CHAPTER 14 - Forward and Futures Markets End-of...

CHAPTER 14 - Forward and Futures Markets

End-of-Chapter Problems

1. Explain why an investor might take an illiquid position in a forward contract rather than using an exchange-traded futures contract?

Solution: This is a simple tradeoff between contract customization and standardization. The exchange-tradedfutures contract is liquid because the standardization of the contracts gives the market sufficient volume and depthto maintain high liquidity. Obviously if the investor is willing to sacrifice the liquidity it is either because it is notimportant, as would be the case if the investor plans to hold the position and consummate the forward contract, orbecause it has attractive specialized characteristics in the dimensions of maturity, size, etc.

2. Explain in detail what is meant by a “speculator taking a short position in Australian dollar futures.”

Solution: A speculator is an investor looking to profit from forecasts of future spot rates and is not investing in thefutures market to hedge a position. A short position is a liability so the has agreed to sell the asset in question atthe specified price in the future. If the asset is Australian dollars the position is to sell Australian dollars foranother currency, perhaps the US dollar. Thus the short Australian dollar position is equivalent to a long US dollarposition. Finally it is a futures contract rather than a forward contract so it has standardized features making itliquid, most likely traded actively on an exchange.

3. Suppose you are the manager of a municipal electric company that buys electricity on the wholesale market and distibutes it to residential customers and charges by passing on the price of the electricity plus operating costs. Without a fixed-price supply contract, what futures market position could you undertake to hedge the resident’s exposure to higher cooling bills this summer if an electricity shortage were to develop?

Solution: You would want to take a long position in the futures market for electricity in effect locking in a price foryour wholesales purchases. Thus if the spot price increases and you have to pay more to acquire the electricityyou will get a compensating profit on your futures contract. While this is a price hedge the problem is on thevolume. You must meet your customers demand so you must forecast the demand in order to determine the amountto hedge in the futures market.

4. Suppose you are a distributor of canola seed and you observe the spot price of canola to be $7.45 per bushel while the futures price for delivery one month from today is $7.60. Assuming a $0.10 per bushel carrying cost, what would you do to hedge your price uncertainty?

Solution: We see that F S C+> . If you short the futures contract, you can lock in a price for your seed at $7.60 perbushel.

5. As a speculator observing the futures price for hogs to be delivered in six months you see a price of $14 per hundred weight while you believe the spot price for hogs will be $15 in six months. Explain what position you should take and how much profit you expect to make. What are the expected cash flows from this position?

Solution: Your expectation is the spot price in six months will be more than what can be locked in through thefutures market so you should go long in hog futures. Your expectation is that you will see the price of the futurescontract rise as time passes and it fact increase by $1 at the end of six months. So you need only sell the futurescontract and realize your cash flow and profit.

Chapter 14 - 1Copyright ©2009 Pearson Education, Inc. Publishing as Prentice Hall.

Financial Economics Solutions Manual

6. You are a dealer in kryptoni te and are contemplating a trade in a forward contract. You observe that the current spot price per ounce of kryptonite is $180.00, the forward price for delivery of one ounce of kryptonite in one year is $205.20, and annual carrying costs of the metal are 4% of the current spot price.a. Can you infer the annual return on a riskless zero-coupon security implied by the Law of One Price?b. Can you describe a trading strategy that would generate arbitrage profits for you if the annual return on the riskless security is only 5% ? What would your arbitrage profit be, per ounce of kryptonite?

Solution: By no arbitrage, we require that the riskless rate r satisfy:

F0 S0 1 r+ s+( )⋅= S0 180:= F0 205.2:= s 4%:=

Solving for r: rF0− S0+ S0 s⋅+( )−

S0:= r 0.1= 10%

The implicit risk-free rate that you can earn by buying kryptonite, storing it, and selling it forward at $205.20 perounce is 10%.

b. If the riskless borrowing rate is 5%, you should borrow at that rate and invest in hedged kryptonite. If youborrow the $180 you will end up paying 5% interest and 4% carrying cost for a total cost of:

180 1 4 %⋅+ 5 %⋅+( )⋅ 196.200=

Thus the profit per ounce will be $9: 205.2 196.2− 9.000=

7. Infer the spot price of an ounce of gold if you observe the price of one ounce of gold for forward delivery in three months is $435.00, the interest rate on a 91-day Treasury bill is 1% and the quarterly carrying cost as a percentage of the spot price is 0.2% .

Solution: Deduce from the futures price parity condition for gold that F = S0(1 + r + s) so that:

F 435:= r 1%:= s 0.2%:=

S0F

1 r+ s+:= S0 429.842= = $429.84

8. Calculate the implicit cost of carrying an ounce of gold and the implied storage cost per ounce of gold if the current spot price of gold per ounce is $425.00, the forward price of an ounce of gold for delivery in 273 days is $460.00, the yield over 91 days on a zero-coupon Treasury bill is 2% and the term structure of interest rates is flat.

Solution: Over 273 days, the risk free rate is 1 2%+( )3 1− 0.0612= 6.12%. Therefore we have:

F S 1 r+ s+( )⋅:= root 425 1 6.12%+ s+( )⋅ 460− s, [ ] 0.0212= 2.12% for 273 days

root 1APR

4+⎛⎜

⎝⎞⎟⎠

31− 2.12%− APR,

⎡⎢⎣

⎤⎥⎦

0.0281= 2.81% per year is the implied storage cost



The implied cost of carry is: F S− 35.000= for 273 days

As a percentage of the spot price: 35425

0.0824= 8.24% for 273 days

root 1APR

4+⎛⎜

⎝⎞⎟⎠

31− 8.24%− APR,

⎡⎢⎣

⎤⎥⎦

0.1070= 10.69% per year is the implied cost of carry

9. The forward price for a share of stock to be delivered in 182 days is $410.00, whereas the current yield on a 91-day T-bill is 2% . If the term structure of interest rates is flat, what spot price for the stock is implied by the Law of One Price?

Solution: F 410:= r 2%:= SF

1 r+( )2:= S 394.08=

10. On your first day of trading in Vietnamese forward contracts, you observe that the share price of Giap Industries is currently 54,000 dong while the one-year forward price is 60,000 dong . If the yield on a one-year riskless security is fifteen percent, are arbitrage profi ts possible in this market? If not, explain why not. If so, devise an appropriate trading strategy.

Solution: Arbitrage profits would seem to be possible, since the no-arbitrage forward price implied by theseparameters is:

S 54000:= F 60000:= r 15%:=

F S 1 r+( )⋅:= F 62100.000=

The forward contract is under-priced, relative to the no-arbitrage value. Consider taking a long position in theforward contract and simultaneously selling a share of Giap stock and buying a riskless bond with a face valueequal to the observed forward price. The short sale of Giap stock would yield you 54,000 dong today. You usethe forward contract to insure you can purchase the share in one-year to close the short position. Thus you need60,000 dong in one year. The present value of the 60,000 dong is given by:

600001 15 %⋅+

52173.913= dong

Thus your 54,000 dong from the short sale is sufficient to set aside enough today for the long forward position ina year and you make a profit today of:

54000 52173.913− 1826.087= dong

11. Suppose the current spot price of a riskless zero-coupon bond with one year to maturi ty is $94.34 per $100 of face value. If a non-dividend-paying stock is currently selling for $37.50 per share what is implied about its forward price for delivery in one year? Use the forward-spot price-parity relationship.

Solution: The riskless rate is: 10094.34

1− 0.060= 6%

The forward share price should be: F 37.5 1 6 %⋅+( )⋅:= F 39.750=



12. Referring to the information from the previous problem (problem 11) suppose the actual forward price of the stock for delivery in one year is $40. What arbitrage opportunity exists? Demonstrate the cash flows from the strategy.

Solution: The forward price is too high meaning you can deliver the share in a year for less than $40. Thus youshould borrow at the riskless rate to purchase a share today in the spot market an go short in the forward marketobliging yourself to supply a share in a year and collect the forward contract price. Thus you must borrow $37.50 toby the share in the sport market. After a year you must repay the loan principal and interest:

37.5 1 6 %⋅+( )⋅ 39.750= But when you supply the share to close your short forward positionyou will receive $40. Thus you make $0.25 per share.

13. You observe that the one-year forward price of a share of stock in Kramer, Inc., a New York tour-bus company and purveyor of fine clothing, is $45.00 while the spot price of a share is $41.00. If the riskless yield on a one-year zero-coupon government bond is 5% :a. What is the forward price implied by the Law of One Price?b. Can you devise a trading strategy to generate arbitrage profits? How much would you earn per share?

Solution: a. The no-arbitrage value of the forward price is 41 1 5%+( )⋅ 43.050= = $43.05.b. The observed forward price is excessive. Consider short-selling a forward contract and taking a long position in aportfolio consisting of one stock and the sale of a zero-coupon bond with face value of $45. . Future liabilities for thisposition are zero, while the current cash inflow is $1.86.

14. The share price of Schleifer and Associates, a financial consultancy in Moscow, is currently 10,000 roubles whereas the forward price for delivery of a share in 182 days is 11,000 rubles. If the yield on a riskless zero-coupon security with term to maturity of 182 days is 15% , infer the expected dividend to be paid by Schleifer and Associates over the next six months.

Solution: The implied dividend is given by: D S 1 r+( )⋅ F−:= 10000 1 15%+( )⋅ 11000− 500.000= rubles

15. Infer the yield on a 273-day, zero-coupon Japanese government security if the spot price of a share of non-dividend-paying stock in Mifune and Associates is 4,750 yen whereas the forward price for delivery of a share in 273 days is 5,000 yen.

Solution: 50004750

1− 0.0526= 5.26% is the implied yield over the 273 days

16. Suppose a risky non-dividend-paying stock is currently priced at $45 per share. If the stock’s risk premium is 5% and the riskless rate is 5% what is the expected spot price of the share one year hence? What does the forward-s pot price-parity relation imply about the forward price of the share for delivery one year hence?

Solution: The stock must be priced so the expected return is 10%. With a current price of $45 per share this impliesan expected spot price in one year equal to: 45 1 10%+( )⋅ 49.500= = $49.50. But by forward-spot price parity wehave a forward price of: 45 1 5%+( )⋅ 47.250= = $47.25. Thus the forward price is not the expected future price.

17. Refer to problems 11 and 12. How would your ans wers change if the stock is expected to pay a $1 dividend at the end of the coming year?

Solution: S 37.50:= r 6%:= D 1:=



The implied forward price is: F S 1 r+( )⋅ D−:= F 38.750=

The forward price of $40 is too high meaning you can deliver the share in a year for less than $40. Thus you shouldborrow at the riskless rate to purchase a share today in the spot market an go short in the forward market obligingyourself to supply a share in a year and collect the forward contract price. Thus you must borrow $37.50 to by theshare in the sport market. After a year you must repay the loan principal and interest:

37.5 1 6 %⋅+( )⋅ 39.750= But when you supply the share ex-dividend to close your short forwardposition you will receive $40. Thus you make $1.25 per share.

18. Suppose that the Treasury yield curve is flat at an interest rate of 7% per year (compounded semiannually).a. What is the spot price of a 30-year Treasury bond with an 8% coupon rate assuming coupons are paid semiannually?b. What is the forward price of the bond for delivery six months from now?

Solution: a. The spot price of the 30-year Treasury is $112.47: Price 8% 7%, 30, ( ) 112.4724= per $100 of face value

b. The forward price for delivery six months from now should be $112.41 F S 1 r+( )⋅ C−:=

112.4724 1.035⋅ 4− 112.409=

19. Continuing the previous probl em (problem 18) show that if the forward price is $1 less than the ans wer found in problem 18, part b, the there is an arbitrage opportunity. Detail the proceedure by which you would produce an arbitrage profit and calculate the magnitude of the profit.

Solution: If the forward price is only $111.41 then arbitrage profits can be made by selling the bond short and buyingit forward at the low forward price.

Sell short a bond at $112.47; buy it forward at $111.41; invest the proceeds of the short sale to earn 3.5% for 6months yielding in six months 112.4724 1.035⋅ 116.409=

After 6 months, buy the bond at the forward price of 111.41. Use the bond to cover your short sale and pay thecoupon of $4 yielding a profit of 116.41 111.41− 4− 1.000= = $1. This is the one dollar profit from the onedollar forward mispricing of the bond.

20. A stock has a spot price of $100; the riskless interest rate is 7% per year (compounded annually), and the expected dividend on the stock is $3, to be received a year from now. a. What should be the one-year futures price?b. If the futures price is $1 higher than the ans wer found in part a, what might that imply about the expected dividend?

Solution: a. S 100:= r 7%:= D 3:= F S 1 r+( )⋅ D−:= F 104.000=

b. If F is $105, that would imply a D of: root S 1 r+( )⋅ D− 105− D, [ ] 2.000=



21. The spot rate of exchange of yen for Canadian dollars is currently 113 yen per dollar but the one-year forward rate is 110 yen per dollar. Determine the yield on a one-year zero-coupon Canadian government security if the corresponding yield on a Japanese government security is 2.21% .

Solution: The spot rate of exchange of Canadian dollars for yen is: S 113 1−:= S 0.00885=

The one-year forward rate is: F 110 1−:= F 0.00909=

The Japanese riskless rate is: rj 2.21%:=

The interest rate parity condition is given by: FS

1 rc+( )⋅ 1 rj+( )=

Solving for rc: rcFS

1 rj+( )⋅ 1−:= rc 0.04998= 5% is the Canadian riskless rate.

22. Assume the current spot price of the South African rand is $0.0995 and the one-year forward price is $0.0997. If the riskless annual dollar interest rate is 5% what is the implied riskless annual rand interest rate?

Solution: The spot rate of exchange of dollars for rand is: S 0.0995:=

The one-year forward rate is: F 0.0997:=

The US riskless rate is: rus 5%:=

The interest rate parity condition is given by: FS

1 rus+( )⋅ 1 rsa+( )=

Solving for rsa: rsaFS

1 rus+( )⋅ 1−:= rsa 0.05211= 5.21% is the South African riskless rate.

23. Challenge Problem: Suppose that you are planning a trip to England. The trip is a year from now, and you have reserved a hotel room in London at a price of £50 per day. You do not have to pay for the room in advance. The exchange rate is currently $1.50 to the pound sterling.a. Explain several possible ways that you could completely hedge the exchange rate risk in this situation.b. Suppose that ruk=.12 and rus=.08. Because S=$1.50, what must the forward price of the pound be?c. Show that if F is $0.10 higher than in your ans wer to part b, there would be an arbi trage opportunity.

Solution: a. This is a payable in pounds which can be completely hedged via a prepayment, a forward hedge, or amoney market hedge. The prepayment simply means you exchange dollars for pounds today at the sport rate and paythe £50 bill in advance. The forward hedge means signing a forward contract to lock in an exchange rate in a year.The money market hedge means converting dollars to pounds at the current spot rate to invest the present value of the£50 in a pound-denominated riskless asset.

b. With ruk 12 %⋅:= and rus 8 %⋅:= . Since S 1.5:= the forward rate implied by Interest Rate Parity is given by:

F S1 rus+

1 ruk+⋅:= F 1.4464=



c. Consider borrowing dollars to convert spot and invest in a pound-denominated risk-free asset for a year and alsosell the pound proceeds forward at the rate: F 1.5464:= . The costs and revenue from this arbitrage would be:

Cost rus( ) 1 rus+:= Cost rus( ) 1.08= Revenue S F, ruk, ( ) 1S

1 ruk+( )⋅ F⋅:= Revenue S F, ruk, ( ) 1.155=

The arbitrage opportunity would make a profit:

Profit S F, rus, ruk, ( ) Revenue S F, ruk, ( ) Cost rus( )−:= Profit S F, rus, ruk, ( ) 0.0746=

24. Challenge Problem: Suppose the one-year forward price of the dollar is K49.5 (Slovakian koruna) while the spot exchange rate is K46.95. The riskless annual dollar rate of interest is 2.75% . If the expectations hypothesis holds what is the dollar/koruna spot exchange rate expected to be one year hence?

Solution: This is a bit of a teaser. One way to view it is that the statement that "the expectations hypothesis holds"means with regard to the dollar/koruna exchange rate in which case the sport exchange rate expected in one year isequal to the one-year forward rate and this is currently:

F1

49.5:= F 0.02020=

On the other hand if we take the statement that "the expectations hypothesis holds" with regard to the quotedkoruna/dollar exchange rate we know it can't hold with respect to the dollar/koruna exchange rate (see footnote 7 inthe text chapter) and therefore we don't know what the dollar/koruna spot exchange rate is expected to be in oneyear.

25. In the forward market the one-year and two-year forward prices of the euro are $0.901 and $0.903 respectively. A two-year swap with a notional principle of 1 million euros is priced when the dollar riskless rate is 5% per annum. What is the agreed swap rate?

Solution: The present value of the swap payments should be the same as the present value of the amountsequivalent under the forward rates or:

9010001 5%+

903000

1 5%+( )2+ 1000000 F⋅

11 5%+

1

1 5%+( )2+⎡

⎢⎣

⎤⎥⎦

⋅=

Solving for F we obtain: F 0.90198:= dollars per euro



Objectives To learn how to use forward and futures contracts for hedging, speculating, and arbitrage. To understand the relations among spot, forward, and futures prices of commodities, currencies, and

securities. To learn what kinds of useful information can be inferred from the relations among spot and forward

prices.

Contents 14.1 Distinctions between Forward and Futures Contracts 14.2 The Economic Function of Futures Markets 14.3 The Role of Speculators 14.4 Relation between Commodity Spot and Futures Prices 14.5 Extracting Information from Commodity Futures Prices 14.6 Forward-Spot Price Parity for Gold 14.7 Financial Futures 14.8 The “Implied” Riskless Rate 14.9 The Forward Price Is Not a Forecast of the Future Spot Price 14.10 Forward-Spot Price-Parity Relation with Cash Payouts 14.11 “Implied” Dividends 14.12 The Foreign-Exchange Parity Relation 14.13 The Role of Expectations in Determining Exchange Rates

Summary

Futures contracts make it possible to separate the decision of whether to physically store a commodity from the decision to have financial exposure to its price changes.

Speculators in futures markets improve the informational content of futures prices and they make futures markets more liquid than they would otherwise be.

The futures price of wheat cannot exceed the spot price by more than the cost of carry:

F S C− ≤ The forward-spot price-parity relation for gold is that the forward price equals the spot price times one plus the cost of carry:

( )1F r s S= + +

where F is the forward price, S is the spot price, r is the riskless interest rate, and s are storage costs. This relation is maintained by the force of arbitrage.

One can infer the implied cost of carry and the implied storage costs from the observed spot and forward prices and the riskless interest rate.

The forward-spot price-parity relation for stocks is that the forward price equals the spot price times 1 plus the riskless rate less the expected cash dividend:

( )1F S r D= + −

This relation can, therefore, be used to infer the implied dividend from the observed spot and forward prices and the riskless interest rate.

Chapter 14 - 8 Copyright ©2009 Pearson Education, Inc. Publishing as Prentice Hall.


The forward-spot price-parity relation for the dollar/yen exchange rate involves two riskless interest rates:

( ) ( )$1 1F S

r r=

+ + ¥

r¥ $

where F is the forward price of the yen, S is the current spot price, is the yen interest rate, and r is the dollar interest rate.


CHAPTER 15 - Markets for Options and Contingent Claims

End-of-Chapter Problems

1. Which has unlimited downside risk, a long or short position in a call option? What about a long or short position in a put option? Explain your ans wers.

Solution: The short-call position has unlimited downside risk as for each dollar the asset price rises above theexercise price there is an additional dollar loss on the position. This is because when the call option is in the money(S>E) and you have sold the call you are required to sell an asset at a price less than its current market value and asthe sale price is fixed the higher the current market value the more you lose. This is true based upon the marketvalue of the underlying asset and even if you own the asset for delivery it represents an opportunity cost.

The short-put position has limited downside loss. If you have sold the put you have agreed to purchase the assetat the exercise price if the option is exercised. If the put is in the money (S<E) it will cost you the difference betweenthe current price and the exercise price as you will be paying more, the exercise price, for the asset than it iscurrently worth. And for each dollar the asset price drops below the exercise price you loss increases. But the lossis limited as the asset price cannot fall below zero. So the maximal loss you can have when the option is exercisedagainst you is the exercise price - you pay this for something which is worthless.

2. Describe the key defining characteristics of a put or call option on any asset.

Solution: Every option has five distinguishing characteristics. First a specification of the underlying asset to bedelivered if the option is exercises. Second a specification of whether the option is a put , the option to sell theasset, or a call, the option to buy the asset. Third the fixed price at which the option can be executed, the exercise orstrike price. Fourth the date after which the option expires, the expiration or maturity date. Fifth whether the optioncan be exercised only on the expiration date, a European-type option, or any time up to and including the expirationdate, an American-type option.

3. Graph the payoff for a European put option selling for a premium of $5 with exercise price (E) of $50, written on a stock with value S, when:a. You hold a long position (i.e., you buy the put)b. You hold a short position (i.e., you sell the put)

Solution: a.

0 20 40 60 80 10010−

10

20

30

40

50Long Put

Stock Price at Expiration

Long

-Put

Opt

ion

Payo

ff

P−

if S E< E S−, 0, ( ) P−

E

S



b.

0 20 40 60 80 100

50−

40−

30−

20−

10−

10Short Put


Shor

t-Put

Opt

ion

Payo

ff

P

if S E> 0, S E−, ( ) P+

E

S

4. Graph the payoff to a portfolio of one European call option and one European put option, each with the same expiration date and each with exercise price (E) of $25, when both options are on a stock with value S.Assume both the call and put premiums are $1. What is the portfolio payoff if the stock price is $40 at expiration? What if the price is $20?

0 10 20 30 40 50

10−

10

20

30Straddle


Payo

ffs

P

E

Dashed line is the callpayoff, dotted line isthe put payoff and thesolid line is the straddlepayoff equal to the sumof the call and putpayoffs.

Straddle S E, C, P, ( ) if S E< 0, S E−, ( ) C− if S E< E S−, 0, ( ) P−( )+:=

Straddle 40 25, 1, 1, ( ) 13.000= Straddle 20 25, 1, 1, ( ) 3.000=



5. If an investor paid a $4.75 premium to obtain a long position in a put option with an exercise price of $47, the underlying asset break-even point corres ponds to what price? Diagram the position.

Solution:

0 20 40 60 80

10−

10

20

30

40

50Long Put


Long

-Put

Opt

ion

Payo

ff

P−

if S E< E S−, 0, ( ) P−

E

S

The break-even stock price is: root if S E< E S−, 0, ( ) P− S, ( ) 42.250=

6. The risk-free one–year rate of interest is 4% , and the Globalex stock index is at 100. The price of one-year European call options on the Globalex stock index with an exercise price of 104 is 8% of the current price of the index. Assume that the expected dividend yield on the stocks in the Globalex index is zero. You have $1 million to invest for the next year. You plan to invest enough of your money in one-year T-bills to insure that you will at least get back your original $1 million, and you will use the rest of your money to buy Globalex call options. If you think that there is a probability of .5 that the Globalex index a year from now will be up 12% , a probabili ty of .25 that it will be up 40% , and a probability of .25 that it will be down 20% , what is the probability distribution of your portfolio rate of return?

Solution: To insure that you will at least get back your original $1 million, you need to invest in T-bills the following amount:

10000001 4%+

961538.462= $961,538.462

The investment in options produces: 1000000 961538.46−

8% 100⋅4807.69= options

Probability of Globalex Stock Index change 0.5 0.25 0.25Percentage change of Global Stock index 12% 40% -20%Next Year's Globalex Stock Index 112 140 80Payoff of 4807.69 shares of options 38461.52 173076.8 0Payoff of your portfolio 1038462 1173077 1000000Portfolio rate of return 3.85% 17.31% 0



7. Given the following variables: S = $55, E = $75, T = 1 year, r = 5 % , and P = $15; if the call option is selling for $10 (C = $10), what arbi trage opportunity exists? Outline the strategy and the profi t to be realized.

Solution: First check the call-put parity.

S 55:= E 75:= T 1:= r 5%:= P 20:=

SE

1 r+( )T− P+ 3.571= Which is significantly below the call option price of $10.

Thus sell calls and replicate them as follows:

Position Immediate Cash Flow Cash Flow at Maturity

if S<$75 if S>$75

sell a call $10 0 -(S-$75)

replicate the call

buy a share -$55 S S

borrow pv of E75

1 5%+71.429= -$75 -$75

buy a put -$20 $75-S 0

net cash flows 71.43 75− 10+ 6.430= 0 0

Thus the strategy nets $6.43 up front.

8. Using the put-call parity relation:a. Show how one can replicate a one-year pure discount bond with a face value of $100 using a share of stock, a put and a call. b. Suppose that E=$100, S=$100, P=$10, and C=$15. What must be the one-year interest rate? c. Show that if the one-year risk-free interest rate is lower than in your answer to Part b, there would be an arbitrage opportunity.

Solution:a. To replicate a one-year pure discount bond with a face value of $100, buy a share of stock, and a Europeanput with exercise price $100, and sell a European call with an exercise price $100.

b.S 100:= P 10:= C 15:= E 100:=

E1 r+

S P+ C−= Solving for r: rE

S P+ C−1−:= r 0.0526=



c. If r = 4%, then one could make risk-free arbitrage profits by borrowing at 4% and investing in synthetic 1-yearpure discount bonds consisting of a share of stock, a European put with exercise price $100, and a short position in aEuropean call with an exercise price $100. The synthetic bond would cost $95 and pay off $100 at maturity in 1 year.The principal and interest on the $95 it costs to buy this synthetic bond would be 95 1 4%+( )⋅ 98.800= = $98.8.Thus there would be a pure arbitrage profit of $1.20 per bond a year from now with zero initial outlay of funds.

9. A 90-day European call option on a share of the stock of Wimendo is currently trading at 20 euros whereas the current price of the share itself is 24 euros. 90-day zero-coupon securities issued by the government of France are selling for 98.55 euros per 100 euros of face value. Infer the price of a 90-day European put option on this stock if both the call and put have a common exercise price of 20 euros.

Solution: C 20:= S 24:= E 20:=

The 90-day riskless rate is: 10098.55

1− 0.015= r 1.5%:=

PE

1 r+S− C+:= P 15.704= The put option price is $15.704

10. Gordon Gekko has assembled a portfolio consisting of ten 90-day US Treasury bills, each having a face value of $1, 000 and a current price of $990.10, and 200 90-day European call options, each written on a share of Paramount stock and having an exercise price of $50. Gekko is offering to trade you this portfolio for 300 shares of Paramount stock, which is currently valued at $215 a share. If 90-day European put options on Paramount stock with a $50 exercise price are currently valued at $25,a. Infer the value of the calls in Gekko’s portfolio.b. Determine whether you should accept Gekko’s offer.

Solution: a. the 90-day riskless rate is: 1000

990.11− 0.010= r 1%:=

E 50:= S 215:= P 25:= C P S+E

1 r+−:= C 190.495=

b. The value of Gekko's portfolio is: 10 990.1⋅ 200 C⋅+ 48000.010=

The value of the shares is: 300 215⋅ 64500.000= Tell Gordo no way!

11. The stock of Kakkonen, Ltd., a hot tuna distributor, currently lists for $50 a share, whereas one-year European call options on this stock, with an exercise price of $20, sell for $40 and European put options with a similar expiration date and exercise price sell for $8.457.a. Infer the yield on a one-year, zero-coupon US government bond sold today.b. If this yield is actually at 9% , construct a profitable trade to exploit the potential for arbitrage.

Solution: a. The riskless yield implied from the call-put parity is:

S 50:= P 8.457:= C 40:= E 20:= rE

S P+ C−1−:= r 0.0836= 8.36%



b. There are many ways to exploit the violation of the Law of One Price to make arbitrage profits. Since the risk-freeinterest rate is 9%, and the implied interest rate on the replicating portfolio is 8.36%, we could go short the replicatingportfolio and invest the proceeds in T-bills. For example, at current prices, short-sell a “unit” portfolio, which consistsof long positions in one put and one share and writing one call, to earn immediate revenue of $184.57. The portfolioyou sold short requires payment of $200 one year from now. If you invest the $184.57 in one-year T-bills you willhave 184.57 1 9%+( )⋅ 201.18= = $201.18 a year from now. Thus you will earn a risk-free arbitrage profit of $1.18with no outlay of your own money.

12. Which has more value, (1) a portfolio composed of call options on ten different stocks or (2) a single call option on a portfolio composed of the same ten stocks? Assume the exercise price of (2) is equal to the sum of the exercise prices of the call options in (1) and all option expiration dates are identical. Explain you answer.

Solution: (1) will be more valuable than (2). The keys are the volatilities of the underlying assets in the two cases.We know that the value a call option is directly related to the volatility of the price of the underlying asset. Aportfolio of stocks (except in the most unlikely case of perfect positive correlation between all pairs of the stocks)will have less volatility than the sum of the individual volatilities. This is a simple result from portfolio theory thatas you increase the number of less than perfectly correlated stocks held in a portfolio the diversification effectreduces the risk of the portfolio. Thus the value of the call option on the portfolio of stocks will be less than thesum of the values of the call options on the individual stocks.

13. The share value of Drummond, Griffin and McNabb, a New Orleans publishing house, is currently trading at $100 but is expected, 90 days from today, to rise to $150 or to decline to $50, depending on critical reviews of its new biography of Ezra Pound. Assuming the risk-free interest rate over the coming90-day period is 1% , can you value a European call option written on a share of DGM stock if the option carries an exercise price of $85? What is the hedge ratio?

Solution:

S0 100:= s 50 %⋅:= E 85:= r 1 %⋅:=

S1U S0 1 s+( )⋅:=

S1L S0 1 s−( )⋅:= C1 S1( ) if S1 E> S1 E−, 0, ( ):=

h1C1 S1U( ) C1 S1L( )−

S1U S1L−:= C0 S0 h1⋅

h1 S1U⋅ C1 S1U( )−

1 r+−:=

S1U 150.000= C1 S1U( ) 65.000=

S0 100.000= C0 32.822=

h1 0.650=S1L 50.000= C1 S1L( ) 0.000=



14. Challenge Problem: Let s be the fixed percentage change by which the stock price rises or falls over a given period. Derive the formula for the price of a put option using a one-period two-state model. Refering to the previous problem (problem 13), use the model to price a put option on DGM with the same maturity and exercise price as the call option. Use the call-put parity to verify the ans wers for the values of the call and the put.

Solution: To price the put option, we create a synthetic put option by selling short a fraction of a share h, andlending $b at the riskless rate. Denote the current price of the stock S0, the current price of the put option P0, thestock price when it is up over the period to be S1U, the stock price when it is lower over the period to be S1L, thepayoffs of the put option in each state P1U and P1L, the percentage change up or down is s, and the risklessinterest r.

The synthetic put satisfies: h S1U⋅ b 1 r+( )⋅+ P1U= h S1L⋅ b 1 r+( )⋅+ P1L=

Solving for b from the fist equation: Substituting in the second equation:

bh S1U⋅ P1U−( )−

1 r+( )= h S1L⋅ h S1U⋅− P1U+ P1L=

Solving for h: Solving for b:

hP1U P1L−

S1U S1L−:= b

S1U P1L⋅ P1U S1L⋅−( )S1L− S1U+( ) 1 r+( )⋅

:=

The present value of the synthetic putshould be the current value of the put:

P0 h S0⋅ b+=

Substitution for h and b results in: P0P1U P1L−( )S1L− S1U+( )

S0⋅S1U P1L⋅ P1U S1L⋅−( )

S1L− S1U+( ) 1 r+( )⋅+=

Where: S1L 1 s−( ) S0⋅:= S1U 1 s+( ) S0⋅:=

P1L if S1L E< E S1L−, 0, ( ):= P1U if S1U E< E S1U−, 0, ( ):=

P0P1U P1L−( )S1L− S1U+( )

S0⋅S1U P1L⋅ P1U S1L⋅−( )

S1L− S1U+( ) 1 r+( )⋅+:=



S0 100:= s 50 %⋅:= E 85:= r 1 %⋅:=

S1U 150.000= P1U 0.000=

S0 100.000= P0 16.980=

h1 0.650= S1L 50.000= P1L 35.000=

b 51.980=

Check the answers using call-put parity: SE

1 r+( )T− P+ C=

S0E

1 r+− P0+ 32.822= which is the call option value calculated in Problem 13.

15. Using the binomial call option model to find the current value of a call option with a $40 exercise price on a stock currently priced at $50. Assume the option expires at the end of two periods, the riskless interest rate is 5 percent per period and the share price will rise or fall by 10 percent per period. What are the hedge ratios?

Solution:

S0 50:= s 10 %⋅:= E 40:= r 5 %⋅:=

S2U S0 1 s+( )2⋅:=S1U S0 1 s+( )⋅:=

S2M S0 1 s+( )⋅ 1 s−( )⋅:=S1L S0 1 s−( )⋅:=

S2L S0 1 s−( )2⋅:= C2 S2( ) if S2 E> S2 E−, 0, ( ):=

h2UC2 S2U( ) C2 S2M( )−

S2U S2M−:= C1U S1U h2U⋅

h2U S2U⋅ C2 S2U( )−

1 r+−:=

h2LC2 S2M( ) C2 S2L( )−

S2M S2L−:= C1L S1L h2L⋅

h2L S2M⋅ C2 S2M( )−

1 r+−:=

h1C1U C1L−

S1U S1L−:= C0 S0 h1⋅

h1 S1U⋅ C1U−

1 r+−:=



S2U 60.500=

C2 S2U( ) 20.500=

S1U 55.000= C1U 16.905=

h2U 1.000= S2M 49.500=

S0 50.000= C0 13.719= C2 S2M( ) 9.500=

h1 1.000= S1L 45.000= C1L 6.905=

h2L 1.000= S2L 40.500=

C2 S2L( ) 0.500=

16. Challenge Problem: In the particular case in the previous problem (problem 15) the option is so deep in the money relative to the limited price movement possible over the remaining time to maturity that we are sure the option will be exercised at expiration. In this case the current value of the call option has a simple expression in terms of three variables: the current underlying stock price (S0), the exercise price (E), and the riskless rate of interest (r). Please formulate the expression, explain it and use it to verify the call option price found in the previous problem.

Solution: We know that buying the call option today will lead to the purchase of the stock at the exercise price atmaturity. This is equivalent to buying a share today since by buying the call option and paying the exercise price atmaturity we will also own a share. If there are no dividends and with no uncertainty these two ways to purchase ashare are identical and by the Law of One Price they should have the same present value.

S0 C0E

1 r+( )T+= or C0 S0

E

1 r+( )T−=

So for a deep-in-the-money call option which we know will remain in the money and be exercised the current price ofthe call option should equal the current stock price less the present value of the exercise price. In the example fromProblem 15 we have:

S0 50:= E 40:= r 5%:= T 2:= 5040

1 5%+( )2− 13.719=

17. Using the Black-Scholes model find the premium on a call option with an exercise price of $35 on a share currently priced at $40. Assume the riskless rate of 10% per annum and the option has six-months to expiration. The risk of the stock is measured by a σ value of 0.25. Decompose the premium into intrinsic value and time value?

Solution:

d1 S E, r, σ, T, ( )ln

SE

⎛⎜⎝

⎞⎟⎠

rσ

2

2+

⎛⎜⎝

⎞⎟⎠

T⋅+

σ T⋅:= d2 S E, r, σ, T, ( ) d1 S E, r, σ, T, ( ) σ T⋅−:=

N d( ) 2 π⋅( )

1

2−

0

d

uexp12

− u2⋅⎛⎜

⎝⎞⎟⎠

⌠⎮⎮⌡

d⋅12

+:=



C S E, r, σ, T, ( ) S N d1 S E, r, σ, T, ( )( )⋅ E N d2 S E, r, σ, T, ( )( )⋅ e r− T⋅⋅−:=

C 40 35, 10%, 25%, 12

, ⎛⎜⎝

⎞⎟⎠

7.205= Since the share price is currently $5 above the exercise price the call has $5of intrinsic value and therefore $2.205 of time value.

18. Use the Black-Scholes formula to find the price of a 3-month European call option on a non-dividend-paying stock with a current price of $50. Assume the exercise price is $51, the continuously compounded risk-free interest rate is 8% per year, and σ is 0.4.a. What is the composition of the initial replicating portfolio for this call option?b. Use the put-call pari ty relation to find the Black -Scholes formula for the price of the corresponding put option.

Solution: a. The call option is priced at: C 50 51, 8%, 40%, 14

, ⎛⎜⎝

⎞⎟⎠

3.987= The hedge ratio, which is the number of

shares of stock you must buy, equals N(d1), and the amount to borrow is N(d2) times the PV of the exercise price. Thehedge ratio is .54, which means you would buy .54 of a share of stock for $27. The amount to borrow is $23.024.

N d1 50 51, 8%, 40%, 14

, ⎛⎜⎝

⎞⎟⎠

⎛⎜⎝

⎞⎟⎠

0.540= N d2 50 51, 8%, 40%, 14

, ⎛⎜⎝

⎞⎟⎠

⎛⎜⎝

⎞⎟⎠

51 e8− %

1

4⋅

⋅ 23.024=

.54 50⋅ 27.000=

b. using call-put paritythe put option price is:

P C 50 51, 8%, 40%, 14

, ⎛⎜⎝

⎞⎟⎠

50− 51 e8− %⋅

1

4⋅

⋅+:= P 3.977=

19. As a financial analyst at Yew and Associates, a Singaporean investment house, you are asked by a client if she should purchase European call options on Rattan, Ltd. stock, which are currently selling in US dollars for $30. These options have an exercise price of $50. Rattan stock currently exhibits a share price of $55, and the estimated rate of return variance of the stock is 0.04. If these options expire in 25 days and the risk-free interest rate over that period is 5% per year, what do you advise your client to do?

Solution: We can apply the Black-Scholes formula, where S = $55, E = $50, σ = .2, T = 25/365, r = .05. We find thatC = $5.20. This is a lot less than $30, so clearly the options are not worth buying.

C 55 50, 5%, 20%, 25365

, ⎛⎜⎝

⎞⎟⎠

5.202=

20. Use the Black-Scholes model to infer the volatility of the returns on the underlying share of stock paying no cash dividend. Assume the following parameters: S = 90, E = 100, r = 10% , T = ½. What is σ if the call option premium is $3.05? What if the premium is $5.52?

Solution: root C 90 100, 10%, σ, 12

, ⎛⎜⎝

⎞⎟⎠

3.05− σ, ⎛⎜⎝

⎞⎟⎠

0.200= σ 20%=

root C 90 100, 10%, σ, 12

, ⎛⎜⎝

⎞⎟⎠

5.52− σ, ⎛⎜⎝

⎞⎟⎠

0.300= σ 30%=



21. Challenge Problem: Five years ago you purchased at face value a newly issued Zurich Insurance Corporation fixed-rate bond with a 5% annual coupon and a six-year maturity. The bond was structured with a put feature which allows you to exercise the option at a strike price of 98 one year before maturity, i.e., you have the option to sell the bond back to ZIC at the strike price. Currently the one-year yields on short-term bonds with similar credit risks are 8% and if you exercised the option you could take the proceeds and invest in the short-term bonds. Explain whether you should you exercise the option, i.e., sell the bond back to the company at 98 and invest the money for a year at 8% as opposed to keeping the bond until maturi ty? Given your decision, how would you calculate the effective annual rate of return have you earned on your six-year investment? How would you calculate the effective annual cost of funds paid by ZIC to finance its borrowings over the six-year period? A cash-flow diagram for ZIC’s borrowings and option dealings would be in order.

Solution: At the current point in time you have the option to put the bond back to the company and receive 98 or waitone year and receive the principal and final-year interest for a total of 105. There are two ways to look at it. Movingthe alternatives into the future, the 98 received from exercising the put can be invested at 8% over the coming yearand would grow to equal: 98 1 8%+( )⋅ 105.840= This is higher than holding the bond for the final year. Theother alternative is to discount the payments to a present value. The present value of receiving 105 in one year when

going interest rates are 8% is equal to: 105

1 8%+97.222= So it is better to put the bond back to ZIC and get 98.

Exercising the put option is best.

Assuming you paid par when the bond was first issued, we can find the IRR (internal rate of return) on your six-yearinvestment from the following equation:

100

1

4

t

5

1 IRR+( )t∑=

105.84

1 IRR+( )5+=

Which has the solution: root

1

4

t

5

1 IRR+( )t∑=

105.84

1 IRR+( )5+ 100− IRR,

⎡⎢⎢⎣

⎤⎥⎥⎦

0.0515= IRR 5.15%=

This is approximated by 5.2% or the average of the 5% return for 4 years and the 6% return for 1 year. The 6%one-year return is made up of the 8% interest less the 2% loss from face value when exercising the option at 98.

From the point of view of ZIC they repay the principal at a discount of 2% at the end of the fourth year and thenmust reborrow the principal and pay 8% on it over the final year. The IRR on the total borrowing is found in thefollowing equation:

100

1

3

t

5

1 IRR+( )t∑=

5 98+ 100−

1 IRR+( )4+

108

1 IRR+( )5+=

Which has the solution: root

1

3

t

5

1 IRR+( )t∑=

5 98+ 100−

1 IRR+( )4+

108

1 IRR+( )5+ 100− IRR,

⎡⎢⎢⎣

⎤⎥⎥⎦

0.0516=

IRR 5.16%=

This is approximated by 5.2% or the average of the 5% paid for 4 years and the 6% return for 1 year. The 6%one-year return is made up of the 8% interest less the 2% discount from face value when the option is exercised at98.



22. Suppose Notaboek Tablets Company has a current market value balance sheet with $250 million. Suppose the highly leveraged company has its entire debt financed with a single issue of zero-coupon bonds with aggregate face value of $240 million. These bonds mature in one year. Use the Black-Scholes model to calculate the aggregate market value of the firm’s equity if the riskless interest rate is 8 % and the standard deviation of the rate of return on the firm’s assets is 10% .

Solution: C 250 240, 8%, 10%, 1, ( ) 29.749=

23. Referring to the previous problem (problem 22), suppose the company is stunned to learn that the government has just canceled a major supply contract. If this immediately decreases the market value of Notaboek’s assets while also increasing the standard deviation of the rate of return on the firm’s assets. How do you expect this to impact on the equity value of the firm?

Solution: A decrease in the firm's assets will drop the equity value of the firm. For example if the market value of theassets were marked down by 10% to $225 million the equity value would fall to:

C 225 240, 8%, 10%, 1, ( ) 10.736=

An increase in the standard deviation of the rate of return on the firm's assets will raise the equity value of the firm.If for example the risk went up by 10% from 10% to 11% the equity value would rise to:

C 250 240, 8%, 11%, 1, ( ) 30.232=

So whether the equity value would on balance rise or fall depends on the magnitude of the effects.

24. Suppose Wally’s World is a small real estate developer with a total market value of assets equal to $100,000. The assets are financed with a commercial paper issue (short-term zero coupon bonds) paying a face value of $50,000 at maturity in 90 days. An appraisal of Wally’s real estate portfolio estimates the assets will be worth $170,000 if real estate zoning changes are granted in 90 days or only $45,000 if the changes are not approved. In effect Wally’s shareholders possess a 90-day European call option on the firm’s assets with an exercise price equal to the face value of the debt. Assume the riskless 90-day interest rate is 2.5 percent. Using the two-state option pricing model value the firm’s equity.

Solution: The aggregate value of the firm's equity in 90 days is E1 max V1 B− 0, ( )= where E1 and V1 are,

respectively, the aggregate values in 90 days of the firm's equity and assets, and where B is the aggregate face valueof the firm's debt.

The current value of the equity can be expressed as: E x V⋅ y−= where x is the fraction of the value of the firm thatone must purchase to replicate the payoffs from the equity, and y is the amount that must be borrowed. We find thevalues for x and y by setting up two equations, one for each of the possible payoffs of the equity 90 days from now:

170000 x⋅ y 1 r+( )⋅− 120000=45000x y 1 r+( )⋅− 0=

The solution to this set of two equations is x120125

:= = .96 and y432001 r+

:= where r is the risk-free 90-day interest

rate. Thus, we can replicate the equity by buying 96% of the firm's assets (at a cost of $96,000) and borrowing thepresent value of $43,200. The current value of the equity is therefore:

E 9600043200

1 2.5%+−:= E 53853.659=



25. Given the information from the previous problem (problem 24), what is the market price of a payment guarantee to the Wally’s World commercial paper holders?

Solution: We know the market value of the assets and equity of Wally's so the market value of the debt is:

100000 E− 46146.341=

The value of riskless bonds promising the same cash flows as Wally's would be: 500001 2.5%+

48780.488=

The difference is the value of the guarantee: 48780.49 46146.34− 2634.150=



Objectives How to use options to modify one’s exposure to investment risk. To understand the pricing relationships that exist among calls, puts, stocks, and bonds. To explain the binomial and Black-Scholes option-pricing models and apply them to the valuation of

corporate bonds and other contingent claims. To explore the range of financial decisions that can be fruitfully analyzed in terms of options.

Contents

15.1 How Options Work 15.2 Investing with Options 15.3 The Put-Call Parity Relation 15.4 Volatility and Option Prices 15.5 Two-State (Binomial) Option Pricing 15.6 Dynamic Replication and the Binomial Model 15.7 The Black-Scholes Model 15.8 Implied Volatility 15.9 Contingent Claims Analysis of Corporate Debt and Equity 15.10 Credit Guarantees 15.11 Other Applications of Option-Pricing Methodology

Summary

Options can be used to modify an investor’s exposure to investment risk. By combining the risk-free asset and stock-index call options, an investor can achieve a guaranteed minimum rate of return plus substantial upside participation in the stock market.

A portfolio consisting of a stock plus a European put option is equivalent to a riskless bond with a face value equal to the option’s exercise price plus a European call option. Therefore, by the Law of One Price, we get the put-call parity relation:

( )1 T

ES P Cr

+ =+

+ (15.1)

where S is the stock price, P the price of the put, r the riskless interest rate, T the maturity of the option, and C the price of the call.

One can create a synthetic option from the underlying stock and the riskless asset through a dynamic replication strategy that is self-financing after the initial investment. By the Law of One Price, the option’s price is given by the formula:

( ) ( )( ) ( )

1 2

2

1

2 1

1 / / 2

dT rTC N d Se N d Ee

n S E r d Td

Td d T

σ

σσ

− −= −

+ − +=

= −

(15.5)



where: = price of the call = price of the stock = exercise price = riskless interest rate (the annualized continuously compounded rate on a safe asset with the same maturity

as the option) = time to mat

CSEr

T urity of the option in years = standard deviation of the annualized continuously compounded rate of return on the stock = continuous dividend yield on the stock

ln = natural logarithm = the base of

d

e

σ

the natural log function (approximately 2.71828)( ) = the probability that a random draw from a standard normal distribution will be

less than N d

d

The same methodology used to price options can be used to value many other contingent claims, including corporate stocks and bonds, loan guarantees, and the real options embedded in investments in research and development and flexible manufacturing technology.


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