Question: Prove the result in equation (11.11). (

> A stock price is currently $80. It is known that at the end of 4 months it will be either $75 or $85. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a 4-month European put option with a strike price of $80?

> A stock price is currently $40. It is known that at the end of 1 month it will be either $42 or $38. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a 1-month European call option with a strike price of $39?

> A currency swap has a remaining life of 15 months. It involves exchanging interest at 10% on £20 million for interest at 6% on $30 million once a year. The term structure of riskfree interest rates in the United Kingdom is flat at 7% and the term structu

> Explain how an aggressive bear spread can be created using put options.

> Use put–call parity to relate the initial investment for a bull spread created using calls to the initial investment for a bull spread created using puts.

> Explain what CVA and DVA measure.

> A trader buys a call option with a strike price of $45 and a put option with a strike price of $40. Both options have the same maturity. The call costs $3 and the put costs $4. Draw a diagram showing the variation of the trader’s profit with the asset pr

> What is the difference between a strangle and a straddle?

> What is meant by a protective put? What position in call options is equivalent to a protective put?

> The table below gives Treasury zero rates and cash flows on a Treasury bond. Zero rates are continuously compounded. (a) What is the bond’s theoretical price? (b) What is the bond’s yield assuming it sells for its theo

> What is a lower bound for the price of a 6-month call option on a non-dividend-paying stock when the stock price is $80, the strike price is $75, and the risk-free interest rate is 10% per annum?

> The price of a non-dividend-paying stock is $19 and the price of a 3-month European call option on the stock with a strike price of $20 is $1. The risk-free rate is 4% per annum. What is the price of a 3-month European put option with a strike price of $

> What is a lower bound for the price of a 1-month European put option on a non-dividend- paying stock when the stock price is $12, the strike price is $15, and the risk-free interest rate is 6% per annum?

> What is a lower bound for the price of a 4-month call option on a non-dividend-paying stock when the stock price is $28, the strike price is $25, and the risk-free interest rate is 8% per annum?

> A stock is expected to pay a dividend of $1 per share in 2 months and in 5 months. The stock price is $50, and the risk-free rate of interest is 8% per annum with continuous compounding for all maturities. An investor has just taken a short position in a

> The price of a European call that expires in 6 months and has a strike price of $30 is $2. The underlying stock price is $29, and a dividend of $0.50 is expected in 2 months and again in 5 months. Risk-free interest rates (all maturities) are 10%. What i

> The spot price of oil is $50 per barrel and the cost of storing a barrel of oil for one year is $3, payable at the end of the year. The risk-free interest rate is 5% per annum continuously compounded. What is an upper bound for the one-year futures price

> What is a lower bound for the price of a 2-month European put option on a non-dividend- paying stock when the stock price is $58, the strike price is $65, and the risk free interest rate is 5% per annum?

> What is the effect of an unexpected cash dividend on (a) a call option price and (b) a put option price?

> List the six factors that affect stock option prices.

> Explain the statement at the end of Section 12.1 that, when dividends are zero, the principal protected note cannot be profitable for the bank no matter how long it lasts.

> Companies X and Y have been offered the following rates per annum on a $5 million 10-year investment: Company X requires a fixed-rate investment; company Y requires a floating-rate investment. Design a swap that will net a bank, acting as intermediary, 0

> A company declares a 2-for-1 stock split. Explain how the terms change for a call option with a strike price of $60.

> A stock option is on a February, May, August, and November cycle. What options trade on (a) April 1 and (b) May 30?

> An interest rate is quoted as 5% per annum with semiannual compounding. What is the equivalent rate with (a) annual compounding, (b) monthly compounding, and (c) continuous compounding.

> An investor sells a European call option with strike price of K and maturity T and buys a put with the same strike price and maturity. Describe the investor’s position.

> In early 2012, the spot exchange rate between the Swiss Franc and U.S. dollar was 1.0404 ($ per franc). Interest rates in the United States and Switzerland were 0.25% and 0% per annum, respectively, with continuous compounding. The 3-month forward exchan

> Options on General Motors stock are on a March, June, September, and December cycle. What options trade on (a) March 1, (b) June 30, and (c) August 5?

> Consider an exchange-traded call option contract to buy 500 shares with a strike price of $40 and maturity in 4 months. Explain how the terms of the option contract change when there is: (a) a 10% stock dividend; (b) a 10% cash dividend; and (c) a 4-for-

> In Business Snapshot 17.1, what is the cost of a guarantee that the return on the fund will not be negative over the next 10 years? Business Snapshot 17.1 Can We Guarantee that Stocks Will Beat Bonds in the Long Run? It is often said that if you are

> Explain how you would value a swap that is the exchange of a floating rate in one currency for a fixed rate in another currency.

> The price of a stock is $40. The price of a 1-year European put option on the stock with a strike price of $30 is quoted as $7 and the price of a 1-year European call option on the stock with a strike price of $50 is quoted as $5. Suppose that an investo

> Explain why FVA can be calculated for a transaction without considering the portfolio to which the transaction belongs, but that the same is not true of MVA.

> Explain the difference between the views of financial economists and most practitioners on how KVA should be calculated.

> ‘‘Nonfinancial companies with high credit risks are the ones that cannot access fixed-rate markets directly. They are the companies that are most likely to be paying fixed and receiving floating in an interest rate swap.’’ Assume that this statement is t

> 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 ti

> After it hedges its foreign exchange risk using forward contracts, is the financial institution’s average spread in Figure 7.11 likely to be greater than or less than 20 basis points? Explain your answer. Figure 7.11 A currency swap

> Explain what is meant by (a) an ABS and (b) an ABS CDO.

> Companies A and B face the following interest rates (adjusted for the differential impact of taxes): Assume that A wants to borrow U.S. dollars at a floating rate of interest and B wants to borrow Canadian dollars at a fixed rate of interest. A financial

> A financial institution has entered into a 10-year currency swap with company Y. Under the terms of the swap, the financial institution receives interest at 3% per annum in Swiss francs and pays interest at 8% per annum in U.S. dollars. Interest payments

> A financial institution has entered into an interest rate swap with company X. Under the terms of the swap, it receives 4% per annum and pays six-month LIBOR on a principal of $10 million for five years. Payments are made every six months. Suppose that c

> Assume that the price of currency A expressed in terms of the price of currency B follows the process
where rA is the risk-free interest rate in currency A and rB is the risk-free interest rate in currency B. What is the process fol

> Companies A and B have been offered the following rates per annum on a $20 million five-year loan: Company A requires a floating-rate loan; Company B requires a fixed-rate loan. Design a swap that will net a bank, acting as intermediary, 0.1% per annum a

> What is the purpose of the convexity adjustment made to Eurodollar futures rates? Why is the convexity adjustment necessary?

> A Eurodollar futures price changes from 96.76 to 96.82. What is the gain or loss to a trader who is long two contracts?

> How is the conversion factor of a bond calculated by the CME Group? How is it used?

> Explain why the forward interest rate is less than the corresponding futures interest rate calculated from a Eurodollar futures contract.

> 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.

> 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?

> 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.)