Giải bài tập môn thị trường tài chính phái sinh

Option Traders The market maker is an independent operator whose objective is to buy options at one price and sell them for a higher price.. Index Options If a call stock option is exerc

CHAPTER 1: INTRODUCTION

END-OF-CHAPTER QUESTIONS AND PROBLEMS

1 (Market Efficiency and Theoretical Fair Value) An efficient market is one in which prices reflect the

true economic values of the assets trading therein In efficient markets, no one can earn returns that aremore than commensurate with the level of risk Efficient markets are characterized by low transactioncosts and by the rapid rate at which new information is incorporated into prices

2 (Arbitrage and the Law of One Price) Arbitrage is a type of investment transaction that seeks to profit

when identical goods are priced differently Buying an item at one price and immediately selling it atanother is a type of arbitrage Because of the combined activities of arbitrageurs, identical goods,primarily financial assets, cannot sell for different prices for long This is the law of one price Arbitragehelps make our markets efficient by assuring that prices are in line with what they are supposed to be Inshort, we cannot get something for nothing A situation involving two identical goods or portfolios thatare not priced equivalently would be exploited by arbitrageurs until their prices were equal The "oneprice" that an asset must be is called the “theoretical fair value.”

3 (Arbitrage and the Law of One Price) The law of one price is violated if the same good is selling at

different prices On the surface it may appear as if that is the case; however, it is important to lookbeneath the surface to determine if the goods are identical Part of the cost of the good is convenienceand customer service Some consumers might be willing to pay more because the dealer is located in amore desirable section of town Also, the higher priced dealer may have a better reputation for serviceand customer satisfaction Buyers may be willing to pay more if they feel that the premium they payhelps assure them that they are getting a fair deal It is important to note that many goods are indeedidentical and, if so, they should sell at the same price, but the Law of One Price is not violated if the pricedifferential accounts for some economic value

4 (The Storage Mechanism) Storage is simply holding the asset Some assets, like commodities, require

considerable storage space and entail significant storage costs Others, like stocks and bonds, do notconsume much space but, as we shall see later, do incur costs Storage enables us to more adequatelymeet our consumption needs and, thus, provides for a more efficient alteration of our consumptionpatterns across time For example, we can store grains for the winter In the case of stocks and bonds,

we can store them and sell them later The proceeds from the sale of the securities can be used to meetconsumption needs at the later time Likewise, storage enables speculators to hold goods and securities

in the hope of selling them later at a profit In addition, storage plays an important role in defining therelationship between spot instruments and derivatives

5 (Delivery and Settlement) In futures markets, delivery seldom occurs Since delivery is always possible,

however, an expiring futures contract will be priced like the spot instrument The knowledge that futuresprices will eventually converge to spot prices is important to the pricing of futures contracts

6 (The Role of Derivative Markets) Derivative markets provide a means of adjusting the risk of spot

market investments to a more acceptable level and identifying the consensus market beliefs They maketrading easier and less costly and spot markets more efficient These markets also provide a means ofspeculating

7 (Criticisms of Derivatives Markets) On the surface, it may be difficult to distinguish speculation from

gambling Both entail high risk with the expectation of high gain The major difference that makesspeculation somewhat more socially acceptable is that it offers benefits to society not conveyed bygambling For example, speculators are necessary to assume the risk not wanted by others In gambling,there is no risk being hedged Gamblers simply accept risk without there being a concomitant reduction

in someone else's risk

8 (Misuses of Derivatives) Derivatives can be misused by speculating when one should be hedging, by not

having acquired the requisite knowledge to use them properly by acting irresponsibly when usingderivatives such as by being overly confident of one’s ability to forecast the direction of the market

9 (The Role of Derivative Markets) The existence of derivative markets in the United States economy and

indeed throughout most modern countries of the world undoubtedly leads to a much higher degree ofmarket efficiency Derivatives facilitate the activities of individual arbitrageurs so that unequal prices ofidentical goods are arbitraged until they are equal Because of the large number of arbitrageurs, this is aquick and efficient process Arbitrage on this large a scale makes markets less capable of beingmanipulated, less costly to trade in, and therefore more attractive to investors (The opportunity tohedge also makes the markets more attractive to investors in managing risk.) This is not to say that aneconomy without derivative markets would be inefficient, but it would not have the advantage of thisarbitrage on a large scale

It is important to note that the derivative markets do not necessarily make the U.S or world economy anylarger or wealthier The basic wealth, expected returns, and risks of the economy would be about thesame without these markets Derivatives simply create lower cost opportunities for investors to aligntheir risks at more satisfactory levels This may not necessarily make them wealthier, but to the extentthat it makes them more satisfied with their positions, it serves a valuable purpose

10 (Return and Risk) Return is the numerical measure of investment performance There are two main

measures of return, dollar return and percentage return Dollar return measures investment performance

as total dollar profit or loss For example, the dollar return for stocks is the dollar profit from the change

in stock price plus any cash dividends paid It represents the absolute performance Percentage return

measures investment performance per dollar invested It represents the percentage increase in theinvestor’s wealth that results from making the investment In the case of stocks, the return is thepercentage change in price plus the dividend yield The concept of return also applies to options, but, as

we shall see later, the definition of the return on a futures or forward contract is somewhat unclear

11 (Repurchase Agreements) A repurchase agreement (known as repos) is a legal contract between a

seller and a buyer, the seller agrees to sell a specified asset to the buyer currently as well as buy it backusually at a specified time in the future at an agreed future price The seller is effectively borrowingmoney from the buyer at an implied interest rate Typically, repos involve low risk securities, such as U

S Treasury bills Repos are useful because they provide a great deal of flexibility to both the borrowerand lender

Derivatives traders often need to be able to borrow and lend money in the most cost-effective mannerpossible Repos are often a very low cost way of borrowing money, particularly if the firm holdsgovernment securities Repos are a way to earn interest on short-term funds with minimal risk (forbuyers) and repos are a way to borrow for short-term needs at a relatively low cost (for sellers)

buyer and a seller—that gives the buyer the right, but not the obligation, to

purchase or sell something at a later date at a price agreed upon today The option buyer pays the seller a sum of money called the price or premium The option seller stands ready to sell or buy according to the contract terms if and when the buyer so

desires An option to buy something is referred to as a call; an option to sell

something is called a put

A forward contract is a contract between two parties—a buyer and a seller—to

purchase or sell something at a later date at a price agreed upon today A forward contract sounds a lot like an option, but an option carries the right, not the

obligation, to go through with the transaction If the price of the underlying good changes, the option holder may decide to forgo buying or selling at the fixed price

On the other hand, the two parties in a forward contract incur the obligation to ultimately buy and sell the good

13 (The Underlying Asset) Because all derivatives are based on the random performance of

something, the word “derivative” is appropriate The derivative derives its value

from the performance of something else That “something else” is often referred to

as the underlying asset The term underlying asset, however, is somewhat confusingand misleading For instance, the underlying asset might be a stock, bond, currency,

or commodity, all of which are assets However, the underlying “asset” might also

be some other random element such as the weather, which is not an asset It might even be another derivative, such as a futures contract or an option

CHAPTER 2: STRUCTURE OF OPTIONS MARKETS

END-OF-CHAPTER QUESTIONS AND PROBLEMS

1 (Option Price Quotations) The option is on AT&T stock It expires in January If it is an

exchange-listed option, it expires the Saturday following the third Friday in January The option is a call with anexercise price of $65 a share In other words, the option gives the right to buy AT&T stock at $65 a share

up to the expiration day in January

2 (Contract Size) a One contract would now cover 110 shares with an exercise price of 60/1.10

or 54.55 This would be rounded to the nearest eighth for 54.50

b Buyers and writers of outstanding contracts are credited with two contracts for every one

previously owned or written The exercise price is changed to 12.50 The contract size is still100

c One contract would now cover 100(4/3) or 133 shares with an exercise price of 85(3/4) or 63.75

Note: In the context of options, a 4-for-3 stock split is the same as a 33 percent stock dividend

d No changes to any contract terms

4 (Position and Exercise Limits) Short puts and long calls are both strategies designed to profit in a

bullish market Thus, they are considered to be "on the same side of the market."

5 (Option Traders) The market maker is an independent operator whose objective is to buy options at one

price and sell them for a higher price A broker is in business to generate commissions on eachtransaction A broker does not have to try to guess where the market is going or whether he can earn thebid-ask spread

CBOE rules allow an individual to be both a market maker and a floor broker but not on the same day.The reason is the potential for a conflict of interest For example, suppose a situation arises in which thetrader has to decide whether to execute a personal transaction or a customer transaction Whichevertransaction is done will bring large profits to the holder of the position The trader could obviously betempted to put personal interests ahead of the customer's interests The practice of trading as both amarket maker and a floor broker is called dual trading

6 (Order Book Official) Consider a limit order to buy an option at no more than 3 If there is no offer to

sell for 3 or less, the OBO takes the limit order, adds it to the other limit orders, and makes the highest

bids known to the traders If any market maker or broker is willing to lower the ask price to 3 or below,the OBO executes the order Because the price may not fall to 3, the limit order may never be filled.

7 (Other Option Trading Systems) In the market maker system, an individual trader who is not a broker

is required to be a market maker That is, the trader must be willing to quote a bid and an ask price oncertain options In the specialist system, there is an individual called the specialist who is charged withmaking a market in certain options In addition, there are registered option traders who trade on theirown but are not required to make a market The market maker system puts the role of the specialist andregistered option trader into one person, the market maker Exchanges using the specialist system claimthat it has the advantage of specialized expertise in keeping the market fair and orderly Proponents ofthe market maker system argue that because the specialist is a monopolist, the cost to the public is muchhigher than under the market maker system, which encourages market makers to compete with each otherfor the public's business

8 (Mechanics of Trading) Since each contract covers 100 shares, your 20 calls cover 2,000 shares Thus,

your premium is $4,500 You pay your premium to your broker Your broker's firm must clear its optiontrades through a clearing firm, which is a member of the Options Clearing Corporation (OCC) Yourbroker's firm sends the money to the clearing firm, which deposits it with the OCC The clearing firmdoes not actually have to deposit your money with the OCC It is allowed to consolidate its accounts andusing a predetermined formula, it deposits the required amount with the clearinghouse

9 (Index Options) If a call stock option is exercised, the writer delivers the stock to the buyer and receives

the exercise price If a put is exercised, the buyer delivers the stock to the writer and receives the exerciseprice If a call index option is exercised, the writer pays the buyer the difference between the stock priceand the exercise price For a put, the writer pays the buyer the difference between the exercise price andthe stock price The major advantage of exercising an index option rather than a stock option is nothaving to handle the stock This results in significantly lower transaction costs

10 (Option Price Quotations) Besides the fact that stock and option prices are already dated by the time

they appear in newspapers, these prices are not synchronized The prices shown are only the prices of thelast trade The last trade of the stock may not have taken place at the same time as the last trade of theoption In addition, the stock and option markets do not even close at the same time Moreover, theprices appearing in the newspapers do not indicate whether the last trade was at a bid price or an askprice Also, printed newspapers provide prices for only the most active options (though moreinformation can usually be obtained from the newspapers’ web sites) Web sites of the exchangesprovide much more current information and in some cases include information not provided by thenewspapers, such as the bid and ask prices

11 (Mechanics of Trading) An option position can be terminated by simply executing an offsetting order in

the market For example, suppose in January you bought a Microsoft March 90 call for 5 3/8 In themiddle of February it is selling for 6 1/4 and you would like to take your profit You simply sell aMicrosoft March 90 call, which offsets your long position An option can also be closed by exercising it.You would simply notify your broker that you want to buy the stock at the exercise price (if a call) or sell

it at the exercise price (if a put) The third way an option position can be terminated is by expiring of-the-money If it is not advantageous to exercise it by the expiration, the option simply expires andyour position is terminated In the over-the-counter market, you can certainly exercise the option or have

out-it expire out-of-the-money While you can effectively offset a posout-ition by opening up a new but opposout-itecontract, the procedure is technically somewhat different than in the exchange-listed options market Inthe latter market, the contracts cancel each other and no further obligation is incurred In the over-the-counter market, both contracts remain in force and consequently each is subject to default on the part ofthe writer

12 (Other Types of Options) Among the option-like instruments are warrants, convertibles and callable

bonds A warrant is an option offered by the firm on its own stock A convertible is a bond or preferredstock that can be converted into common stock at a fixed rate at the holder's discretion Callable bondsare bonds that can be retired early at a specific price at the discretion of the issuing firm In additionfirms issue options similar to warrants to executives and employees Finally, we should note that stockitself is like a call option held by the stockholders and written by the bondholders

13 (Real Options) Real options are options that corporations hold when they invest in certain projects and

includes options to expand projects, contract projects, temporarily shut them down, terminate them, orsell them to other companies These options do not trade in open markets like exchange-listed and over-the-counter options, but they possess the characteristics of ordinary options such as having an exerciseprice and an expiration

14 (Transaction Costs in Option Trading) Floor trading and clearing fees run from $0.50 to $1.00 These

represent the costs of paperwork involved in processing the trade as well as the exchange's overhead.Commissions, which reflect the cost of the labor involved in arranging the trade, vary and depend on thetype of broker (discount or full service) The bid-ask spread is the cost of providing liquidity to themarket The public and floor brokers representing the public incur all of these costs while market makersincur floor trading and clearing fees and may incur the bid-ask spread if they have to deal with othermarket makers instead of the public

Transactions in the OTC market do not generally incur commissions and floor trading and clearing fees.They do incur costs of paperwork and, in particular, the legal expenses of laying out the rights of eachparty Since transactions in the OTC market are generally executed through dealers, they incur thedealer's bid-ask spread

15 (Over-the-Counter Options Markets) Exchange-traded options are regulated by the Securities and

Exchange Commission There is essentially no regulation of OTC option transactions Firms that trade

in the OTC market, however, are typically regulated by the National Association of Securities Dealers or,

if they are banks, by banking regulators

APPENDIX 2A QUESTIONS AND PROBLEMS

e The stock price exceeds the exercise price so only 1,000($45)(0.5) or $22,500 can be borrowed

The call premium, however, can be applied so the investor must come up with only $27,500

-$7,000 = $20,500

f 20($500) = $10,000 One-hundred percent margin must be posted on all option purchase

transactions if the expiration is less than nine months

APPENDIX 2B QUESTIONS AND PROBLEMS

1 a $450 - $600 = -$150 The $150 loss applies against other taxable income and reduces

taxes by $150(0.28) = $42

b $650 - $600 = $50 The tax is $50(0.28) = $14

c The stock is treated as having been purchased for $25 + $6 = $31 The taxable gain is $3,500

-$3,100 = $400 The tax is $400(0.28) = $112

d The call would be exercised You deliver the stock and receive $25 for it The sale price of the

stock for tax purposes is $25 + $6 = $31 You purchased the stock at $30 The tax is ($3,100

-$3,000)(0.28) = $28

e The taxable gain is $600 - $350 = $250 The tax is $250(0.28) = $70

2 a Your loss is 100($15 - $12) = $300 This is netted against other gains for a tax saving

of $300(0.31) = $93 The after-tax profit is -$300 + $93 = -$207

b You exercise the call and receive 100($441.35 - $425) = $1,635 Your profit is $1,635 - $1,500

= $135 The tax is $135(0.244) = $32.94 The after-tax profit is $135 - $32.94 = $102.06

c The call expires worthless Your loss is $1,500 This is netted against other gains for tax

savings of $1,500(0.31) = $465 The after-tax profit is -$1,500 + $465 = -$1,035

d Taxes are paid at the end of the year on all trading profits whether the positions are closed out or

not Thus, in a and b., if the end of the year came before you sold or exercised the call, youwould owe taxes on any profits or be able to deduct any losses accumulated up to that time

3 a This would be a wash sale You replaced the stock with a call option within the 61-day

period The loss on the stock is not deductible for tax purposes

b This would be a wash sale because you acquired the call within a 61-day period surrounding the

sale of the stock It does not matter that you acquired the call before you sold the stock

c This is not a wash sale The wash sale rule pertains only to cases where the stock is sold at a

loss The rule prohibits deducting the loss In this case, the stock was sold at a gain so the washsale rule has no effect

CHAPTER 3: PRINCIPLES OF OPTION PRICING

END-OF-CHAPTER QUESTIONS AND PROBLEMS

1 (Basic Notation and Terminology) The average of the bid and ask discounts is 8.22.

Discount = 8.22(68/360) = 1.5527Price = 100 – 1.5527 = 98.4473Yield = (100/98.4473) (365/68) – 1 = 0.0876Note that even though the T-bill matured in 67 days, we must use 68 days since that is the option's time

to expiration

2 (Minimum Value of a Call) This would create an arbitrage opportunity The call would be purchased

and immediately exercised For example, suppose S0 = 44, X = 40, and the call price is $3 Then aninvestor would buy the call and immediately exercise it This would cost $3 for the call and $40 for thestock Then the stock would be immediately sold for $44, netting a risk-free profit of $1 In other words,the investor could obtain a $44 stock for $43 Since everyone would do this, it would drive the price of

the call up to at least $4 If the call were European, however, immediate exercise would not be possible(unless, of course, it was the expiration day), so the European call could technically sell for less than theintrinsic value of the American call We saw, though, that the European call has a lower bound of thestock price minus the present value of the exercise price (assuming no dividends) Since this is greaterthan the intrinsic value, the European call would sell for more than the intrinsic value Then atexpiration, it would sell for the intrinsic value.

3 (Lower Bound of a European Call) The call is underpriced, so buy the call, sell short the stock, and buy

risk-free bonds with face value of X The cash received from the stock is greater than the cost of the calland bonds Thus, there is a positive cash flow up front The payoffs from the portfolio at expiration are

4 (Effect of Time to Expiration) Time value is a measure of the amount of uncertainty in an

option Uncertainty relates to whether the option will expire in- or out-of-the-money When the option isdeep in-the-money, there is little uncertainty about the fact that the option will expire in-the-money Theoption will then begin to behave about like the stock When the option is deep-out-the-money, there isalso little uncertainty since it is likely to finish out-of-the-money When the option is at-the-money there

is considerable uncertainty about how it will finish

5 (Effect of Exercise Price) Assuming the stock pays no dividends, there is no reason to exercise a call

early (this obviously presumes the call is American) The tendency to believe that exercising an optionbecause the stock can go up no further ignores the fact that an option can generally be sold Exercising

an option throws away any chance that the stock can go up further If the stock falls, the option holderwould be hurt, but if the option holder exercised and became a stock holder, he would also be hurt by afalling stock price There is simply no reason to give away the time value that arises because of thepossibility that the stock can always go further upward In simple, mathematical terms, exercisingcaptures only the intrinsic value S0 – X The call can always be sold for at least S0 – X (1 + r) -T

6 (Effect of Stock Volatility) The paradox is resolved by recalling that if the option expires

out-of-the-money, it does not matter how far out-of-the-money it is The loss to the option holder is limited to thepremium paid For example, suppose the stock price is $24, the exercise price is $20, and the call price is

$6 Higher volatility increases the chance of greater gains to the holder of the call It also increases thechance of a larger stock price decrease If, however, the stock price does end up below $20, theinvestor's loss is the same regardless of whether the stock price at expiration is $19 or $1 If the stockwere purchased instead of the call, the loss would obviously be greater if the stock price went to $1 than

if it went to $19 For this reason, holders of stocks dislike volatility, while holders of calls like volatility

A similar argument applies to puts

7 (American Put Versus European Put) The minimum value of an American put is Max(0, X – S0) This

is always higher than the lower bound of a European put, X (1 + r) -T – S0, except at expiration when thetwo are equal

8 (Effect of Interest Rates) When buying a call option, one hopes to exercise it at a later date Thus, the

exercise price will be paid out later If interest rates are higher, additional interest can be earned on themoney that will eventually be paid out as the exercise price When buying a put option, one hopes toexercise it later, thus receiving the exercise price If interest rates are higher, the put is less valuablebecause the holder is foregoing interest by having to wait to exercise the put Higher interest rates makethe present value of the exercise price be lower In the case of the call, this is good because the callholder anticipates having to pay out the exercise price For the put holder this is bad because the putholder anticipates receiving the exercise price

9 (Put-Call Parity) If the put price is higher than predicted by the model, the put is overpriced Then the

put should be sold The funds should be used to construct a portfolio consisting of a long call, a shortposition in the stock, and a long position in risk-free bonds with face value of X The payoffs atexpiration of this strategy are shown below:

10 (Early Exercise of American Puts) An American call is exercised early only to capture a dividend.

When a stock goes ex-dividend, the call will lose value as the stock drops This will cause a loss in value

to the holder of the call The call holder knows this loss will be incurred as soon as the stock goes dividend If the call were exercised just before the stock goes ex-dividend, however, the call holderwould capture the stock and the dividend, which might be enough to offset the otherwise loss in the value

ex-of the call For a put, however, dividends are not necessary to make the argument that it might beoptimal to exercise early The holder of an American put faces a situation in which the gains are limited

to the exercise price Since the stock price can go down only to zero, early exercise of a put on abankrupt firm would obviously be advisable But the firm does not have to go bankrupt If the stockprice is low enough, the gains from waiting for it to go lower are not worth the wait If dividends wereadded to the picture, however, they would discourage early exercise The more dividends paid, the lowerthe stock price is driven and the more valuable it is to hold on to the put

11 (Principles of Call Option Pricing) a. July 160

Intrinsic value = Max (0, 165.13 – 160) = 5.13Time value = 6 – 5.13 = 0.87

Lower bound:

T = 11/365 = 0.0301(1 + r)-T = (1.0516)-0.0301 = 0.9985Lower bound = Max[0, 165.13 – 160(0.9985)] = 5.37

Lower bound:

T = 102/365 = 2795(1 + r)-T = (1.0588)-0.2795 – 1 = 0.9842Lower bound = Max[0, 165.13 – 155(0.9842)] = 12.579

Intrinsic value = Max(0, 165 – 165.13) = 0Time value = 2.35 – 0 = 2.35

Lower bound = Max(0, 165(0.9985) – 165.13) = 0

P + S0 = 2.75 + 165.13 = 167.88

C + X (1 + r)-T = 8.10 + 160(0.9933) = 167.023Difference = 0.857

P + S0 = 9 + 165.13 = 174.13

C + X (1 + r)-T = 6 + 170(0.9842) = 173.314Difference = 0.8160

These values are supposed to be zero If arbitrage could be executed at a cost less than the indicateddifference, it would be advisable to do so Consider the October 170 combination The difference of0.8160 suggests that a portfolio of short put, long call, short stock, and long risk-free bonds wouldgenerate a cash inflow of 0.8160 with no cash outflow at expiration, assuming, of course, that there is noearly exercise.

14 (Put-Call Parity) In each case we compute the value of C + X, P + S0, and C + X(1 + r)-T The values

should line up in descending order

C + X = 10.5 + 155 = 165.5

P + S0 = 0.20 + 165.13 = 165.33

C + X(1 + r)-T = 10.5 + 155(0.9985) = 165.2675These align correctly

C + X = 8.10 + 160 = 168.10

P + S0 = 2.75 + 165.13 = 167.88

C + X(1 + r)-T = 8.10 + 160(0.9933) = 167.028These align correctly

C + X = 6 + 170 = 176

P + S0 = 9 + 165.13 = 174.13

C + X(1 + r)-T = 6 + 170(0.9842) = 173.314These align correctly

15 (Effect of Exercise Price) The difference in premiums of American calls should not exceed the

difference in exercise prices

Neither pair represents a violation

16 (Effect of Exercise Price) The difference in premiums of American puts should not exceed the

difference in exercise prices

Neither pair represents a violation

17 (Principles of Call/Put Option Pricing) The time period is from December 9 to January 13 is 35 days.

So

T = 35/365 = 0.0959

a Intrinsic Value = Max(0, S0 – X)

18 (Lower Bound of a European Call) The most likely arbitrage opportunity is a violation of a lower

boundary condition With dividends, the lower boundary for a call is expressed as:

 S , T , X  Max  0 , S D X  1 r   Max  0 , 100 1 100  1 0 05   3 76

Note that the present value of the dividend is D, because the dividend is paid in an instant.Because the lower boundary is higher than the quoted call price of $3.5, there is an arbitrage opportunity.One method to assess the appropriate trading strategy is to rearrange the boundary condition such thatone side is greater than zero In this case, the boundary is non-zero and it is violated, therefore

 1 r  C  S , T , X  0 X

D

or in this case

1 0 05 3 5 0 26 0 100

1

To generate $0.26 in cash flow today, execute the trades suggested by their symbols, short sellstock, lend the present value of the strike price, and buy the call option In the next instant, the shortseller must pay the dividend One way to demonstrate that this is an arbitrage is to create a cash flowtable.

Strategy Today At Expiration: S T < X At Expiration: S T  X

19 (Lower Bound of a European Put) The most likely arbitrage opportunity is a violation of a lower

boundary condition With dividends, the lower boundary for a put is expressed as:

 S , T , X  Max  0 , X  1 r  S D  Max  0 , 100  1 0 05  90 1  6 24

Again we see that the present value of the dividend is D, because it occurs in an instant.Because the lower boundary is higher than the quoted call price of $6, there is an arbitrage opportunity.One method to assess the appropriate trading strategy is to rearrange the boundary condition such thatone side is greater than zero In this case, the boundary is non-zero and it is violated, therefore

Strategy Today At Expiration: S T < X At Expiration: S T  X

its price up and buying pressure on the put will drive its price up Hence, this trading activity willeventually eliminate the arbitrage opportunity.

20 (Lower Bound of a European Call) Recall

0 0

1 r 0X

S ln

 1 r 

ln T X

S

ln   0    

Thus

10 0 1 ) 095310

0 exp(

1 1 110

100 ln exp 1 T X

S ln exp

S0 = Ce(S0,T,X) – Pe(S0,T,X) + X(1 + r) -T

So the equivalent of a position in a "point" would be to buy a call, sell a put and invest X(1+r)-T dollars inrisk-free bonds, paying X dollars at the end of the game This is demonstrated below where ST is thenumber of points at the end of the game

in-the-At any given time, the above formula for S0 must equal the number of points Thus at the beginning ofthe game, S0 = 0 and

Ce(S0,T,X) = Pe(S0,T,X) – X(1 + r)-T

23 (Put-Call Parity) Remember that put call parity says that Pe(S0,T,X) + S0 = Ce(S0,T,X) + X(1 + r)-T To

isolate a short position in the stock, we need to get – S0 on one side of the equation This would be

Pe(S0,T,X) – Ce(S0,T,X) – X(1 + r)-T = – S0 So to do the equivalent of a short sale you would buy a put,sell a call and borrow the present value of the exercise price The table below demonstrates this point

Short call 0 –(S T – X)

CHAPTER 4: OPTION PRICING MODELS: THE BINOMIAL MODEL

END-OF-CHAPTER QUESTIONS AND PROBLEMS

1 (Two-Period Binomial Model) In a recombining tree, if the underlying first moves up and then down, it

will be at the same price as in the case when it first moves down and then up In a non-recombining tree,the underlying will not be at the same price in those two cases A recombining tree will have far fewerpossible prices of the underlying For a recombining tree of n time steps, there will be n + 1 possiblefinal prices For a non-recombining tree of n time steps, there will be 2n possible final prices It is farmore practical to work with a recombining tree, because the number of time steps is likely to be a moremanageable number

2 (One-Period Binomial Model) The up and down factors reflect a spread between the next two possible

stock prices That spread is an indication of the volatility It is easy to see that if we increase u and/ordecrease d, we increase the volatility

3 (Extending the Binomial Model to n Periods) If the up and down parameters were not adjusted, the

stock would have unreasonable volatility For example, suppose u = 1.25 and d = 0.80 and the period oftime is one year Then one year later, a $100 stock would be at $125 or $80 If we now went to a two-period model, dividing the option’s life into two six-month periods, we could not maintain u and d attheir current values Otherwise, the stock could get as high as $100(1.25)2 = $156.25 or as low as

$100(0.80)(0.80) = $64 If, however, we are letting the binomial period be one year but then want towork with a two-year option, we would maintain u and d at 1.25 and 0.80 since it would then be realistic

to allow values as high as $156.25 and $64 over two years While this problem might not appear to bethat significant in a one- or two-period situation, if we are using many periods, we are admitting anunreasonable degree of volatility if we do not adjust u and d

4 (Dividends) There are two ways using discrete dividends and one way using continuous dividends For

discrete dividends, we can specify that the dividend is a constant percentage of the stock price At eachtime step, the stock price moves up or down and then makes an immediate fall by the amount of thedividend In other words, let us say a $100 stock could move up or down by 10 percent and that there is a

5 percent dividend So the next period the stock moves up to $110 or down to $90 Without leaving thattime point, however, it immediately drops to 0.95($110) = $104.50 or 0.95($90) = $85.50 So wereplaced $110 by $104.50 and $90 by $85.50 The tree would still recombine From $104.50 it couldmove down to (0.90)$104.50 = $94.05, and from $85.50, it could move up to (1.10)($85.50) = $94.05.Alternatively, we could specify that the dividend is a fixed dollar amount Unfortunately, if we do it this way, the tree will no longer recombine That is, up followed by down is no longer the same as down followed by up For example, in the above case, let the dividend just be $5 Then the stock goes to $110 or $90 Replace these values by the ex-dividend values of $105 and $85 Now let it move again up

10 percent or down 10 percent Then from $105 it would go down to 0.90($105) = $94.50 From $85 it would move up to (1.10)($85) =

$93.50.

An alternative approach that would handle this problem is to let the up and down factors apply only to thestock price minus the present value of the dividends Let the risk-free rate be 5% Then the presentvalue of the $5 dividend is $5/1.05 = $4.76 Now the stock price minus the present value of thedividends is $100 – $4.76 = $95.24 It can now go up to $95.24(1.10) = $104.76 or down to $95.24(0.90)

= $85.72 At those points the actual stock price is really $104.76 + $5 = $109.76 and $85.72 + $5 =

$90.72 Then when the stock goes ex-dividend, the price falls to $104.76 or $85.72 Then from $104.76

it can go down to (0.90)$104.76 = $94.28 From $85.72 it can go up to (1.10)($85.72) = $94.29, with thedifference being only a rounding error

5 (One-Period Binomial Model) a 25(1.15) = 28.75 25(0.85) = 21.25

Max(0, 21.25 – 25) = 0

c p = (1.10 – 0.85)/(1.15 – 0.85) = 0.8333, 1 – p = 0.1667

2.841.10

(0.1667)075

45(1.10)(

Sud

36.4545(.9)

Sd40.5,45(.9)

Sd

54.4545(1.10)

Su49.5,45(1.10)

Su45,S

2 2

2 2

4.5540)

5Max(0,44.5C

14.4540)

5Max(0,54.4C

2 2

d ud u

0(.25))/1

(4.55(.75)

r)p))/(1(1

Cp(CC

11.40/1.05

4.55(.25)))

(14.45(.75

r)p))/(1(1

Cp(CC

.25.751p1

.75.9).9)/(1.10(1.05

d)d)/(ur)

((1p

2 2

d ud d

ud u

3.25(.25)))

(11.40(.75

r)p))/(1(1

Cp(C

)25.340.11(

Sd))/(SuC(C

)55.445.14(

Sud))/(Su

C(C

h Buy 906 shares and write 1,000 calls

Value of portfolio today:

906(45) – 1,000(8.92) = 31,850Value of portfolio one period later:

If stock goes up,

906(49.50) – 1,000(11.40) = 33,447

If the stock goes down,

906(40.50) – 1,000(3.25) = 33,443These two amounts are essentially equivalent

Return over one period = (33,447/31,850) – 1 ≈ 0.05

If stock goes up to 49.5, hu = 1 Then buy 94 calls at 11.40 for $1,072 Borrow the money at therisk-free rate Now you have 906 shares and 906 calls, which is a hedge ratio of 1 Yourportfolio is:

33,447Value of portfolio one period later:

If stock goes up,

.561736.45)

0)/(44.55(4.55

)Sd)/(SudC

33,443Value of the portfolio at the end of the second period:

If stock goes up,

562(44.55) – 1,000(4.55) + 13,932(1.05) = 35,116

If stock goes down,

562(36.45) + 13,932(1.05) = 35,114The difference between 35,114 and 35,116 is due to rounding

Return over one period = 35 , 114 / 31 , 850– 1 ≈ 0.05

If it were overpriced, the investor should establish the same riskless hedge by buying 906 sharesand writing 1,000 calls If it were underpriced, the investor should buy 1,000 calls and sell short

906 shares This would create a type of loan in which money is received today and paid backlater The effective rate on the loan would be less than the risk-free rate

1

)39.2(4.)

1

)0(4.)19

25-34.50for call

the

exercise

Sodividend.-

exgoes

it before34.50

30(1.15)

at isstock thestate, top

.4(4.19)+

=

C

4.19

=25)-29.19Max(0,

=C12.29,

=25)-37.29Max(0,

=

C

29.19)

25.38(1.15

or 32.43(.9)

=

S

22.84

=25.38(.9)

=Sd37.29,

=)32.43(1.15

=

Su

25.38

=.06)-30(.9)(1

=Sd32.43,

=.06)-30(1.15)(1

=

Su

u u

d

ud u

ud

2 2

Sud

55.9658.90(.95)

Sd

75.02)

68.20(1.10Su

58.9062(.95)

Sd

68.2062(1.10)

.13(5.21)+

.87(0)

=P

14.04

=55.96)-

70Max(0,

=

P

5.21

=64.79)-

70Max(0,

=P

0

=75.02)-

70Max(0,

=P

.8667

=.95)-.95)/(1.10-

.13(14.04)+

=1.08

.13(11.10)+

.87(1.80)

=P

but worth Max(0,70 – 62) = 8 if exercised so P = 8.

38.4060(0.80)

Sd

55.20.80)

60(1.15)(0Sud

79.3560(1.15)

Su

4860(0.80)Sd

6960(1.15)Su

2 2

2 2

5.2050)0Max(0,55.2C

29.3550)

5Max(0,79.3C

.8570.80)50.80)/(1.1(1.10

p

2 2

d ud u

.143(4.05))

.857(23.54C

4.051.10

.143(0.0).857(5.20)

C

23.541.10

.143(5.20))

.857(29.35C

d u

0.05.20h

1.0055.2079.35

5.2029.35h

.92848

69

4.0523.54h

At time 2 when the stock goes from 69 to 79.35, the portfolio is worth

At time 1 when the stock is 48, the portfolio is worth

928(48) – 1,000(4.05) = 40,494The new hedge ratio is 0.310 Let us sell the shares to generate 618(48) = 29,664 and invest this in bonds Our position is now 310 shares,

1000 short calls and 29,664 invested in bonds.

At time 2 when the stock goes from 48 to 55.20, the portfolio is worth

11 (Extending the Binomial Model to n Periods)

Inserting the proper values into the spreadsheet gives the following:

Adjusting the risk-free rate: r = (1.0456)0.0959/2 – 1 = 0.0021

The up and down factors:

0.83381/1.1993

d

1.1993e