Why do you think the increase in house prices during the 2000 to 2007 period is referred to as a bubble?
> Suppose the firm’s labor demand curve is given by w = 20 - 0.01E where w is the hourly wage and E is the level of employment. Suppose also that the union’s utility function is given by U = w × E It is easy to show that the marginal utility of the wage fo
> Use DerivaGem to check that equation (19.4) is satisfied for the option considered in Section 19.1. (Note: DerivaGem produces a value of theta ‘‘per calendar day.’’ The theta in equation (19.4) is ‘‘per year.’’)
> Suppose the risk-free rates are as in Problem 4.30. What is the value of an FRA where the holder pays LIBOR and receives 7% (semiannually compounded) for a six-month period beginning in 18 months? The current forward rate for this period is 6% (semiannua
> A trader has a put option contract to sell 100 shares of a stock for a strike price of $60. What is the effect on the terms of the contract of (a) A $2 dividend being declared (b) A $2 dividend being paid (c) A 5-for-2 stock split (d) A 5% stock dividend
> Explain why collateral requirements will increase in the OTC market as a result of new regulations introduced since the 2008 credit crisis.
> A bank offers a corporate client a choice between borrowing cash at 11% per annum and borrowing gold at 2% per annum. (If gold is borrowed, interest must be repaid in gold. Thus, 100 ounces borrowed today would require 102 ounces to be repaid in 1 year.)
> A stock price is $29. A trader buys one call option contract on the stock with a strike price of $30 and sells a call option contract on the stock with a strike price of $32.50. The market prices of the options are $2.75 and $1.50, respectively. The opti
> Call options on a stock are available with strike prices of $15, $1712 , and $20, and expiration dates in 3 months. Their prices are $4, $2, and $12 , respectively. Explain how the options can be used to create a butterfly spread. Construct a table showi
> A trader sells a put option with a strike price of $40 for $5. What is the trader’s maximum gain and maximum loss? How does your answer change if it is a call option?
> What does a stop order to sell at $2 mean? When might it be used? What does a limit order to sell at $2 mean? When might it be used?
> Explain two ways in which a bear spread can be created.
> Use a three-step tree to value an American futures put option when the futures price is 50, the life of the option is 9 months, the strike price is 50, the risk-free rate is 3%, and the volatility is 25%.
> An index provides a dividend yield of 1% and has a volatility of 20%. The risk-free interest rate is 4%. How long does a principal-protected note, created as in Example 12.1, have to last for it to be profitable for the bank issuing it? Use DerivaGem. E
> A foreign currency is currently worth $0.64. A 1-year butterfly spread is set up using European call options with strike prices of $0.60, $0.65, and $0.70. The risk-free interest rates in the United States and the foreign country are 5% and 4% respective
> What is the result if the strike price of the put is higher than the strike price of the call in a strangle?
> A 6-month American call option on a stock is expected to pay dividends of $1 per share at the end of the second month and the fifth month. The current stock price is $30, the exercise price is $34, the risk-free interest rate is 10% per annum, and the vo
> Company X wishes to borrow U.S. dollars at a fixed rate of interest. Company Y wishes to borrow Japanese yen at a fixed rate of interest. The amounts required by the two companies are roughly the same at the current exchange rate. The companies have been
> A stock price is currently $100. Over each of the next two 6-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a 1-year European call option with a s
> What trading strategy creates a reverse calendar spread?
> On July 1, 2017, a company enters into a forward contract to buy 10 million Japanese yen on January 1, 2018. On September 1, 2017, it enters into a forward contract to sell 10 million Japanese yen on January 1, 2018. Describe the payoff from this strateg
> Suppose that in Table 3.5 the company decides to use a hedge ratio of 1.5. How does the decision affect the way the hedge is implemented and the result? Table 3.5 Data for the example on rolling oil hedge forward. Date Apr. 2017 Sept. 2017 Feb. 2018
> A bank enters into an interest rate swap with a nonfinancial counterparty using bilaterally clearing where it is paying a fixed rate of 3% and receiving LIBOR. No collateral is posted and no other transactions are outstanding between the bank and the cou
> ‘‘A box spread comprises four options. Two can be combined to create a long forward position and two to create a short forward position.’’ Explain this statement.
> How can a forward contract on a stock with a particular delivery price and delivery date be created from options?
> An investor believes that there will be a big jump in a stock price, but is uncertain as to the direction. Identify six different strategies the investor can follow and explain the differences among them.
> A trader buys a call option with a strike price of $30 for $3. Does the trader ever exercise the option and lose money on the trade? Explain your answer.
> Use put–call parity to show that the cost of a butterfly spread created from European puts is identical to the cost of a butterfly spread created from European calls.
> Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. How can the options be used to create (a) a bull spread and (b) a bear spread? Construct a table that shows the profit and payoff for both spreads.
> The futures price of an asset is currently 78 and the risk-free rate is 3%. A six-month put on the futures with a strike price of 80 is currently worth 6.5. What is the value of a sixmonth call on the futures with a strike price of 80 if both the put and
> A stock price is currently $50. It is known that at the end of 2 months it will be either $53 or $48. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a 2-month European call option with a strike price of $49
> When is it appropriate for an investor to purchase a butterfly spread?
> A corn farmer argues ‘‘I do not use futures contracts for hedging. My real risk is not the price of corn. It is that my whole crop gets wiped out by the weather.’’ Discuss this viewpoint. Should the farmer estimate his or her expected production of corn
> Explain why the arguments leading to put–call parity for European options cannot be used to give a similar result for American options.
> What are the numbers in Table 8.1 for a loss rate of (a) 12% and (b) 15%? Table 8.1 Estimated losses to tranches of ABS CDO in Figure 8.3. Losses on Losses to Losses to Losses to Losses to underlying mezzanine tranche equity tranche mezzanine tranche
> A $100 million interest rate swap has a remaining life of 10 months. Under the terms of the swap, six-month LIBOR is exchanged for 4% per annum (compounded semiannually). Six month LIBOR forward rates for all maturities are 3% (with semiannual compoundin
> Why did mortgage lenders frequently not check on information provided by potential borrowers on mortgage application forms during the 2000 to 2007 period?
> Why is an American call option on a dividend-paying stock always worth at least as much as its intrinsic value. Is the same true of a European call option? Explain your answer.
> ‘‘The early exercise of an American put is a trade-off between the time value of money and the insurance value of a put.’’ Explain this statement.
> Give two reasons why the early exercise of an American call option on a non-dividend paying stock is not optimal. The first reason should involve the time value of money. The second should apply even if interest rates are zero.
> What is the impact (if any) of negative interest rates on: (a) The put–call parity result for European options (b) The result that American call options on non-dividend-paying stocks should never be exercised early (c) The result that American put option
> A 5-year bond provides a coupon of 5% per annum payable semiannually. Its price is 104. What is the bond’s yield? You may find Excel’s Solver useful.
> Use the put–call parity relationship to derive, for a non-dividend-paying stock, the relationship between: (a) The delta of a European call and the delta of a European put (b) The gamma of a European call and the gamma of a European put (c) The vega of a
> ‘‘If there is no basis risk, the minimum variance hedge ratio is always 1.0.’’ Is this statement true? Explain your answer.
> In Problem 13.19, suppose a trader sells 10,000 European call options and the two-step tree describes the behavior of the stock. How many shares of the stock are needed to hedge the 6-month European call for the first and second 3-month period? For the s
> Consider a 5-year call option on a non-dividend-paying stock granted to employees. The option can be exercised at any time after the end of the first year. Unlike a regular exchange-traded call option, the employee stock option cannot be sold. What is th
> Explain the arbitrage opportunities in Problem 11.14 if the European put price is $3. Data from Problem 11.14: The price of a European call that expires in six months and has a strike price of $30 is $2. The underlying stock price is $29, and a dividend
> Describe the terminal value of the following portfolio: a newly entered-into long forward contract on an asset and a long position in a European put option on the asset with the same maturity as the forward contract and a strike price that is equal to th
> Give an intuitive explanation of why the early exercise of an American put becomes more attractive as the risk-free rate increases and volatility decreases.
> A 1-month European put option on a non-dividend-paying stock is currently selling for $2:50. The stock price is $47, the strike price is $50, and the risk-free interest rate is 6% per annum. What opportunities are there for an arbitrageur?
> Portfolio A consists of a 1-year zero-coupon bond with a face value of $2,000 and a 10-year zero-coupon bond with a face value of $6,000. Portfolio B consists of a 5.95-year zero-coupon bond with a face value of $5,000. The current yield on all bonds is
> Six-month LIBOR is 5%. LIBOR forward rates for the 6- to 12-month period and for the 12- to 18-month period are 5.5%. Swap rates for 2- and 3-year semiannual pay swaps are 5.4% and 5.6%, respectively. Estimate the LIBOR forward rates for for 18 months to
> A futures price is currently 40. It is known that at the end of three months the price will be either 35 or 45. What is the value of a three-month European call option on the futures with a strike price of 42 if the risk-free interest rate is 7% per annu
> Suppose that in Example 3.2 of Section 3.3 the company decides to use a hedge ratio of 0.8. How does the decision affect the way in which the hedge is implemented and the result?
> A 4-month European call option on a dividend-paying stock is currently selling for $5. The stock price is $64, the strike price is $60, and a dividend of $0.80 is expected in 1 month. The risk-free interest rate is 12% per annum for all maturities. What
> ‘‘When a futures contract is traded on the floor of the exchange, it may be the case that the open interest increases by one, stays the same, or decreases by one.’’ Explain this statement.
> Suppose that a European call option to buy a share for $100.00 costs $5.00 and is held until maturity. Under what circumstances will the holder of the option make a profit? Under what circumstances will the option be exercised? Draw a diagram illustratin
> A corporate treasurer is designing a hedging program involving foreign currency options. What are the pros and cons of using (a) NASDAQ OMX and (b) the over-the counter market for trading?
> ‘‘Employee stock options issued by a company are different from regular exchangetraded call options on the company’s stock because they can affect the capital structure of the company.’’ Explain this statement.
> Suppose that a European put option to sell a share for $60 costs $8 and is held until maturity. Under what circumstances will the seller of the option (the party with the short position) make a profit? Under what circumstances will the option be exercise
> Explain why margin accounts are required when clients write options but not when they buy options.
> Suppose that USD/sterling spot and forward exchange rates are as follows: What opportunities are open to an arbitrageur in the following situations? (a) A 180-day European call option to buy £1 for $1.52 costs 2 cents. (b) A 90-day European
> A U.S. investor writes five naked call option contracts. The option price is $3.50, the strike price is $60.00, and the stock price is $57.00. What is the initial margin requirement?
> What is a mezzanine tranche?
> An index is 1,200. The three-month risk-free rate is 3% per annum and the dividend yield over the next three months is 1.2% per annum. The six-month risk-free rate is 3.5% per annum and the dividend yield over the next six months is 1% per annum. Estimat
> Imagine you are the treasurer of a Japanese company exporting electronic equipment to the United States. Discuss how you would design a foreign exchange hedging strategy and the arguments you would use to sell the strategy to your fellow executives.
> What was the role of GNMA (Ginnie Mae) in the mortgage-backed securities market of the 1970s?
> Under what circumstances are (a) a short hedge and (b) a long hedge appropriate?
> Explain carefully the difference between selling a call option and buying a put option.
> Add rows in Table 8.1 corresponding to losses on the underlying assets of (a) 2%, (b) 6%, (c) 14%, and (d) 18%. Table 8.1 Estimated losses to tranches of ABS CDO in Figure 8.3. Losses on Losses to Losses to Losses to Losses to underlying mezzanine tr
> What differences exist in the way prices are quoted in the foreign exchange futures market, the foreign exchange spot market, and the foreign exchange forward market?
> What is the difference between the operation of the margin accounts administered by a clearing house and those administered by a broker?
> A company has issued a 3-year convertible bond that has a face value of $25 and can be exchanged for two of the company’s shares at any time. The company can call the issue, forcing conversion, when the share price is greater than or equal to $18. Assumi
> Suppose that in September 2018 a company takes a long position in a contract on May 2019 crude oil futures. It closes out its position in March 2019. The futures price (per barrel) is $48.30 when it enters into the contract, $50.50 when it closes out its
> Explain how CCPs work. What are the advantages to the financial system of requiring CCPs to be used for all standardized derivatives transactions between financial institutions?
> What is a subprime mortgage?
> Data for a number of stock indices are provided on the author’s Web site: http://www-2.rotman.utoronto.ca/~hull/data Choose an index and test whether a three standard deviation down movement happens more often than a three standard deviation up movement
> It is July 2017. A mining company has just discovered a small deposit of gold. It will take 6 months to construct the mine. The gold will then be extracted on a more or less continuous basis for 1 year. Futures contracts on gold are available with delive
> What is the price of a European put option on a non-dividend-paying stock when the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the volatility is 35% per annum, and the time to maturity is 6 months?
> Construct a table showing the payoff from a bull spread when puts with strike prices K1 and K2, with K2 > K1, are used.
> A trader owns a commodity that provides no income and has no storage costs as part of a long-term investment portfolio. The trader can buy the commodity for $1,250 per ounce and sell it for $1,249 per ounce. The trader can borrow funds at 6% per year and
> What is the price of a European call option on a non-dividend-paying stock when the stock price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the volatility is 30% per annum, and the time to maturity is 3 months?
> What does the Black–Scholes–Merton stock option pricing model assume about the probability distribution of the stock price in one year? What does it assume about the probability distribution of the continuously compounded rate of return on the stock duri
> Explain why the AAA-rated tranche of an ABS CDO is more risky than the AAA-rated tranche of an ABS.
> Suppose that G is a function of a stock price S and time. Suppose that and are the volatilities of S and G. Show that, when the expected return of S increases by , the growth rate of G increases by , where is a constant.
> A trader creates a bear spread by selling a 6-month put option with a $25 strike price for $2.15 and buying a 6-month put option with a $29 strike price for $4.75. What is the initial investment? What is the total payoff (excluding the initial investment
> Suppose that x is the yield to maturity with continuous compounding on a zero-coupon bond that pays off $1 at time T. Assume that x follows the process where a, , and s are positive constants and dz is a Wiener process. What is the process followed by
> Explain why the market maker’s bid–offer spread represents a real cost to options investors.
> Suppose that a stock price S follows geometric Brownian motion with expected return µ and volatility σ: What is the process followed by the variable Sn? Show that Sn also follows geometric Brownian motion.
> A cattle farmer expects to have 120,000 pounds of live cattle to sell in 3 months. The live cattle futures contract traded by the CME Group is for the delivery of 40,000 pounds of cattle. How can the farmer use the contract for hedging? From the farmer’s
> A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle. For what range of stock prices would the straddle lead to a loss?
> Consider the situation in which stock price movements during the life of a European option are governed by a two-step binomial tree. Explain why it is not possible to set up a position in the stock and the option that remains riskless for the whole of th
> What are the formulas for u and d in terms of volatility?
> A company that is uncertain about the exact date when it will pay or receive a foreign currency may try to negotiate with its bank a forward contract that specifies a period during which delivery can be made. The company wants to reserve the right to cho
> What is meant by the ‘‘delta’’ of a stock option?
> It is July 16. A company has a portfolio of stocks worth $100 million. The beta of the portfolio is 1.2. The company would like to use the December futures contract on a stock index to change the beta of the portfolio to 0.5 during the period July 16 to
> Explain why an American option is always worth at least as much as its intrinsic value.
> ‘‘Speculation in futures markets is pure gambling. It is not in the public interest to allow speculators to trade on a futures exchange.’’ Discuss this viewpoint.
> Explain why an American option is always worth at least as much as a European option on the same asset with the same strike price and exercise date.
> Suppose that a bank buys an option from a client. The option is uncollateralized and there are no other transactions outstanding with the client. The expected values of the option at the midpoint of years 1, 2, and 3 are 6, 5, and 4. The probability of t
> Calculate u, d, and p when a binomial tree is constructed to value an option on a foreign currency. The tree step size is 1 month, the domestic interest rate is 5% per annum, the foreign interest rate is 8% per annum, and the volatility is 12% per annum.
