Question: Why are U.S. Treasury rates significantly

> If the volatility of a stock is 18% per annum, estimate the standard deviation of the percentage price change in (a) 1 day, (b) 1 week, and (c) 1 month.

> Prove the result in equation (11.11). (Hint: For the first part of the relationship, consider (a) a portfolio consisting of a European call plus an amount of cash equal to D þ K, and (b) a portfolio consisting of an American put option plus one share.)

> What do you think would happen if an exchange started trading a contract in which the quality of the underlying asset was incompletely specified?

> An exchange rate is currently 1.0 and the implied volatilities of 6-month European options with strike prices 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3 are 13%, 12%, 11%, 10%, 11%, 12%, 13%. The domestic and foreign risk-free rates are both 2.5%. Calculate the i

> The 3-month Eurodollar futures price for a contract maturing in 6 years is quoted as 95.20. The standard deviation of the change in the short-term interest rate in 1 year is 1.1%. Estimate the forward LIBOR interest rate for the period between 6.00 and 6

> Between October 30, 2018, and November 1, 2018, you have a choice between owning a U.S. government bond paying a 12% coupon and a U.S. corporate bond paying a 12% coupon. Consider carefully the day count conventions discussed in this chapter and decide w

> “The impact of DVA on earnings volatility is generally greater than that of CVA.” Explain this statement.

> Suppose that it is February 20 and a treasurer realizes that on July 17 the company will have to issue $5 million of commercial paper with a maturity of 180 days. If the paper were issued today, the company would realize $4,820,000. (In other words, the

> Suppose that the 300-day LIBOR zero rate is 4% and Eurodollar quotes for contracts maturing in 300, 398, and 489 days are 95.83, 95.62, and 95.48. Calculate 398-day and 489-day LIBOR zero rates. Assume no difference between forward and futures rates for

> Suppose that the 9-month LIBOR interest rate is 8% per annum and the 6-month LIBOR interest rate is 7.5% per annum (both with actual/365 and continuous compounding). Estimate the 3-month Eurodollar futures price quote for a contract maturing in 6 months.

> A trader is looking for arbitrage opportunities in the Treasury bond futures market. What complications are created by the fact that the party with a short position can choose to deliver any bond with a maturity between 15 and 25 years?

> It is July 30, 2018. The cheapest-to-deliver bond in a September 2018 Treasury bond futures contract is a 13% coupon bond, and delivery is expected to be made on September 30, 2018. Coupon payments on the bond are made on February 4 and August 4 each yea

> Suppose the current USD/euro exchange rate is 1.2000 dollar per euro. The six-month forward exchange rate is 1.1950. The six-month USD interest rate is 1% per annum continuously compounded. Estimate the six-month euro interest rate.

> A 1-year long forward contract on a non-dividend-paying stock is entered into when the stock price is $40 and the risk-free rate of interest is 5% per annum with continuous compounding. (a) What are the forward price and the initial value of the forward

> Is the futures price of a stock index greater than or less than the expected future value of the index? Explain your answer.

> Explain the difference between bilateral and central clearing for OTC derivatives.

> What trading position is created from a long strangle and a short straddle when both have the same time to maturity? Assume that the strike price in the straddle is halfway between the two strike prices of the strangle.

> Explain why a foreign currency can be treated as an asset providing a known yield.

> If the market considers that the default probability for a bank has increased, what happens to its DVA? What happens to the income it reports?

> Consider again the situation in Problem 19.24. Suppose that a second traded option with a delta of 0.1, a gamma of 0.5, and a vega of 0.6 is available. How could the portfolio be made delta, gamma, and vega neutral? Data from Problem 19.24

> Explain carefully the meaning of the terms convenience yield and cost of carry. What is the relationship between futures price, spot price, convenience yield, and cost of carry?

> Explain carefully why the futures price of gold can be calculated from its spot price and other observable variables whereas the futures price of copper cannot.

> What is meant by (a) an investment asset and (b) a consumption asset? Why is the distinction between investment and consumption assets important in the determination of forward and futures prices?

> Explain why the end-of-year bonus is sometimes referred to as ‘‘short-term compensation.’’

> A U.S. company is interested in using the futures contracts traded by the CME Group to hedge its Australian dollar exposure. Define r as the interest rate (all maturities) on the U.S. dollar and rf as the interest rate (all maturities) on the Australian

> The Value Line Index is designed to reflect changes in the value of a portfolio of over 1,600 equally weighted stocks. Prior to March 9, 1988, the change in the index from one day to the next was calculated as the geometric average of the changes in the

> Explain carefully what is meant by the expected price of a commodity on a particular future date. Suppose that the futures price for crude oil declines with the maturity of the contract at the rate of 2% per year. Assume that speculators tend to be short

> Show that equation (5.3) is true by considering an investment in the asset combined with a short position in a futures contract. Assume that all income from the asset is reinvested in the asset. Use an argument similar to that in footnotes 2 and 4 of thi

> A trader enters into a short forward contract on 100 million yen. The forward exchange rate is $0.0090 per yen. How much does the trader gain or lose if the exchange rate at the end of the contract is (a) $0.0084 per yen and (b) $0.0101 per yen?

> What is the difference between the forward price and the value of a forward contract?

> How were the risks in ABS CDOs misjudged by the market?

> Draw a diagram showing the variation of an investor’s profit and loss with the terminal stock price for a portfolio consisting of : (a) One share and a short position in one call option (b) Two shares and a short position in one call option (c) One share

> Show that the growth rate in an index futures price equals the excess return on the portfolio underlying the index over the risk-free rate. Assume that the risk-free interest rate and the dividend yield are constant.

> When a known future cash outflow in a foreign currency is hedged by a company using a forward contract, there is no foreign exchange risk. When it is hedged using futures contracts, the daily settlement process does leave the company exposed to some risk

> How is an ABS CDO created? What was the motivation to create ABS CDOs?

> When compounded annually an interest rate is 11%. What is the rate when expressed with (a) semiannual compounding, (b) quarterly compounding, (c) monthly compounding, (d) weekly compounding, and (e) daily compounding.

> Suppose that F1 and F2 are two futures contracts on the same commodity with times to maturity, t1 and t2, where t2 > t1. Prove that where r is the interest rate (assumed constant) and there are no storage costs. For the purposes of this problem, assu

> Estimate the difference between short-term interest rates in Japan and the United States on May 3, 2016, from the information in Table 5.4. Table 5.4 Futures quotes for a selection of CME Group contracts on foreign currencies on May 3, 2016. Open Hig

> Explain what happens when an investor shorts a certain share.

> What does duration tell you about the sensitivity of a bond portfolio to interest rates. What are the limitations of the duration measure?

> An investor receives $1,100 in one year in return for an investment of $1,000 now. Calculate the percentage return per annum with: (a) Annual compounding (b) Semiannual compounding (c) Monthly compounding (d) Continuous compounding.

> A U.S. company expects to have to pay 1 million Canadian dollars in 6 months. Explain how the exchange rate risk can be hedged using (a) a forward contract and (b) an option.

> How much is gained from exercising early at the lowest node at the 9-month point in Example 21.4?

> The cash prices of 6-month and 1-year Treasury bills are 94.0 and 89.0. A 1.5-year Treasury bond that will pay coupons of $4 every 6 months currently sells for $94.84. A 2-year Treasury bond that will pay coupons of $5 every 6 months currently sells for

> A 5-year bond with a yield of 7% (continuously compounded) pays an 8% coupon at the end of each year. (a) What is the bond’s price? (b) What is the bond’s duration? (c) Use the duration to calculate the effect on the bond’s price of a 0.2% decrease in it

> Explain the difference between a market-if-touched order and a stop order.

> The five-year swap rate when cash flows are exchanged semiannually is 4%. A company wants a swap where it receives payments at 4.2% per annum on a principal of $10 million. The OIS zero curve is flat at 3.6%. How much should a derivatives dealer charge t

> Explain why an FRA is equivalent to the exchange of a floating rate of interest for a fixed rate of interest.

> Why does a loan in the repo market involve very little credit risk?

> A diagonal spread is created by buying a call with strike price K2 and exercise date T2 and selling a call with strike price K1 and exercise date T1, where T2 > T1. Draw a diagram showing the profit from the spread at time T1 when (a) K2 > K1 and (b) K2

> ‘‘When the zero curve is upward-sloping, the zero rate for a particular maturity is greater than the par yield for that maturity. When the zero curve is downward-sloping the reverse is true.’’ Explain why this is so.

> Explain carefully why liquidity preference theory is consistent with the observation that the term structure of interest rates tends to be upward-sloping more often than it is downward-sloping.

> A 10-year 8% coupon bond currently sells for $90. A 10-year 4% coupon bond currently sells for $80. What is the 10-year zero rate? (Hint: Consider taking a long position in two of the 4% coupon bonds and a short position in one of the 8% coupon bonds.)

> What is the waterfall in a securitization?

> Consider a European call option on a non-dividend-paying stock where the stock price is $40, the strike price is $40, the risk-free rate is 4% per annum, the volatility is 30% per annum, and the time to maturity is 6 months. (a) Calculate u, d, and p for

> A corporate treasurer tells you that he has just negotiated a five-year loan at a competitive fixed rate of interest of 5.2%. The treasurer explains that he achieved the 5.2% rate by borrowing at six-month LIBOR plus 150 basis points and swapping LIBOR f

> The forward price of the Swiss franc for delivery in 45 days is quoted as 1.1000. The futures price for a contract that will be delivered in 45 days is 0.9000. Explain these two quotes. Which is more favorable for a trader wanting to sell Swiss francs?

> Suppose that the 6-month, 12-month, 18-month, and 24-month zero rates are 5%, 6%, 6.5%, and 7%, respectively. What is the 2-year par yield?

> A 3-year bond provides a coupon of 8% semiannually and has a cash price of 104. What is the bond’s yield?

> Give three reasons why the treasurer of a company might not hedge the company’s exposure to a particular risk.

> The author’s website (www-2.rotman.utoronto.ca/hull/data) contains daily closing prices for the crude oil futures contract and the gold futures contract. You are required to download the data for crude oil and answer the following: (a) Assuming that dai

> For the situation considered in Problem 13.5, what is the value of a 1-year European put option with a strike price of $100? Verify that the European call and European put prices, satisfy put–call parity. Data from Problem 13.5: A stock price is currentl

> Explain what is meant by a perfect hedge. Does a perfect hedge always lead to a better outcome than an imperfect hedge? Explain your answer.

> Suppose that the 1-year gold lease rate is 1.5% and the 1-year risk-free rate is 5.0%. Both rates are compounded annually. Use the discussion in Business Snapshot 3.1 to calculate the maximum 1-year gold forward price Goldman Sachs should quote to the go

> An airline executive has argued: ‘‘There is no point in our using oil futures. There is just as much chance that the price of oil in the future will be less than the futures price as there is that it will be greater than this price.’’ Discuss the executi

> A futures contract is used for hedging. Explain why the daily settlement of the contract can give rise to cash-flow problems.

> It is May and a trader writes a September call option with a strike price of $20. The stock price is $18 and the option price is $2. Describe the trader’s cash flows if the option is held until September and the stock price is $25 at that time.

> What would it mean to assert that the temperature at a certain place follows a Markov process? Do you think that temperatures do, in fact, follow a Markov process?

> 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

> Suppose that you write a put contract with a strike price of $40 and an expiration date in 3 months. The current stock price is $41 and the contract is on 100 shares. What have you committed yourself to? How much could you gain or lose?

> Suppose that a March call option to buy a share for $50 costs $2.50 and is held until March. Under what circumstances will the holder of the option make a profit? Under what circumstances will the option be exercised? Draw a diagram illustrating how the

> Explain why a futures contract can be used for either speculation or hedging.

> What is the cost of carry for: (a) a non-dividend-paying stock (b) a stock index (c) a commodity with storage costs (d) a foreign currency.

> It is sometimes argued that a forward exchange rate is an unbiased predictor of future exchange rates. Under what circumstances is this so?

> The spot price of silver is $25 per ounce. The storage costs are $0.24 per ounce per year payable quarterly in advance. Assuming that interest rates are 5% per annum for all maturities, calculate the futures price of silver for delivery in 9 months.

> The 2-month interest rates in Switzerland and the United States are, respectively, 1% and 2% per annum with continuous compounding. The spot price of the Swiss franc is $1.0500. The futures price for a contract deliverable in 2 months is also $1.0500. Wh

> Suppose that the risk-free interest rate is 6% per annum with continuous compounding and that the dividend yield on a stock index is 4% per annum. The index is standing at 400, and the futures price for a contract deliverable in four months is 405. What

> Assume that the risk-free interest rate is 4% per annum with continuous compounding and that the dividend yield on a stock index varies throughout the year. In February, May, August, and November, dividends are paid at a rate of 5% per annum. In other mo

> In an interest rate swap, a financial institution has agreed to pay 3.6% per annum and to receive three-month LIBOR in return on a notional principal of $100 million with payments being exchanged every three months. The swap has a remaining life of 14 mo

> Section 11.1 gives an example of a situation where the value of a European call option decreases as the time to maturity is increased. Give an example of a situation where the same thing happens for a European put option.

> The risk-free rate of interest is 7% per annum with continuous compounding, and the dividend yield on a stock index is 3.2% per annum. The current value of the index is 150. What is the 6-month futures price?

> A stock index currently stands at 350. The risk-free interest rate is 4% per annum (with continuous compounding) and the dividend yield on the index is 3% per annum. What should the futures price for a 4-month contract be?

> Suppose that you enter into a 6-month forward contract on a non-dividend-paying stock when the stock price is $30 and the risk-free interest rate (with continuous compounding) is 5% per annum. What is the forward price?

> Explain the difference between value at risk and expected shortfall.

> Explain how a forward contract to sell foreign currency is mapped into a portfolio of zero-coupon bonds with standard maturities for the purposes of a VaR calculation.

> Suppose that the daily change in the value of a portfolio is, to a good approximation, linearly dependent on two factors, calculated from a principal components analysis. The delta of a portfolio with respect to the first factor is 6 and the delta with r

> Suppose you know that the gamma of the portfolio in the previous question is 16.2. How does this change your estimate of the relationship between the change in the portfolio value and the percentage change in the exchange rate?

> What volatility smile is likely to be observed for 6-month options when the volatility is uncertain and positively correlated to the stock price?

> A stock price is currently $20. Tomorrow, news is expected to be announced that will either increase the price by $5 or decrease the price by $5. What are the problems in using Black–Scholes–Merton to value 1-month options on the stock?

> A European call and put option have the same strike price and time to maturity. The call has an implied volatility of 30% and the put has an implied volatility of 25%. What trades would you do?

> A trader enters into a short cotton futures contract when the futures price is 50 cents per pound. The contract is for the delivery of 50,000 pounds. How much does the trader gain or lose if the cotton price at the end of the contract is (a) 48.20 cents

> Data for a number of foreign currencies are provided on the author’s website: http://www-2.rotman.utoronto.ca/_hull/data Choose a currency and use the data to produce a table similar to Table 20.1. Table 20.1 Percentage of days wh

> What volatility smile is likely to be caused by jumps in the underlying asset price? Is the pattern likely to be more pronounced for a 2-year option than for a 3-month option?

> What volatility smile is observed for equities?

> What volatility smile is likely to be observed when: (a) Both tails of the stock price distribution are less heavy than those of the lognormal distribution? (b) The right tail is heavier, and the left tail is less heavy, than that of a lognormal distribu

> Calculate the delta of an at-the-money six-month European call option on a non-dividend- paying stock when the risk-free interest rate is 10% per annum and the stock price volatility is 25% per annum.

> Suppose you buy a put option contract on October gold futures with a strike price of $1,400 per ounce. Each contract is for the delivery of 100 ounces. What happens if you exercise when the October futures price is $1,380?

> How does the put–call parity formula for a futures option differ from put–call parity for an option on a non-dividend-paying stock?

> A futures price is currently 50. At the end of six months it will be either 56 or 46. The risk-free interest rate is 6% per annum. What is the value of a six-month European call option on the futures with a strike price of 50?

> ‘‘A futures price is like a stock paying a dividend yield.’’ What is the dividend yield?