arbitrage?
2b) Suppose that the 90 day forward rate for 90 day LIBOR is 6.80%. Is there an arbitrage opportunity? Why?
If so, describe exactly how what transactions you would undertake at these prices to lock in the arbitrage
11. A Canadian company that is planning to buy some equipment from a German manufacturer in 90 days. The
cost of this machinery is 20 million Euros. You have been asked to analyze the consequences of entering
into a forward contract to reduce the companys exposure to foreign exchange risk. The current quotes on
Euros are: Spot exchange rate (CA$/Euro) = CA$1.38 (you have to pay CA$1.38 to get 1 Euro) and 90-day
forward rate (CA$/Euro) = $1.48.
a) Based on the information above, are interest rates higher in Canada (r) or in Europe (r )? Shof the
b) Skip ahead 90 days in time and examine what might happen if the company does not enter into the forward
contract. Suppose first that the spot exchange in 90 days (S ) iT CA$1.35, and then suppose that it is
CA$1.65. What will be the cost in CA$ in 90 days for the equipment in both cases?
c) Briefly explain which forward strategy may be appropriate for the company, which positions should be
taken and what will be the total cost in CA$ in 90 days.
a) Assume now that you enter into the strategy proposed in (c) and that after 60 days the management of the
company decides to buy immediately the equipment from the German manufacturer. If at this time the spot
exchange rate (CA$/Euro) = $1.58, the 30-day forward rate (CA$/Euro) = $1.66, and the one-month
Canadian interest rate is 5%, what is the effective total cost in CA$ for the equipment?
12. You identify two futures on individual stock that you think are mispriced:
1-year futures on stock X: F X = $34, Spot price of stock A: S X = $30