CHAPTER 1: INTRODUCTION END-OF-CHAPTER QUESTIONS AND PROBLEMS (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 are more than commensurate with the level of risk Efficient markets are characterized by low transaction costs and by the rapid rate at which new information is incorporated into prices (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 at another 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 Arbitrage helps make our markets efficient by assuring that prices are in line with what they are supposed to be In short, we cannot get something for nothing A situation involving two identical goods or portfolios that are not priced equivalently would be exploited by arbitrageurs until their prices were equal The "one price" that an asset must be is called the “theoretical fair value.” (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 look beneath the surface to determine if the goods are identical Part of the cost of the good is convenience and customer service Some consumers might be willing to pay more because the dealer is located in a more desirable section of town Also, the higher priced dealer may have a better reputation for service and customer satisfaction Buyers may be willing to pay more if they feel that the premium they pay helps assure them that they are getting a fair deal It is important to note that many goods are indeed identical and, if so, they should sell at the same price, but the Law of One Price is not violated if the price differential accounts for some economic value (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, not consume much space but, as we shall see later, incur costs Storage enables us to more adequately meet our consumption needs and, thus, provides for a more efficient alteration of our consumption patterns 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 meet consumption 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 the relationship between spot instruments and derivatives (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 futures prices will eventually converge to spot prices is important to the pricing of futures contracts (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 make trading easier and less costly and spot markets more efficient These markets also provide a means of speculating (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 makes speculation somewhat more socially acceptable is that it offers benefits to society not conveyed by gambling 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 Chapter 1 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part (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 using derivatives such as by being overly confident of one’s ability to forecast the direction of the market (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 of market efficiency Derivatives facilitate the activities of individual arbitrageurs so that unequal prices of identical goods are arbitraged until they are equal Because of the large number of arbitrageurs, this is a quick and efficient process Arbitrage on this large a scale makes markets less capable of being manipulated, less costly to trade in, and therefore more attractive to investors (The opportunity to hedge also makes the markets more attractive to investors in managing risk.) This is not to say that an economy without derivative markets would be inefficient, but it would not have the advantage of this arbitrage on a large scale It is important to note that the derivative markets not necessarily make the U.S or world economy any larger or wealthier The basic wealth, expected returns, and risks of the economy would be about the same without these markets Derivatives simply create lower cost opportunities for investors to align their risks at more satisfactory levels This may not necessarily make them wealthier, but to the extent that 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 the investor’s wealth that results from making the investment In the case of stocks, the return is the percentage 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 back usually at a specified time in the future at an agreed future price The seller is effectively borrowing money 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 borrower and lender Derivatives traders often need to be able to borrow and lend money in the most cost-effective manner possible Repos are often a very low cost way of borrowing money, particularly if the firm holds government securities Repos are a way to earn interest on short-term funds with minimal risk (for buyers) and repos are a way to borrow for short-term needs at a relatively low cost (for sellers) 12 (Derivative Markets and Instruments) An option is a contract between two parties—a 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 Chapter End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part On the other hand, the two parties in a forward contract incur the obligation to ultimately buy and sell the good Chapter End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 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 confusing and 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 (Option Price Quotations) The option is on AT&T stock It expires in January If it is an exchangelisted option, it expires the Saturday following the third Friday in January The option is a call with an exercise 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 (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 still 100 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 (Expiration Dates) Jan cycle a b c Feb, Mar, Apr, Jul Jul, Aug, Oct, Jan Dec, Jan, Apr, Jul Feb cycle Feb, Mar, May, Aug Jul, Aug, Nov, Feb Dec, Jan, Feb, May March cycle Feb, Mar, Jun, Sep Jul, Aug, Sep, Dec Dec, Jan, Mar, Jun (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." (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 each transaction A broker does not have to try to guess where the market is going or whether he can earn the bid-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 the trader has to decide whether to execute a personal transaction or a customer transaction Whichever transaction is done will bring large profits to the holder of the position The trader could obviously be tempted to put personal interests ahead of the customer's interests The practice of trading as both a market maker and a floor broker is called dual trading (Order Book Official) Consider a limit order to buy an option at no more than If there is no offer to sell for or less, the OBO takes the limit order, adds it to the other limit orders, and makes the highest Chapter End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part bids known to the traders If any market maker or broker is willing to lower the ask price to or below, the OBO executes the order Because the price may not fall to 3, the limit order may never be filled (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 on certain options In the specialist system, there is an individual called the specialist who is charged with making a market in certain options In addition, there are registered option traders who trade on their own but are not required to make a market The market maker system puts the role of the specialist and registered option trader into one person, the market maker Exchanges using the specialist system claim that it has the advantage of specialized expertise in keeping the market fair and orderly Proponents of the market maker system argue that because the specialist is a monopolist, the cost to the public is much higher than under the market maker system, which encourages market makers to compete with each other for the public's business (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 option trades through a clearing firm, which is a member of the Options Clearing Corporation (OCC) Your broker's firm sends the money to the clearing firm, which deposits it with the OCC The clearing firm does not actually have to deposit your money with the OCC It is allowed to consolidate its accounts and using a predetermined formula, it deposits the required amount with the clearinghouse (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 exercise price If a call index option is exercised, the writer pays the buyer the difference between the stock price and the exercise price For a put, the writer pays the buyer the difference between the exercise price and the stock price The major advantage of exercising an index option rather than a stock option is not having 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 the last trade The last trade of the stock may not have taken place at the same time as the last trade of the option In addition, the stock and option markets not even close at the same time Moreover, the prices appearing in the newspapers not indicate whether the last trade was at a bid price or an ask price Also, printed newspapers provide prices for only the most active options (though more information can usually be obtained from the newspapers’ web sites) Web sites of the exchanges provide much more current information and in some cases include information not provided by the newspapers, 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 3/8 In the middle of February it is selling for 1/4 and you would like to take your profit You simply sell a Microsoft 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 out-of-the-money If it is not advantageous to exercise it by the expiration, the option simply expires and your position is terminated In the over-the-counter market, you can certainly exercise the option or have it expire out-of-the-money While you can effectively offset a position by opening up a new but opposite contract, the procedure is technically somewhat different than in the exchange-listed options market In the 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 of the 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 preferred stock that can be converted into common stock at a fixed rate at the holder's discretion Callable bonds are bonds that can be retired early at a specific price at the discretion of the issuing firm In addition Chapter End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part firms issue options similar to warrants to executives and employees Finally, we should note that stock itself 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, or sell them to other companies These options not trade in open markets like exchange-listed and overthe-counter options, but they possess the characteristics of ordinary options such as having an exercise price 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 the type of broker (discount or full service) The bid-ask spread is the cost of providing liquidity to the market The public and floor brokers representing the public incur all of these costs while market makers incur floor trading and clearing fees and may incur the bid-ask spread if they have to deal with other market makers instead of the public Transactions in the OTC market not generally incur commissions and floor trading and clearing fees They incur costs of paperwork and, in particular, the legal expenses of laying out the rights of each party Since transactions in the OTC market are generally executed through dealers, they incur the dealer'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 a 10[0.2($5,000) + $700] = $17,000 b 10[0.2($5,000) - ($5,500 - $5,000) + $300] = $8,000 This amount equals the minimum of 10[$300 + 0.1($5,000)] = $8,000 c 10[0.2($5,000) - ($5,000 - $4,500) + $300] = $8,000 This amount exceeds the minimum of 10[$300 + 0.1($4,500)] = $7,500 d 10[0.2($5,000) + $700] = $17,000 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 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 Chapter End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 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 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, you would owe taxes on any profits or be able to deduct any losses accumulated up to that time 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 wash sale rule has no effect CHAPTER 3: PRINCIPLES OF OPTION PRICING END-OF-CHAPTER QUESTIONS AND PROBLEMS (Basic Notation and Terminology) The average of the bid and ask discounts is 8.22 Discount = 8.22(68/360) = 1.5527 Price = 100 – 1.5527 = 98.4473 Yield = (100/98.4473) (365/68) – = 0.0876 Note that even though the T-bill matured in 67 days, we must use 68 days since that is the option's time to expiration (Minimum Value of a Call) This would create an arbitrage opportunity The call would be purchased and immediately exercised For example, suppose S = 44, X = 40, and the call price is $3 Then an investor would buy the call and immediately exercise it This would cost $3 for the call and $40 for the stock 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 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 the intrinsic value of the American call We saw, though, that the European call has a lower bound of the stock price minus the present value of the exercise price (assuming no dividends) Since this is greater than the intrinsic value, the European call would sell for more than the intrinsic value Then at expiration, it would sell for the intrinsic value (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 Chapter End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part call and bonds Thus, there is a positive cash flow up front The payoffs from the portfolio at expiration are as follows: Transaction Long call Short stock Long bonds Total X Value at Expiration ST < X ST ≥ X ST – X –S T –S T X X – ST Observe that in the case where S T < X, the payoff from this portfolio is X – S T, which is positive In the other outcome, the payoff is zero Thus, this portfolio produces either a positive cash flow at expiration or no cash flow at all Yet, it also produces a positive cash flow at the start So, there is no way to lose money and a guaranteed gain of money at the start with a possibility of more at expiration (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 is deep in-the-money, there is little uncertainty about the fact that the option will expire in-the-money The option will then begin to behave about like the stock When the option is deep-out-the-money, there is also 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 (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 option because 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 holder would be hurt, but if the option holder exercised and became a stock holder, he would also be hurt by a falling stock price There is simply no reason to give away the time value that arises because of the possibility that the stock can always go further upward In simple, mathematical terms, exercising captures only the intrinsic value S – X The call can always be sold for at least S – X (1 + r) -T (Effect of Stock Volatility) The paradox is resolved by recalling that if the option expires out-of-themoney, it does not matter how far out-of-the-money it is The loss to the option holder is limited to the premium 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 the chance of a larger stock price decrease If, however, the stock price does end up below $20, the investor's loss is the same regardless of whether the stock price at expiration is $19 or $1 If the stock were 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 (American Put Versus European Put) The minimum value of an American put is Max(0, X – S 0) This is always higher than the lower bound of a European put, X (1 + r) -T – S0, except at expiration when the two are equal (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 the money that will eventually be paid out as the exercise price When buying a put option, one hopes to exercise it later, thus receiving the exercise price If interest rates are higher, the put is less valuable because the holder is foregoing interest by having to wait to exercise the put Higher interest rates make the present value of the exercise price be lower In the case of the call, this is good because the call holder anticipates having to pay out the exercise price For the put holder this is bad because the put holder anticipates receiving the exercise price (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 short Chapter End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part position in the stock, and a long position in risk-free bonds with face value of X The payoffs at expiration of this strategy are shown below: Transaction Short put Long call Short stock Long bonds Total Value at Expiration ST < X ST ≥ X –(X – ST) 0 ST – X –S T –S T X X 0 Thus, this portfolio has no cash inflow or outflow at expiration, but the sale of the put will produce more cash than the cost to buy the combination of long call, short stock, and long bonds that replicates the purchase of a put 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 ex-dividend If the call were exercised just before the stock goes ex-dividend, however, the call holder would capture the stock and the dividend, which might be enough to offset the otherwise loss in the value of the call For a put, however, dividends are not necessary to make the argument that it might be optimal 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 a bankrupt firm would obviously be advisable But the firm does not have to go bankrupt If the stock price is low enough, the gains from waiting for it to go lower are not worth the wait If dividends were added to the picture, however, they would discourage early exercise The more dividends paid, the lower the 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) = 5.13 Time value = – 5.13 = 0.87 Lower bound: T = 11/365 = 0.0301 (1 + r) -T = (1.0516) -0.0301 = 0.9985 Lower bound = Max[0, 165.13 – 160(0.9985)] = 5.37 b October 155 Intrinsic value = Max(0, 165.13 – 155) = 10.13 Time value = 14 – 10.13 = 3.87 Lower bound: T = 102/365 = 2795 (1 + r) -T = (1.0588) -0.2795 – = 0.9842 Lower bound = Max[0, 165.13 – 155(0.9842)] = 12.579 c August 170 Intrinsic value = Max(0, 165.13 – 170) = Time value = 3.20 – = 3.20 Lower bound: T = 46/365 = 0.1260 (1 + r) -T = (1.0550) -0.1260 = 0.9933 Lower bound = Max[0, 165.13 – 170(0.9933)] = All prices conform to the boundary conditions, so there are no profitable opportunities Chapter End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 12 (Principles of Put Option Pricing) a July 165 Intrinsic value = Max(0, 165 – 165.13) = Time value = 2.35 – = 2.35 Lower bound = Max(0, 165(0.9985) – 165.13) = b August 160 Intrinsic value = Max(0, 160 – 165.13) = Time value = 2.75 – = 2.75 Lower bound = Max(0, 160(0.9933) – 165.13) = c October 170 Intrinsic value = Max(0, 170 – 165.13) = 4.87 Time value = – 4.87 = 4.13 Lower bound = Max(0, 170(0.9842) – 165.13) = 2.184 All prices conform to the boundary conditions, so there are no profitable opportunities 13 (Put-Call Parity) In each case, we compute the value of P + S and compare it to the value C + X (1 + r)-T a July 155 P + S0 = 0.20 + 165.13 = 165.33 C + X (1 + r) -T = 10.5 + 155(0.9985) = 165.2675 Difference = 0625 b August 160 P + S0 = 2.75 + 165.13 = 167.88 C + X (1 + r) -T = 8.10 + 160(0.9933) = 167.023 Difference = 0.857 c October 170 P + S0 = + 165.13 = 174.13 C + X (1 + r) -T = + 170(0.9842) = 173.314 Difference = 0.8160 These values are supposed to be zero If arbitrage could be executed at a cost less than the indicated difference, it would be advisable to so Consider the October 170 combination The difference of 0.8160 suggests that a portfolio of short put, long call, short stock, and long risk-free bonds would generate a cash inflow of 0.8160 with no cash outflow at expiration, assuming, of course, that there is no early exercise 14 (Put-Call Parity) In each case we compute the value of C + X, P + S 0, and C + X(1 + r) -T The values should line up in descending order a July 155 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) = 165.2675 These align correctly b Chapter August 160 C + X = 8.10 + 160 = 168.10 10 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part d1 = ln(0.4712/46) + (.071 + (0.145)2 /2)(0.0959) 0.145 0.0959 = 0.7098 d = 0.7098 − 0.145 0.0959 = 0.6649 Look up the normal probabilities: N(0.71) = 0.7611 N(0.66) = 0.7454 Then calculate the option price: c = 0.4712(0.7611) – 0.46e -0.071(0.0959)(0.7454) = 0.0181 The call should be selling for $0.0181 so it is underpriced at $0.0163 16 (Managing the Risk of Options) Plugging into the Black-Scholes-Merton model S = 100, X = 100, r c = 0.045, T = 1, σ = 40, we obtain C = 17.80 and a delta of 0.6227 Sell 10,000 calls and buy 6,227 shares The value of the portfolio is now V = 6,227(100) – 10,000(17.80) = 444,700 a Now at the end of Day 1, S = 99, T = 364/365 = 0.9973 Plugging into the Black-Scholes-Merton model, we obtain C = 17.1559 Our portfolio is now worth V1 = 6,227(99) – 10,000(17.1559) = 444,914 The new delta from the Black-Scholes-Merton model would be 0.6129 So we need 6,129 shares We sell 6,227 – 6,129 = 98, raising 98(99) = 9,702, which we invest in risk-free bonds Now at the end of Day 2, S = 102, T = 363/365 = 0.9945 Plugging into the Black-Scholes-Merton model we obtain 19.0108 Our portfolio is now worth V2 = 6,129(102) – 10,000(19.0108) + 9,702e 0.045(1/365) = 444,753 The amount we should have had is the initial amount invested plus two days interest: 444,700e 0.045(2/365) = 444,810 The target was not achieved because a delta hedge works well only for very small changes in the stock price The discrepancy between the amount we obtained and the target is due to the gamma risk, which reflects large stock price moves b Going back to Day 0, on the original option, we would obtain a gamma of 0.0095 Now we add a new option Its Black-Scholes-Merton value is 15.6929, its delta is 0.5756 and its gamma is 0.0098 Remember that we have 10,000 of the first option The number of the second is 10,000(0.0095/0.0098) = 9,694 The number of shares of stock we need is 10,000(0.6227 – (0.0095/0.0098).5756) = 647 So now our portfolio value is Chapter 29 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part V1 = 647(100) – 10,000(17.80) + 9,694(15.6929) = 38,827 Now on Day 1, the stock is at 99 From the Black-Scholes-Merton formula, the first call price is 17.1559, its delta is 0.6129 and its gamma is 0.0097 The second call is at 15.0958, its delta is 0.5654 and its gamma is 0.0100 The value of the portfolio is V2 = 647(99) – 10,000(17.1559) + 9,694(15.0958) = 38,833 Now we revise the number of the second call to 10,000(0.0097/0.0100) = 9,700 We revise the number of shares to 10,000(0.6129 – (0.0097/.0100).5654) = 645 So we need to buy 9,700 – 9,694 = more of the second call and sell 647 – 645 = shares This will generate 2(99) – 6(15.0958) = 107, which we invest in bonds Now on Day 2, the stock is at 102 The Black-Scholes-Merton value is 19.0108 for the first call and 16.8092 for the second The portfolio is now worth V2 = 644(102) – 10,000(19.0108) + 9,700(16.8092) + 107e 0.045(1/365) = 38,838 The amount we should have is 38,827 plus two days interest: 38,827e0.045(2/365) = 38,837 We see that we have nearly a perfect hedge The target was achieved because gamma hedging is able to eliminate risk caused by the larger price movement better than delta hedging alone 17 (A Nobel Formula) Using either software package and the inputs S = 82, X = 80, r = 0.04 (continous), σ = 0.3, and T = 1, the current call price is 15.32 If the option pays off 150% of the value of an ordinary option on this stock, then it is equivalent to an option on a stock currently priced at 82(1.50) = 123 with an exercise price of 80(1.5) = 120 That is, the ordinary option pays S T – X or This new option pays 1.50(S T – X) or 1.5(0) = This is just like multiplying the stock price and exercise price by 1.5 Plugging these values (123 for stock price and 120 for exercise price) into the model gives a call option value of 22.98, which is 150% of the value of the ordinary option 18 (A Nobel Formula) Using either software package and the inputs of S = 100, X = 100, r = 0.05 (continuous), σ = 0.3, and T = 1, the current call option price is $14.2312 and the put option price is $9.3542 After the stock split, the inputs are S = 50, X = 50, r = 0.05 (continuous), σ = 0.3, and T = 1, and the current call price is $7.1156 and the current put price is $4.6771 Because the number of options contracts are doubled, the original option holders are in the same financial position as before 19 (Exercise Price) Using BSMbin8e.xls, the call and put prices for a stock option are the same when the current stock price is $100, the exercise price is $105.1271, the risk-free interest rate is percent (continuously compounded), the volatility is 30 percent, and the time to expiration is year This result is based on put-call parity Recall that the difference between the call and put price (assuming the same underlying asset, same exercise price, and same time to maturity) should be the current stock price minus the present value of the exercise price Notice in this case that the present value of the exercise price is $100 Therefore, the difference between the put and call price is zero ($11.94 in each case) Chapter 30 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 20 (Risk-Free Rate) Using BSMbin8e.xls, the call and put prices for a stock option are the same when the current stock price is $100, exercise price is $100, risk-free interest rate is percent (continuously compounded), volatility is 30 percent, and time to expiration is year This result is based on put-call parity As with the previous problem, the difference between the call and put price (assuming the same underlying asset, same exercise price, and same time to maturity) should be the current stock price minus the present value of the exercise price Notice in this case that the present value of the exercise price is $100 when the interest rate is zero Therefore, the difference between the put and call price is zero ($11.94 in each case) Notice that moving the exercise price lower based on the present value function as the interest rate is lower results in the same call and put prices Therefore, interest rates play an important, but secondary, role in option valuation 21 (Stock Price) The value of h shares of stock and one put is V = hS + P The change in the value of the portfolio is ∆V = h∆S + ∆P Divide by ∆S and set to zero to establish a hedge: ∆V/∆S = h(∆S/∆S) + (∆P/∆S) = Solve for h: h = –∆P/∆S This is the put delta From the chapter we know that the put delta equals N(d 1) – So h = –(N(d 1) – 1) = – N(d 1) Since N(d 1) is less than 1, then h > That means we will buy stock This should make sense If we are long puts, we need to buy stock to hedge because the put will lose money when the stock price rises 22 (A Nobel Formula) a Inserting the numbers into BSMbin8e.xls gives a call value of 6.0544 b The call is underpriced so buy it, thereby investing $500 c The call price is calculated below using a time to expiration of T = 2/12 = 0.167, the stock price shown on the same line, and the same other inputs Stock price 60 70 80 90 100 0.0942 1.1151 4.8772 11.9136 20.9455 Call price Profit 100(0.0942 – 5) = –490.58 100(1.1151 – 5) = –388.49 100(4.8772 – 5) = –12.28 100(11.9136 – 5) = 691.36 100(20.9455 – 5) = 1594.55 CHAPTER 8: THE STRUCTURE OF FORWARD AND FUTURES MARKETS END-OF-CHAPTER QUESTIONS AND PROBLEMS (General Classes of Futures Traders) Locals are in business for themselves They attempt to profit by buying at low prices and selling at high prices In so doing, they provide liquidity to the public Commission brokers simply execute transactions for other parties who not have access to the trading Chapter 31 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part floor They make their income by the commissions they receive on each transaction A futures commission merchant is a firm that solicits public orders It may have a commission broker on the floor of the exchange or it may engage an independent broker to execute its trades (Mechanics of Futures Trading) An open outcry system involves traders on the floor of an exchange who call out bids and offers On an electronic system, traders are off the floor of the exchange and communicate their bids and offers by a computer link In addition some electronic systems actually have the computer match bids with offers (Classification by Trading Strategy) If a position in the futures market is accompanied by an opposite position in the spot market, the transaction is a hedge The hedger does not necessarily have to have a long or short position in the spot market A hedge can be established if the hedger is reasonably certain of taking a future position in the spot market The hedge protects against price changes in the interim period until the spot transaction is made A speculative strategy is not normally accompanied by a transaction or contemplated transaction in the spot market (Classification by Trading Strategy) A spread strategy is a long position in one futures contract and a short position in another futures contract The prices of the two contracts are normally highly correlated so that the gains on one contract are at least partially offset by the losses on the other The objective is to take a small amount of risk in the hope of a small profit An arbitrage strategy involves a near riskless transaction in one or more futures contracts and possibly a spot transaction Arbitrage trading is usually triggered by a deviation from the theoretical relationship between the prices of two instruments Both transactions can be viewed as hedges A hedge is a position in the spot market and an opposite position in the futures market Thus, it is similar to a spread in that the gain on one position is at least partially offset by the loss on the other Arbitrage is like hedging in that it is designed to have low risk and it often involves a position in the spot market and an opposite position in the futures market (General Classes of Futures Traders) These three types of futures traders differ primarily in the length of time they hold their positions Scalpers attempt to profit from small changes in the price of the contract They hold their positions for very short time intervals, sometimes less than a minute Day traders usually hold their positions for less than a day Near the end of the trading day they close out their positions so that they have no open positions overnight Position traders hold their transactions open for different lengths of time This could be several days or weeks They attempt to profit by capitalizing on trends that typically last longer than a day (Costs and Profitability of Exchange Membership) An individual can buy a full membership, which provides the right to go onto the trading floor and engage in futures transactions Some exchanges also offer limited memberships which permit trading in certain contracts only Alternatively, an individual can lease a seat from another individual already owning a seat (Contract Terms and Conditions) Daily price limits determine the maximum and minimum price at which a contract can trade during a day At the end of a given day, the settlement rice plus or minus the daily price limit establishes the maximum and minimum prices for trades the following day There are rules, however, that relax the limits under certain conditions The purpose of daily price limits is to prevent the margin accounts from being depleted so quickly that losses cannot be covered (Daily Price Limits and Trading Halts) Circuit breakers are rules that restrict trading after prices have moved by a specified amount They were instituted after the crash of 1987 They are designed to permit markets to "cool off" and, in some cases, additional margin to be collected Generally, the idea is that by prohibiting panic trading, investors are encouraged to absorb and analyze information before trading, something they are supposed to anyway Circuit breakers have the disadvantage, however, of simply disguising the true equilibrium price and can actually induce more panic If a building is on fire, locking the doors might reduce injuries due to trampling but would hardly put out the fire (Role of the Clearinghouse) The clearinghouse intervenes in each contract, guaranteeing to the buyer that the seller's losses will be covered and guaranteeing to the seller that the buyer's losses will be Chapter 32 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part covered This allows a trader to enter into a transaction without having to check the creditworthiness of the other party The clearinghouse requires that each trader maintain a margin account to cover losses The clearinghouse also maintains a cash reserve to cover losses in the event of a failure to cover a loss by a trader or firm As a last resort, the clearinghouse can assess the member firms a charge to make up any losses not already covered 10 (Daily Settlement) An offsetting trade means to simply take an opposite position in the same contract For example, a trader who buys a gold futures contract can offset the trade by selling a gold futures contract with the same expiration month This establishes a long and short position in the same contract, which is equivalent to not having a position at all A cash settlement is permitted at expiration on certain contracts The settlement price on the last day of trading is automatically equal to the spot price The account is marked-to-market on the last day and all open positions are automatically closed If the contract provides for delivery, the holder of the short position must deliver the commodity to the holder of the long position who pays the futures price on that day, subject to some adjustments provided in certain contracts Forward contracts are designed to be held to expiration The terms of the contract are written so as to accommodate delivery if that is the intention of the party Many forward contracts, however, are cash settled at expiration If the holder of a forward contract decides to terminate the position early, he would simply re-enter the forward market and request a new offsetting contract While this is similar to offsetting a futures contract, the forward market may not necessarily have the same liquidity as it did when the contract was opened While the contract can generally be offset, it may end up being very costly to offset In addition since both contracts still exist, credit risk remains Chapter 33 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 11 (Daily Settlement) Date Settlement Price Settlement Price ($) Mark-toMarket Other Entries Account Balance 7/1 453.95 226,975 850 9,000 9,850 7/2 454.50 227,250 275 10,125 7/3 452.00 226,000 –1,250 8,875 7/7 443.55 221,775 –4,225 4,650 7/8 441.65 220,825 –950 7/9 442.85 221,425 600 8,650 7/10 444.15 222,075 650 9,300 7/11 442.25 221,125 –950 8,350 7/14 438.30 219,150 –1,975 6,375 7/15 435.05 217,525 –1,625 4,750 7/16 435.50 217,750 225 +4,350 4,250 8,050 9,225 Explanation of Other Entries: 7/1: Initial margin deposit of $9,000 7/8: Balance on 7/7 was $4,650, which is below $6,000 maintenance margin Required to deposit $4,350 to bring balance up to initial margin of $9,000 7/16: Balance on 7/15 was $4,750, which is below $6,000 maintenance margin Required to deposit $4,250 to bring balance up to initial margin of $9,000 12 (Daily Settlement) You start off with $3,375 in your account It can drop to $2,500, a difference of $875, before you get a margin call The price changes in increments of $0.01 Since you have a contract on 1,000 barrels, each move of $0.01 is a change in the margin account balance of 1,000($0.01) = $10 To loss $875, there must be $875/$10 = 87.5 moves of $0.01, or a decrease in the contract price of $0.875 Since each move is a minimum of $0.01, it must fall by $0.88 That would take the price from $27.42 to $27.42 – $0.88 = $26.54 13 (Hedge Funds/Managed Funds) In many respects a futures fund and a hedge fund are quite similar A hedge fund is a fund that uses futures and other derivatives to invest typically in highly risky positions A futures fund, however, normally does not use over-the-counter derivatives, whereas a hedge fund often does Hedge funds tend to be fairly secretive, often registered offshore, and appeal primarily to wealthy investors The hedge fund industry is much larger than the futures fund industry 14 (Regulation of Futures and Forward Markets) The objective of federal regulation of the futures markets is to authorize futures exchanges, to approve new contracts and modifications of existing contracts, to ensure that price information is available to the public, to authorize individuals to provide services related to futures trading, and to oversee the markets to prevent manipulation Chapter 34 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 15 (Regulation of Futures and Forward Markets) An industry self-regulatory organization in the futures markets exists to provide a means in which the futures industry regulates itself This takes some of the burden of regulation off of the federal government The industry provides a means of licensing participants, policing its members, and adjudicating disputes 16 (Transaction Costs in Forward and Futures Trading) There are three types of futures trading costs, commissions, bid-asked spreads, and delivery costs All three costs influence the profitability of various futures trading strategies Commissions paid by the public to brokers are assessed on the basis of a dollar charge per contract The commission is paid at the order’s initiation and includes both the opening and closing commissions; that is, a round-trip commission is charged regardless of whether the trader ultimately closes out the contract, makes or takes delivery, or makes a cash settlement There are also exchange fees and NFA fees Many floor traders quote prices at which they are willing to simultaneously buy at the bid price and sell at the ask price The bid-ask spread is the cost to the public of liquidity — the ability to buy and sell quickly without a large price concession A futures trader who holds a position to delivery faces the potential for incurring a substantial delivery cost In the case of most financial instruments, this cost is rather small For commodities, however, it is necessary to arrange for the commodity’s physical transportation, delivery, and storage 17 (Delivery and Cash Settlement) All contracts eventually expire Physical settlement requires the seller to make physical delivery of the appropriate underlying instrument Most futures contracts allow for more than one deliverable instrument The contract usually specifies that the price paid by the long to the short be adjusted to reflect a difference in the quality of the deliverable good On cash-settled contracts, such as stock index futures, the settlement price on the last trading day is fixed at the closing spot price of the underlying instrument, such as the stock index All contracts are marked to market on that day, and the positions are deemed to be closed One exception to this procedure is the Chicago Mercantile Exchange’s S&P 500 futures contract, which closes trading on the Thursday before the third Friday of the expiration month but bases the final settlement price on the opening stock price on Friday morning This procedure was installed to avoid some problems created when a contract settles at the closing prices 18 (Off-Floor Futures Traders) An introducing broker (IB) is an individual who solicits orders from public customers to trade futures contracts IBs not execute orders themselves, nor their firms; rather, they subcontract with futures commission merchants (FCMs) to this The IB and the FCM divide the commission A commodity trading advisor (CTA) is an individual or firm that analyzes futures markets and issues reports, gives advice, and makes recommendations on the purchase and sale of contracts CTAs earn fees for their services but not necessarily trade contracts themselves A commodity pool operator (CPO) is an individual or firm that solicits funds from the public, pools them, and uses them to trade futures contracts The CPO profits by collecting a percentage of the assets in the fund and sometimes through sales commissions A CPO essentially is the operator of a futures fund, a topic discussed later in this chapter Some commodity pools are privately operated, however, and are not open for public participation An associated person (AP) is an individual associated with any of the above individuals or institutions or any other firm engaged in the futures business APs include directors, partners, officers, and employees but not clerical personnel Chapter 35 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 19 (Forward Market Traders) The forward market is dominated by large institutions, such as banks and corporations A typical forward market trader is an individual sitting at a desk with a telephone and a computer terminal Using the computer or telephone, the trader finds out the current prices available in the market The trader can then agree upon a price with another trader at another firm The trader may represent his or her own firm or may execute a trade for a client such as a corporation or hedge fund The trade may be a hedge, a spread, or an arbitrage In fact, it is the thousands of traders off the floor whose arbitrage activities play a crucial role in making the market so efficient APPENDIX 8B SOLUTIONS First year Price at year end = 422.40($500) = $211,200 Taxable gain: $211,200 – $205,150 = $6,050 Tax: ($6,050)(0.6)(0.20) + ($6,050)(0.4)(0.31) = $1,476.20 Second year Price when sold: 427.30($500) = $213,650 Taxable gain: $213,650 – $211,200 = $2,450 Tax: ($2,450)(0.6)(0.20) + ($2,450)(0.4)(0.31) = $597.80 Total tax on the transaction: $476.20 + $597.80 = $2,074 First year Taxable gain: $10,500 – $10,000 = $500 Tax: ($500)(0.6)(0.20) + ($500)(0.4)(0.31) = $122 Second year It is assumed that you bought the commodity at the price at $11,200 You would have to pay tax on the accrued profit since the end of the year Taxable gain: $11,200 – $10,500 = $700 Tax: ($700)(0.6)(0.20) + ($700)(0.4)(0.31) = $170.80 You would not pay tax on the commodity until it is sold CHAPTER 9: PRINCIPLES OF PRICING FORWARDS, FUTURES, AND OPTIONS ON FUTURES END OF CHAPTER QUESTIONS AND PROBLEMS (Forward/Futures Pricing Revisited) The futures price will not be the expected spot price in September because the dominance of the long hedgers will induce a risk premium Thus, the futures price of $2.76 is biased low Without information on the magnitude of the risk premium, it is impossible to come up with a precise estimate The expected spot price in September, however, is no less than $2.76 Of course, the actual spot price in September could be far less (Early Exercise of Call and Put Option on Futures) In Chapters 3, and we covered American call options on the spot and explained that in the absence of dividends they will not be exercised early They will always sell for at least the lower bound, which is higher than the intrinsic value, and usually more Call options on the futures, however, might be exercised early If the price of the underlying instrument is extremely high, the call will begin to behave like the underlying instrument For an option on a futures, this means that the call will behave like the futures, changing almost dollar-for-dollar with the Chapter 36 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part futures price For an option on the spot, the call will behave like the spot, changing almost one-for-one with the price of the spot Exercise of the futures call will release funds tied up in the call and provide a position in the futures Exercise of the call on the spot does not, however, release funds, since the investor has to purchase the spot instrument (Black Futures Option Pricing Model) The Black model is not an American option on futures pricing model and Eurodollar options on futures are American Also the Black model assumes constant interest rates Since Eurodollars are interest-sensitive instruments, they violate an assumption of the model (Black Futures Option Pricing Model) A spot option pricing model such as Black-Scholes-Merton is a model for pricing options on instruments with a cost of carry of r c (or r c – δ if there is a dividend yield) Since a futures requires no outlay of funds nor does it incur a storage cost, it has no cost of carry The cost of carry relevant to the futures price is the cost of carry of the underlying spot instrument An option on a futures is, therefore, an option on an instrument with a zero cost of carry The futures option pricing model is the same as the spot option pricing model where the spot price is replaced with the futures price and the cost of carry is zero The latter is established by assuming a dividend yield equal to the interest rate (Value of a Forward Contract) The value of the forward contract can be found by subtracting the present value of the forward price from the current spot price Thus, the value of the contract is $52 – $45(1.10) -0.5 = $9.09 This is the correct value of the contract at this point, six months into the life of the contract, because it is the value of a portfolio that could be constructed at this time to produce the same result six months later That is, you could buy the asset costing $52 and take out a loan, promising to pay $45 in six months This combination would guarantee that you would receive at time T, six months later, the value of the asset ST minus the $45 loan repayment, which is the value of the forward contract when it expires (Value of a Futures Contract) The value at the opening is 899.70 – 899.30 = 0.40 In dollars, this is 0.40($250) = $100 An instant before the close, the value is (899.10 – 899.30) = –0.20 In dollars, that is –0.20($250) = –$50 After the market has closed, the contract is marked-to-market, the gain or loss is distributed, and the value is zero (Price of a Futures Contract) Consider a futures contract on a stock Say you sell short the stock and buy the futures You receive S up front and during the holding period you earn interest of iS If the stock pays dividends, a short seller has to make them up so you would incur a cost of D T where DT is the compound future value of the dividends At expiration, you accept delivery of the stock and pay f Thus your profit is S0 – f0(T) + iS0 – DT Chapter 37 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Since this must equal zero, f0(T) = S0 + iS0 – DT (Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets ) (a) In a market with risk premiums, the futures price underestimates the spot price at expiration by the amount of the risk premium Therefore, the expected spot price in December is $3.64 + $0.035 = $3.675 (b) Arbitrage assures us that whether or not a risk premium exists, the futures price equals the spot price plus the cost of carry This is confirmed by noting that the spot price of $3.5225 plus the cost of carry of $.1175 equals the futures price of $3.64 (c) The answer is apparent in part (a) The expected price of wheat in December exceeds the futures price by the risk premium (d) If there is a risk premium, holders of long futures contracts expect to sell them for a profit equal to the risk premium Thus, the expected futures price at expiration is $3.64 + $.035 = $3.675, which is also the expected spot price at expiration (e) Speculators who take long positions in futures earn the risk premium They so because they are supplying insurance to the hedgers and, therefore, expect to receive a return in compensation for their willingness to take the risk (Stock Indices and Dividends) (a) T = 73/365 = 0.2 f0(T) = 956.49e(0.0596 - 0.0275)(0.2) = 962.65 At 960.50, it is underpriced (b) f0(T) = 956.49(1 + 0.0596)0.2 – 5.27= 962.36 At 960.50, it is underpriced The main difference is compounding of interest Annual compounding results in lower proceeds than continuous, hence the annual compounded carrying cost is lower than the continuous compounding 10 (Futures Prices and Risk Premia) E(S T) = 60, E(Φ) = 4, θ = 5.50 E(S T) = E(fT(T)) = 60 f0(T) = E(fT(T)) – E(Φ) = 60 – = 56 11 (Foreign Exchange) (a) S0 = $0.009313, F = $0.010475, r = 0.0615, ρ = 0.0364 With annual compounding, the forward rate should be 1.07(730/365) ($0.009313) 1.01(730/365) = $0.01045 So the forward rate should be $0.01045 but is actually $0.010475 Thus, the forward contract is overpriced You should buy the yen in the spot market and sell it in the forward market (b) With continuous compounding, the forward rate should be $0.009313e ( 0.07 − 0.01)( 730 / 365 ) = $0.01050 Chapter 38 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part So the forward rate should be $0.01050 but is actually $0.010475 Thus, the forward contract is underpriced You should sell the yen in the spot market and buy it in the forward market 12 (Put-Call Parity of Options on Futures) 100 days between September 12 and December 21, T = 100/365 = 0.2740 C(f0(T),T,X) – P(f0(T),T,X) = 26.25 – 3.25 = 23 (f0(T) – X)(1 + r) -T = (423.70 – 400)(1.0275) -0.2740 = 23.52 We can view the futures as overpriced and assume the call and put are correctly priced We sell the futures, buy a call and sell a put Payoffs at Expiration ST ≤ X –(S T – f0(T)) –(X – ST) f0(T) – X Short futures Long call Short put ST > X –(S T – f0(T)) ST – X f0(T) – X f0(T) – X = 423.70 – 400 = 23.70 The present value of this is 23.70(1.0275) -0.2740 = 23.52 The portfolio will cost 26.25 – 3.25 = 23.00 Thus, you will earn a present value of 23.52 – 23 = 0.52 13 (Pricing Options on Futures) The option’s life is January 31 to March 18, so T = 46/365 = 0.1260 a Intrinsic Value = Max(0, f0 – X) = Max(0, 483.10 – 480) = 3.10 b Time Value = Call Price – Intrinsic Value = 6.95 – 3.10 = 3.85 c Lower bound = Max[0, (f0 – X)(1 + r) -T] = Max[0, (483.10 – 480)(1.0284) -0.1260] = 3.09 d Intrinsic Value = Max(0, X – f0) = Max(0, 480 – 483.10) =0 e Time Value = Put Price – Intrinsic Value = 5.25 – = 5.25 f Lower bound = Max[0, (X – f0)(1 + r) -T] = Max[(0, (480 – 483.10)(1.0284) -0.1260] =0 (Note: the lower bound applies only to European puts.) Chapter 39 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part g C = P + (f0 – X)(1 + r) -T = 5.25 + (483.10 – 480)(1.0284) -0.1260 = 8.34 The actual call price is 6.95, so put-call parity does not hold 14 (Put-Call Parity of Options on Futures) (f0 – X)(1 + r) -T = (102 – 100)(1.10) -0.25 = 1.95 C – P = – 1.75 = 2.25 C – P is too high so the call is overpriced and/or the put is underpriced (or we could assume the futures is underpriced) So sell the call, buy the put, and buy the futures At expiration the payoffs will be Short call Long put Long futures fT ≤ X X – fT fT – f X – f0 fT > X –(fT – X) fT – f X – f0 This is equivalent to a risk-free loan, as a lender if X > f or as a borrower if f0 > X Here f0 > X so you are a borrower The present value should be (X – f 0)((1 + r) –T = (102 – 100)(1.10) -0.25 = –1.95 Thus you sell the call for and buy the put for –1.75 for a net inflow of 2.25 At expiration, you pay back 2.00 15 (Black Futures Option Pricing Model) First find the continuously compounded risk-free rate: r c = ln(1.0284) = 0.0280 Then price the option: d1 = = ln(f /X) + (σ /2)T σ T ln(483.10/480) + ((.08 ) /2).1260 08 1260 = 0.2409 d = d1 - σ T = 0.2409 - 08 1260 = 2125 N(d1 ) = N(.24) = 5948 N(d ) = N(.21) = 5832 C = e-rc T [f N(d1 ) - XN(d )] = e-.0280(.1260) [483.10(.5948) - 480(.5832)] = 7.39 The option appears to be underpriced You could sell e-rcT N(d1 ) = 0.5927 futures and buy one call, adjusting the hedge ratio through time and earn an arbitrage profit 16 (Black Futures Option Pricing Model) P = e - rc T [1 - N( d )] - f e - rc T [1 - N( d1 )] We already know that N(0.24) = 0.5948 and N(0.21) = 0.5832 Then P = 480e -0.0280(0.1260)[1 – 0.5832] – 483.10e -0.0280(0.1260)[1 – 0.5948] = 4.30 17 (Foreign Currencies and Foreign Interest Rates: Interest Rate Parity) The correct forward price is given by F( 0, T ) = S0 Chapter (1 + r ) T (1 + r ) f T = 1.665(1.015)/(1.02)= 1.6568 40 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Because the forward price is higher than the model price, we will sell the forward contract If transaction costs could be covered, you would buy the foreign currency in the spot market at $1.665 and sell it in the forward market at $1.664 You would earn interest at the foreign interest rate of percent By selling it forward, you could then convert back to dollars at the rate of $1.664 In other words, $1.665 would be used to buy unit of the foreign currency, which would grow to 1.02 units (the percent foreign rate) Then 1.02 units would be converted back to 1.02($1.664) = $1.69728 This would be a return of $1.69728/$1.665 – = 0.019387 or 1.9 percent, which is better than the U S rate 18 (Lower Bound of a European Option on Futures) f0(T) = $100, X = 90, and r = 5% The lower bound on a futures option is Ce(f0(T), T, X) ≥ Max[0, (f0(T) – X)(1 + r) –T] = Max[0, (100 – 90)(1 + 0.05) -1] = 9.5238 The quoted price of $9.40 violates the lower bound and the quoted price is low Therefore, we would buy the futures call option and hedge the resulting risk as illustrated in the following cash flow table Expiration (f T(T) ≤ X) Today (t=0) Strategy Buy futures call option –C = –$9.40 $0 Sell futures contract $0 (only margin required) Borrow +(f0(T) – X)(1 + r) –T = +$9.5238 +f0(T) – fT(T) = 100 – fT(T) –(f0(T) – X) *(1 + r) –T(1 + r) T = –$10 X – fT(T) (non-negative because fT(T) ≤ X) NET CASH FLOW 19 $0.1238 Expiration (f T(T)>X) fT(T) – X = fT(T) – 90 +f0(T) – fT(T) = 100 – fT(T) –(f0(T) – X) *(1 + r) –T(1 + r) T = –$10 $0 (Put-Call Party) We now have three versions of put-call parity Put-call parity with options on the underlying: Ce ( S0 , T, X ) = S0 − X(1 + r ) −T + Pe ( S0 , T, X ) Put-call parity with options on futures: Ce ( f ( T ) , T, X ) = ( f ( T ) − X )(1 + r ) −T + Pe ( f ( T ) , T, X ) Put-call futures parity: C e ( S0 , T, X ) = ( f ( T ) − X )(1 + r ) −T + Pe ( S0 , T, X ) 20 (Black Futures Option Pricing Model) Recall the standard Black-Scholes-Merton option pricing model: Ce ( S0 , T, X ) = S0 N( d1 ) − Xe − rc T N( d ) d1 = Chapter ln S0 + r + σ T X c σ T 41 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part d = d1 − σ T The generic carry formula for forward contracts is f ( T ) = S0 e rc T Solving for S0, S0 = f ( T ) e − rc T Substituting this result into the standard Black-Scholes-Merton option pricing formula results in the Black forward option pricing formula Ce ( f ( T ) , T, X ) = e − rc T [ f ( T ) N ( d1 ) − XN( d ) ] d1 = ln f0 ( T) + σ T X σ T d = d1 − σ T 21 (Stock Indices and Dividends) (a) Find the future value of the dividends 0.75(1.12) (60/365) + 0.85(1.12) (30/365) + 0.90 = 2.522 f0(T) = 100(1.12) (90/365) – 2.522 = 100.312 (b) Since f0(T) = S0 + θ, then θ = f0(T) – S0 so 100.312 – 100 = 0.312 This is the compound future value of the interest lost minus the compound future value of the dividends 22 (Forward/Futures Pricing Revisited) Let the spot price be S 0, the futures price be f0(T), and the margin requirement be M Consider the position of someone who buys the asset and sells a futures contract to form a risk-free hedge Today: Buy the asset, paying S 0, and sell the futures by depositing M dollars in a margin account that earns the rate q where q < r At expiration: The accumulated costs of storage and the interest lost on S dollars add up to θ When the trader delivers the asset, he receives f 0(T) The total amount of cash will be f 0(T) – θ + M + interest on M at the rate q Since the transaction is still risk-free, the amount initially invested must grow at the risk-free rate to equal this future value; however, the spot price does not have to be compounded because the interest on it is already included in the cost of carry Thus, f (T) + M(1 + q ) T (1 + r ) -T - θ = S + M Solving for f0(T) gives Chapter 42 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part f = S + θ + M[1 - (1 + q ) T (1 + r ) -T] The bracketed term is the difference in interest between the rate q and rate r If q is less than r, the whole bracketed term is greater than zero so the futures price will be greater than the spot price plus the cost of carry In other words if the margin account pays interest at less than the risk-free rate, the futures price will be greater than the spot price plus the cost of carry The higher futures price compensates for the loss of interest Chapter 43 End-of-Chapter Solutions © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part