期权期货习题

Future Option and other
derivatives
exercises
1. What5s the differenee between entering into
a long forward contact when the forward price is $50 and taking a long position in a call with a strike price of $50?
2.An investor enters into a short forward contract to sell 100,000British pounds for US dollars at an exchange rate of 1.5000US dollars per pound. How much does the investor gain or lose if the exchange rate at the end of the contract is 1.4900 and 1.5200?
❖3.You would like to speculate on a rise in the price of a certain stock. The current stock price is $29,and a 3-month call with a strike price of $30 costs $2.90.You have $5,800 to in vest. Ide ntify two alter native investment strategies, one in the stock and the other in an option on the stock. What are the potential gains and losses from each?
❖4.Suppose that a March call option to buy a share for $50 costs $2.5 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 profit from a long position in the option depends on the stock price at maturity of the option.
• 5.Explain why a forward contract can be used for either speculation or hedging.
• 6.Suppose that a June put option to sell a share for $60 costs $4 and is held until June. Under what circumstances will the option be exercised? Draw a diagram illustrating how the profit from a short position in the option depends on the stock price at maturity of the option.
• 7」t 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.
❖8.A trader writes a December put option with a strike price of $30. The price of the option is $4. Under what circumstances does the trader make a gain?
❖9.A company knows that it is due to receive a certain amount of a foreign currency in 4 mon ths. What type of optio n con tract is appropriate for hedging?
❖10.The price of gold is currently $500 per ounce. The forward price for delivery in 1 year is $700. An arbitrageur can borrow money at 10% per annum. What should the arbitrageur do? Assume that the cost of storing gold is zero and that gold provides no in come.
11 .Suppose that you enter into a short futures con tract to sell July silver for $5.20 per ounce on the New York Commodity Excha nge. The size of the con tract is 5,000 ounces. The initial margin is $4,000, and the maintenance margin is $3,000. What change in futures price will lead to a margin call? What happens if you do not meet the margin call?
12.An investor enters into two long July futures con tracts on orange juice. Each contract is for the delivery of 15,000 pounds. The current futures price is 160 cents per pound, the initial margin is $6,000 per con tract, and the main tenance margin is $4,500 per con tract. What price cha nge would lead to a margin call? Under what circumstances could $2,000 be withdrawn from the margin account?
13.At the end of one day a clearinghouse member is long 100 con tracts, and the settleme nt price is $50,000 per con tract. The original margin is $2,000 per contract. On the following day the member becomes responsible for clearing an additional 20 long con tracts, en tered into at a price of $51,000 per con tract. The settleme nt price at the end of this day is $50,200. How much does the member have to add to its margin account with the exchange clearinghouse?
❖14.Describe the profit from the following portfolio: a long forward con tract on an asset and a long European put option on the asset with the same maturity as the forward con tract and a strike price that is equal to the forward price of the asset at the time the portfolio is set up.
❖15.The current Price of a stock is $94, and3-month European call options with a strike price of $95 currently sell for $4.70.An investor who feels that the Price of the stock will in crease is trying to decide between buying 100 shares and buying 2,000 call options (=20 contracts). Both strategies in volve an in vestment of $9,400.What advice would you give? How high does the stock price have to rise for the option strategy to be more profitable?
Table Data for the example on rolling oil hedge forward.
Date Apr. 04 Sept.04 Feb.05 i J une05
Oct.04 futures price 18.20
17.40
Mar.O5
futures price
17.00 16.50
July.O5
futures price
16.30 15.90
Spot price 19.00
16.00
• 16.Suppose that the standard deviation of quarterly changes in the prices of a commodity is $0.65, the standard deviation of quarterly changes in a futures price on the commodity is $0.81,a nd the coefficie nt of correlation between the two changes is 0.8. What is the optimal hedge ratio for a 3- month con tract? What does it mean?
❖17/1f the minimunri variance hedge ratio is calculated as 1.0, the hedge must be perfect.5, Is this statement true? Explain your answer.
❖18.The standard deviation of monthly changes in the spot price of live cattle is (in cents per pound)12 The standard deviation of monthly changes in the futures price of live cattle for the closest con tract is 14 The correlation between the futures price changes and the spot price changes is 0・7」t is now October 15. A beef producer is committed to purchasing 200,000 pounds of live cattle on November 15.The producer wants to use the December live cattle futures con tracts to hedge its risk・Each contract is for the delivery of
40,000 pounds of cattle・What strategy should the
beef producer follow?
❖19.A long forward contract on a non・dividend-paying stock was entered into some time ago. It currently has 6 months to maturity. The risk-free rate of interest (with continuous compounding) is 10% per annum, the stock price is $25, and the delivery price is $24.
❖20.Consider a 3-month futures con tract on the S&P500. Suppose that the stocks un derlyi ng the in dex provide a divide nd yield of 1% per Qrinum, that the current value of the index is 800, and that the continuously compounded risk-free interest rate is 6% per Qrinum.
♦21 .Suppose that you enter into a 6・month forward contract on a non-dividend-paying stock when the stock price is $30and the risk-free interest rate (with continuous compounding) is 12% per annuin. What is the forward price?
• 22.A 1 -year long forward con tract on a norv dividend-paying stock is entered into when the stock Price is $40 and the risk-free rate of interest is 10% per annum with continuous compounding.
(a) What are the forward price and the initial value of the forward con tract?
(b) Six months later, the price of the stock is $45 and the risk-free interest rate is still 10% What are the forward price and the value of the forward con tract?
23.Suppose that the risk-free interest rate is 10% 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 4O5.What arbitrage opportunities does this create?
24.Assume that the risk-free interest rate is 9% 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 months, dividends are paid at a rate of 2% per annurn. Suppose that the value of the index on July 31, 2006，is 300.What is the futures price for a contract deliverable on December 31，2006?
25.The 2-month interest rates in Switzerland and the United States are,respectively,3% and 8% per annum with continuous compounding. The spot price of the Swiss franc is $0.6500.The futures price for a contract deliverabe in 2 months is
$0.6600.What arbitrage opportunities does this create?
26.The spot price of silver is $9 per ounce. The-storage costs are $0.24 per ounce; per year payable quarterly in advanee. Assuming that interest rates are 10% per annum for all maturities, calculate the futures price of silver for delivery in 9 months.
• 27.Suppose that the Treasury bond futures price is 101-12. Which of the following four bonds is cheapest to deliver?
Bond Price Conversion factor
1 125-05 1.2131
2 142-15 . 1.3792
3 115-31 1.1149
4 144-02 1.4026
❖28.The price of a 90-day Treasury bill is quoted as 10.OO.What continuously compounded return does an investor earn on the Treasury bill for the 90-day period?
29.lt is May 5，2005.The quoted price of a government bond with a 12% coupon that matures on July 27, 2011, is 110-17.What is the cash price?
• 31 ・Suppose that, in a Treasury bond futures con tract, it is known that the cheapest-to-deliver bond will be a 12% coupon bond with a conversion factor of
1.4000.Suppose also that it is known that delivery will take place in 270days・ Coupons are payable semiannually on the bond・ As illustrated in
Figure5the last coupon date was 60days ago5 the next coupon date is in 122days5 and the coup on date thereafter is in 305days ・ The term structure is flat, and the rate of interest (with continuous compounding) is 10% per annum. Assume the current quoted bond price the proportion of the next coupon payment that accrues to the holder.
Figure. Time chart for Example 31
Coupon Current Coupon Maturity Coupon Payment time payment of payment
futures
contract
60days 122days 148days 35days
A number of factors determine the cheapest-to-deliver bond. When bond yields
are in excess 6%,the conversion factor system tends to favor the delivery of bonds.
❖32lt5s July 30,2005.The cheapest-to-deliver bond in a September 2005 Treasury bond futures contract is a 13% coupon bond, and delivery is expected to be made on September 30,2005.Coupon payments on the bond are made on February 4 and August 4 each year. The term structure is flat, and the rate of interest with semiarmual compounding is 12% per annum. The conversion factor for the bond is 1 ・5.The current quoted bond price is $110.Calculate the quoted futures price for the contract.
• 33.Assume that a bank can borrow or lend money at the same interest rate in the LIBOR market. The 90-day rate is 10% per annum, and the 180-day rate is 10.2% per annum both expressed with continuous compounding and actual/actual day count. The Eurodollar futures price for a con tract maturing in 90days is quoted as 89.5.What arbitrage opportunities are open to the bank?
• 34」t is January 30, You are managing a bond portfolio worth $6 million. The duration of the portfolio in 6 months will be 8.2 years. The September Treasure bond futures price is currently 108-15, and the cheapest-to- deliver bond will have a duration of 7.6 years in September. How should you hedge against changes in interest rates over the n ext 6 mon ths?
❖35.Explain why brokers require margins when clients write options but not when they buy options・
• 36.A company declares a 2・to・1 stock split. Explain how the terms change for a call option with a strike price of $60.
❖37.Consider an exchange・traded call option con tract 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:1 )a 10% stock dividend;
2)a 10% cash dividend; 3)a 4-to-1 stock split.
❖38.A United States 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?
❖39.An investor writes four naked call option con tracts on a stock. The option price is $5, the strike price is $40, and the stock price is $38. Because the option is $2 out of the mon ey. •If the option had been a put, it would be $2 in the money and the margin requirement would be?
•4O.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?
•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?
❖41 .A 1 -month European put option on a non-dividend-paying stock is currently selling for $2.5.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?
❖42.A European call option and put option on a stock both have a strike price of $20 and an expiratio n date in 3 mon ths. Both sell for $3. The risk-free interest rate is 10% per annum, the current stock price is $19,and a $1 divide nd is in 1 month. Ide ntify the arbitrage opportunity open to a trader.
❖43.The price of a European call that expires in 6months and has a strike price of $30 is $2.The underlying stock price is $29, and a divide nd of $0.50 is expected in 2 months and again in
5months. The term structure is flat, with all risk-free interest rates being
10% .What is the price of a European put option that expires in 6months and has a strike price of $30?
♦Explain carefully the arbitrage opportunities in problem 8.4 if the European put price is $3.
• 44.Suppose that oare the prices of European call options with strike prices 技、&、％respectively, where
All options have the same maturity. Show that
What is the result corresponding to that in Problem .5 for European put options?
❖45.The price of an American call on a non・dividend-paying stock is $4.The stock price is $31,the strike price is $30,and the expiration date is in 3 months. The risk-free interest rate is 8%.Derive upper and lower bounds for the price of an American put on the same stock with the same strike price and expiration date.
❖Explain carefully the arbitrage opportunities in Problem45 if the American put price is greater than the calculated upper bound.
❖46.Suppose that put option 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.。