Question: In an interest rate swap, a financial

> 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?

> Why are U.S. Treasury rates significantly lower than other rates that are close to risk-free?

> 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

> 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?

> Calculate the price of a three-month European call option on the spot value of silver. The three-month futures price is $12, the strike price is $13, the risk-free rate is 4% and the volatility of the price of silver is 25%.

> Trader A enters into a forward contract to buy an asset for $1,000 in one year. Trader B buys a call option to buy the asset for $1,000 in one year. The cost of the option is $100. What is the difference between the positions of the traders? Show the pro

> Why are options on bond futures more actively traded than options on bonds?

> A trader sells a strangle by selling a 6-month European call option with a strike price of $50 for $3 and selling a 6-month European put option with a strike price of $40 for $4. For what range of prices of the underlying asset in 6 months does the trade

> Suppose that a futures price is currently 30. The risk-free interest rate is 5% per annum. A three-month American futures call option with a strike price of 28 is worth 4. Calculate bounds for the price of a three-month American futures put option with a

> ‘‘The price of an at-the-money European futures call option always equals the price of a similar at-the-money European futures put option.’’ Explain why this statement is true.

> Consider a four-month futures put option with a strike price of 50 when the risk-free interest rate is 10% per annum. The current futures price is 47. What is a lower bound for the value of the futures option if it is (a) European and (b) American?

> Consider a two-month futures call option with a strike price of 40 when the risk-free interest rate is 10% per annum. The current futures price is 47. What is a lower bound for the value of the futures option if it is (a) European and (b) American?

> A foreign currency is currently worth $1.50. The domestic and foreign risk-free interest rates are 5% and 9%, respectively. Calculate a lower bound for the value of a six-month call option on the currency with a strike price of $1.40 if it is (a) Europea

> Calculate the value of an eight-month European put option on a currency with a strike price of 0.50. The current exchange rate is 0.52, the volatility of the exchange rate is 12%, the domestic risk-free interest rate is 4% per annum, and the foreign risk

> Calculate the value of a three-month at-the-money European call option on a stock index when the index is at 250, the risk-free interest rate is 10% per annum, the volatility of the index is 18% per annum, and the dividend yield on the index is 3% per an

> A currency is currently worth $0.80 and has a volatility of 12%. The domestic and foreign risk-free interest rates are 6% and 8%, respectively. Use a two-step binomial tree to value (a) a European four-month call option with a strike price of 0.79 and (b

> A fund manager has a portfolio worth $50 million with a beta of 0.87. The manager is concerned about the performance of the market over the next 2 months and plans to use 3-month futures contracts on a well-diversified index to hedge its risk. The curren

> A stock index is currently 300, the dividend yield on the index is 3% per annum, and the risk-free interest rate is 8% per annum. What is a lower bound for the price of a six month European call option on the index when the strike price is 290?

> ‘‘Once we know how to value options on a stock paying a dividend yield, we know how to value options on stock indices and currencies.’’ Explain this statement.

> Use the software DerivaGem to verify that Figures 11.1 and 11.2 are correct. Figure 11.1 option prices when So = 50, K = 50, r= 5%, a = 30%, and T = 1. Effect of changes in stock price, strike price, and expiration date on 4 Put option Call option pr

> A total return index tracks the return, including dividends, on a certain portfolio. Explain how you would value (a) forward contracts and (b) European options on the index.

> An index currently stands at 1,500. European call and put options with a strike price of 1,400 and time to maturity of six months have market prices of 154.00 and 34.25, respectively. The six-month risk-free rate is 5%. What is the implied dividend yield

> An index currently stands at 696 and has a volatility of 30% per annum. The risk-free rate of interest is 7% per annum and the index provides a dividend yield of 4% per annum. Calculate the value of a three-month European put with an exercise price of 70

> Consider a stock index currently standing at 250. The dividend yield on the index is 4% per annum, and the risk-free rate is 6% per annum. A three-month European call option on the index with a strike price of 245 is currently worth $10. What is the valu

> Explain why employee stock options on a non-dividend-paying stock are frequently exercised before the end of their lives, whereas an exchange-traded call option on such a stock is never exercised early.

> A company has granted 500,000 options to its executives. The stock price and strike price are both $40. The options last for 12 years and vest after 4 years. The company decides to value the options using an expected life of 5 years and a volatility of 3

> In a Dutch auction of 10,000 options, bids are as follows: A bids $30 for 3,000; B bids $33 for 2,500; C bids $29 for 5,000; D bids $40 for 1,000; E bids $22 for 8,000; and F bids $35 for 6,000. What is the result of the auction? Who buys how many at wha

> The one-year forward price of the Mexican peso is $0.0750 per MXN. The U.S. risk-free rate is 1.25% and the Mexican risk-free rate is 4.5%. The exchange rate volatility is 13%. What are the values of one-year European and American put options with a stri

> The notes accompanying a company’s financial statements say: ‘‘Our executive stock options last 10 years and vest after 4 years. We valued the options granted this year using the Black–Scholes–Merton model with an expected life of 5 years and a volatilit

> Using the notation in this chapter, prove that a 95% confidence interval for ST is between /

> A stock price is currently $40. Assume that the expected return from the stock is 15% and that its volatility is 25%. What is the probability distribution for the rate of return (with continuous compounding) earned over a 2-year period?

> Does a forward contract on a stock index have the same delta as the corresponding futures contract? Explain your answer.

> What difference does it make to your calculations in Problem 15.4 if a dividend of $1.50 is expected in 2 months?

> Calculate the price of a 3-month European put option on a non-dividend-paying stock with a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% per annum, and the volatility is 30% per annum.