page 515 PART FIVE: SPECIAL TOPICS Options and Corporate Finance 17 OPENING CASE On March 31, 2016, the closing stock prices for Reynolds American, Cabela’s, and Royal Gold were $50.31, $48.69, and $51.29, respectively. Each company had a call option trading on the Chicago Board Options Exchange with a $55 strike price and an expiration date of May 20—50 days away. Given how close the stock prices are, you might expect that the prices on these call options would be similar, but they were not. The Reynolds American options sold for $.15, Cabela’s options traded at $.95, and Royal Gold options traded at $1.69. Why would options on these three similarly priced stocks be priced so differently when the strike prices and the time to expiration were exactly the same? A big reason is that the volatility of the underlying stock is an important determinant of an option’s underlying value, and in fact, these three stocks had very different volatilities. In this chapter, we will explore this issue—and many others—in much greater depth using the Nobel prize-winning Black-Scholes option 929 pricing model. Please visit us at corecorporatefinance.blogspot.com for the latest developments in the world of corporate finance. 17.1 OPTIONS An option is a contract giving its owner the right to buy or sell an asset at a fixed price on or before a given date. For example, an option on a building might give the buyer the right to buy the building for $1 million on or anytime before the Saturday prior to the third Wednesday in January 2018. Options are a unique type of financial contract because they give the buyer the right, but not the obligation, to do something. The buyer uses the option only if it is advantageous to do so; otherwise the option can be thrown away. The Options Industry Council has a web page with lots of educational material at www.optionseducation.org. There is a special vocabulary associated with options. Here are some important definitions: 1. Exercising the option. The act of buying or selling the underlying asset via the option contract is referred to as exercising the option. 2. Strike or exercise price. The fixed price in the option contract at which the holder can buy or sell the underlying asset is called the strike price or exercise price. 3. Expiration date. The maturity date of the option is referred to as the expiration date. After this date, the option is dead. 4. American and European options. An American option may be exercised anytime up to the expiration date. A European option differs from an American option in that it can be exercised only on the expiration date. page 516 17.2 CALL OPTIONS 930 The most common type of option is a call option. A call option gives the owner the right to buy an asset at a fixed price during a particular time period. There is no restriction on the kind of asset, but the most common ones traded on exchanges are options on stocks and bonds. For example, call options on IBM stock can be purchased on the Chicago Board Options Exchange. IBM does not issue (that is, sell) call options on its common stock. Instead, individual investors are the original buyers and sellers of call options on IBM common stock. A representative call option on IBM stock enables an investor to buy 100 shares of IBM on or before July 15, at an exercise price of $100. This is a valuable option if there is some probability that the price of IBM common stock will exceed $100 on or before July 15. The Value of a Call Option at Expiration What is the value of a call option contract on common stock at expiration? The answer depends on the value of the underlying stock at expiration. Let’s continue with the IBM example. Suppose the stock price is $130 at expiration. The buyer1 of the call option has the right to buy the underlying stock at the exercise price of $100. In other words, he has the right to exercise the call. Having the right to buy something for $100 when it is worth $130 is obviously a good thing. The value of this right is $30 (= $130 − 100) on the expiration day.2 The call would be worth even more if the stock price were higher on expiration day. For example, if IBM were selling for $150 on the date of expiration, the call would be worth $50 (= $150 − 100) at that time. In fact, the call’s value increases $1 for every $1 rise in the stock price. If the stock price is greater than the exercise price, we say that the call is in the money. Of course, it is also possible that the value of the common stock will turn out to be less than the exercise price. In this case, we say that the call is out of the money. The holder will not exercise in this case. For example, if the stock price at the expiration date is $90, no rational investor would exercise. Why pay $100 for stock worth only $90? Because the option holder has no obligation to exercise the call, she can walk away from the option. As a consequence, if IBM’s stock price is less than $100 on the expiration date, the value of the call option will be $0. In this case, the value of the call option is not the difference between IBM’s stock price and $100, as it would be if the holder of the call option had the obligation to exercise the call. The payoff of a call option at expiration is: 931 PAYOFF ON THE EXPIRATION DATE IF STOCK PRICE IS LESS THAN $100 IF STOCK PRICE IS GREATER THAN $100 $0 Stock price − $100 Call option value Figure 17.1 plots the value of the call at expiration against the value of IBM’s stock. It is referred to as the hockey stick diagram of call option values. If the stock price is less than $100, the call is out of the money and worthless. If the stock price is greater than $100, the call is page 517 in the money and its value rises one-for-one with increases in the stock price. Notice that the call can never have a negative value. It is a limited liability instrument, which means that all the holder can lose is the initial amount she paid for it. FIGURE 17.1 The Value of a Call Option on the Expiration Date EXAMPLE 17.1 Call Option Payoffs Suppose Mr. Optimist holds a one-year call option on TIX common stock. It is a European call option and can be exercised at $150. Assume that the expiration date has arrived. What is the value of the 932 TIX call option on the expiration date? If TIX is selling for $200 per share, Mr. Optimist can exercise the option—purchase TIX at $150— and then immediately sell the share at $200. Mr. Optimist will have made $50 (= $200 − 150). Instead, assume that TIX is selling for $100 per share on the expiration date. If Mr. Optimist still holds the call option, he will throw it out. The value of the TIX call on the expiration date will be $0 in this case. 17.3 PUT OPTIONS A put option can be viewed as the opposite of a call option. Just as a call gives the holder the right to buy the stock at a fixed price, a put gives the holder the right to sell the stock for a fixed exercise price. The Value of a Put Option at Expiration The circumstances that determine the value of the put are the opposite of those for a call option, because a put option gives the holder the right to sell shares. Let us assume that the exercise price of the put is $50 and the stock price at expiration is $40. The owner of this put option has the right to sell the stock for more than it is worth, something that is clearly profitable. That is, he can buy the stock at the market price of $40 and immediately sell it at the exercise price of $50, generating a profit of $10 (= $50 − 40). Thus, the value of the option at expiration must be $10. page 518 FIGURE 17.2 The Value of a Put Option on the Expiration Date 933 The profit would be greater still if the stock price were lower. For example, if the stock price were only $30, the value of the option would be $20 (= $50 − 30). In fact, for every $1 that the stock price declines at expiration, the value of the put rises by $1. However, suppose that the stock at expiration is trading at $60—or any price above the exercise price of $50. The owner of the put would not want to exercise here. It is a losing proposition to sell stock for $50 when it trades in the open market at $60. Instead, the owner of the put will walk away from the option. That is, he will let the put option expire. The payoff of this put option is: PAYOFF ON THE EXPIRATION DATE Put option value IF STOCK PRICE IS LESS THAN $50 IF STOCK PRICE IS GREATER THAN $50 $50 − Stock price $0 Figure 17.2 plots the values of a put option for all possible values of the underlying stock. It is instructive to compare Figure 17.2 with Figure 17.1 for the call option. The call option is valuable whenever the stock is above the exercise price, and the put is valuable when the stock price is below the exercise price. EXAMPLE 17.2 934 Put Option Payoffs Ms. Pessimist feels quite certain that BMI will fall from its current $160 per-share price. She buys a put. Her put option contract gives her the right to sell a share of BMI stock at $150 one year from now. If the price of BMI is $200 on the expiration date, she will tear up the put option contract because it is worthless. That is, she will not want to sell stock worth $200 for the exercise price of $150. On the other hand, if BMI is selling for $100 on the expiration date, she will exercise the option. In this case, she can buy a share of BMI in the market for $100 per share and turn around and sell the share at the exercise price of $150. Her profit will be $50 (= $150 − 100). The value of the put option on the expiration date therefore will be $50. page 519 17.4 SELLING OPTIONS An investor who sells (or writes) a call on common stock promises to deliver shares of the common stock if required to do so by the call option holder. Notice that the seller is obligated to do so. Check out this option exchange: www.cmegroup.com. If, at the expiration date, the price of the common stock is greater than the exercise price, the holder will exercise the call and the seller must give the holder shares of stock in exchange for the exercise price. The seller loses the difference between the stock price and the exercise price. For example, assume that the stock price is $60 and the exercise price is $50. Knowing that exercise is imminent, the option seller buys stock in the open market at $60. Because she is obligated to sell at $50, she loses $10 (= $50 − 60). Conversely, if at the expiration date, the price of the common stock is below the exercise price, the call option will not be exercised and the seller’s liability is zero. Why would the seller of a call place himself in such a precarious position? After all, the seller loses money if the stock price ends up above the exercise price and he merely avoids losing money if the stock price 935 ends up below the exercise price. The answer is that the seller is paid to take this risk. On the day that the option transaction takes place, the seller receives the price that the buyer pays. Now, let’s look at the seller of puts. An investor who sells a put on common stock agrees to purchase shares of common stock if the put holder should so request. The seller loses on this deal if the stock price falls below the exercise price and the holder puts the stock to the seller. For example, assume that the stock price is $40 and the exercise price is $50. The holder of the put will exercise in this case. In other words, he will sell the underlying stock at the exercise price of $50. This means that the seller of the put must buy the underlying stock at the exercise price of $50. Because the stock is only worth $40, the loss here is $10 (= $40 − 50). The values of the “sell-a-call” and “sell-a-put” positions are depicted in Figure 17.3. The graph on the left-hand side of the figure shows that the seller of a call loses nothing when the stock price at expiration is below $50. However, the seller loses a dollar for every dollar that the stock rises above $50. The graph in the center of the figure shows that the seller of a put loses nothing when the stock price at the expiration date is above $50. However, the seller loses a dollar for every dollar that the stock falls below $50. FIGURE 17.3 The Payoffs to Sellers of Calls and Puts and Buyers of Common Stock It is worthwhile to spend a few minutes comparing the graphs in Figure 17.3 to those in Figures 17.1 and 17.2. The graph of selling a call (the graph in the left-hand side of Figure 17.3) is the mirror image of the graph page 520 of buying a call (Figure 17.1).3 This occurs because options are a zero-sum game. The seller of a call loses what the buyer makes. Similarly, the graph of selling a put (the middle graph in Figure 17.3) is the mirror image of the graph of buying a put (Figure 17.2). 936 Again, the seller of a put loses what the buyer makes. Figure 17.3 also shows the value at expiration of simply buying common stock. Notice that buying the stock is the same as buying a call option on the stock with an exercise price of $0. This is not surprising. If the exercise price is $0, the call holder can buy the stock for nothing, which is really the same as owning it. 17.5 OPTION QUOTES Now that we understand the definitions for calls and puts, let’s see how these options are quoted. Table 17.1 presents information on Intel Corporation options expiring in April 2016, obtained from finance.yahoo.com. At the time of these quotes, Intel was selling for $32.45. For more on option ticker symbols, go to the “Symbol Directory” link under “Trading Resources” at www.cboe.com. TABLE 17.1 Information on the options of Intel Corporation 937 On the left-hand side of the table are the available strike prices. The top half of the table presents call option quotes; put option quotes are featured in the section below. The second column contains page 521 ticker symbols, which uniquely indicate the underlying stock, the type of option, the expiration date, and the strike price. Next, we have the most recent prices on the options (“Last”), the bid and ask prices, and the change from the previous day (“Change”). Note that option prices are quoted on a per-option basis, but trading actually occurs in standardized contracts, where each contract calls for the purchase (for 938 calls) or sale (for puts) of 100 shares. Thus, the call option with a strike price of $33 last traded at $1.09 per option, or $109 per contract. Volume shows the number of contracts traded this day, and Open Interest is the number of contracts currently outstanding. 17.6 COMBINATIONS OF OPTIONS Puts and calls can serve as building blocks for more complex option contracts. For example, Figure 17.4 illustrates the payoff from buying a put option on a stock and simultaneously buying the stock. ExcelMaster coverage online www.mhhe.com/RossCore5e If the share price is greater than the exercise price, the put option is worthless, and the value of the combined position is equal to the value of the common stock. If instead the exercise price is greater than the share price, the decline in the value of the shares will be exactly offset by the rise in value of the put. The strategy of buying a put and buying the underlying stock is called a protective put. It is as if one is buying insurance for the stock. The stock can always be sold at the exercise price, regardless of how far the market price of the stock falls. Note that the combination of buying a put and buying the underlying stock has the same shape in Figure 17.4 as the call purchase in Figure 17.1. To pursue this point, let’s consider the graph for buying a call, which is shown at the far left of Figure 17.5. FIGURE 17.4 Payoff to the Combination of Buying a Put and Buying the Underlying Stock 939 FIGURE 17.5 Payoff to the Combination of Buying a Call and Buying a Zero Coupon Bond page 522 This graph is the same as Figure 17.1, except that the exercise price is $50 here. Now let’s try the strategy of: (Leg A) Buying a call. (Leg B) Buying a risk-free, zero coupon bond (i.e., a T-bill), with a face value of $50 that matures on the same day that the option expires. We have drawn the graph of Leg A of this strategy at the far left of Figure 17.5, but what does the graph of Leg B look like? It looks like the middle graph of the figure. That is, anyone buying this zero coupon bond will be guaranteed to get $50, regardless of the price of the stock at expiration. What does the graph of simultaneously buying both Leg A and Leg B of this strategy look like? It looks like the far-right graph of Figure 17.5. That is, the investor receives a guaranteed $50 from the bond, regardless of what happens to the stock. In addition, the investor receives a payoff from the call of $1 for every $1 that the price of the stock rises above the exercise price of $50. The far-right graph of Figure 17.5 looks exactly like the far-right graph of Figure 17.4. Thus, an investor gets the same payoff from the strategy of Figure 17.4 and the strategy of Figure 17.5, regardless of what happens to the price of the underlying stock. In other words, the investor gets the same payoff from: 1. Buying a put and buying the underlying stock. 2. Buying a call and buying a risk-free, zero coupon bond. 940 If investors have the same payoffs from the two strategies, the two strategies must have the same cost. Otherwise, all investors will choose the strategy with the lower cost and avoid the strategy with the higher cost. This leads to the interesting result that: This relationship is known as put–call parity and is one of the most fundamental relationships concerning options. It says that there are two ways of buying a protective put. You can buy a put and buy the underlying stock simultaneously. Here, your total cost is the price of the underlying stock plus the price of the put. Or, you can buy a call and buy a zero coupon bond. Here, your total cost is the price of the call plus the price of the zero coupon bond. The price of the zero coupon bond is equal to the present value of the exercise price, i.e., the present value of $50 in our example. Equation 17.1 is a very precise relationship. It holds only if the put and the call have both the same exercise price and the same expiration date. In addition, the maturity date of the zero coupon bond must be the same as the expiration date of the options. To see how fundamental put–call parity is, let’s rearrange the formula, yielding: This relationship now states that you can replicate the purchase of a share of stock by buying a call, selling a put, and buying a zero coupon bond. (Note that, because a minus sign comes before “Price of put,” the put is sold, not bought.) Investors in this three-legged strategy are said to have purchased a synthetic stock. Let’s do one more transformation: page 523 FIGURE 17.6 Payoff to the Combination of Buying a Stock and Selling a Call 941 Many investors like to buy a stock and write a call on the stock simultaneously. This is a conservative strategy known as selling a covered call. The preceding put–call parity relationship tells us that this strategy is equivalent to selling a put and buying a zero coupon bond. Figure 17.6 develops the graph for the covered call. You can verify that the covered call can be replicated by selling a put and simultaneously buying a zero coupon bond. Of course, there are other ways of rearranging the basic put–call relationship. For each rearrangement, the strategy on the left-hand side is equivalent to the strategy on the right-hand side. The beauty of put–call parity is that it shows how any strategy in options can be achieved in two different ways. To test your understanding of put–call parity, suppose shares of stock in Joseph-Belmont, Inc., are selling for $80. A three-month call option with an $85 strike price goes for $6. The risk-free rate is .5 percent per month. What’s the value of a three-month put option with an $85 strike price? We can rearrange the put–call parity relationship to solve for the price of the put as follows: As illustrated, the value of the put is $9.74. EXAMPLE 17.3 A Synthetic T-bill 942 Suppose that shares of stock in Smolira Corp. are selling for $110. A call option on Smolira with one year to maturity and a $110 strike price sells for $15. A put with the same terms sells for $5. What’s the risk-free rate? To answer, we need to use put–call parity to determine the price of a risk-free, zero coupon bond: Plugging in the numbers, we get: Since the present value of the $110 strike price is $100, the implied risk-free rate is obviously 10 percent. page 524 17.7 VALUING OPTIONS In the last section, we determined what options are worth on the expiration date. Now, we wish to determine the value of options when you buy them well before expiration.4 We begin by considering the lower and upper bounds on the value of a call. Bounding the Value of a Call LOWER BOUND Consider an American call that is in the money prior to expiration. For example, assume that the stock price is $60 and the exercise price is $50. In this case, the option cannot sell below $10. To see this, note the simple strategy if the option sells at, say, $9. DATE TRANSACTION Today (1) Buy call. −$ 9 Today (2) Exercise call—that is, buy underlying stock at exercise price. −$50 +$60 Today (3) Sell stock at current +$ 1 943 market price. Arbitrage profit The type of profit that is described in this transaction is an arbitrage profit. Arbitrage profits come from transactions that have no risk or cost and cannot occur regularly in normal, well-functioning financial markets. The excess demand for these options would quickly force the option price up to at least $10 (= $60 − 50). Of course, the price of the option is likely to be above $10. Investors will rationally pay more than $10 because of the possibility that the stock will rise above $60 before expiration. For example, suppose the call actually sells for $12. In this case, we say that the intrinsic value of the option is $10, meaning it must always be worth at least this much. The remaining $12 − 10 = $2 is sometimes called the time premium, and it represents the extra that investors are willing to pay because of the possibility that the stock price will rise before the option expires. UPPER BOUND Is there an upper boundary for the option price as well? It turns out that the upper boundary is the price of the underlying stock. That is, an option to buy common stock cannot have a greater value than the common stock itself. A call option can be used to buy common stock with a payment of an exercise price. It would be foolish to buy stock this way if the stock could be purchased directly at a lower price. The upper and lower bounds are represented in Figure 17.7. In addition, these bounds are summarized in the bottom half of Table 17.2. The Factors Determining Call Option Values The previous discussion indicated that the price of a call option must fall somewhere in the shaded region of Figure 17.7. We now will determine more precisely where in the shaded region it should be. The factors that determine a call’s value can be broken into two sets. The first set contains the features of the option contract. The two basic contractual features are the expiration date and the exercise price. The second set of factors affecting the call price concerns characteristics of the stock and the market. EXERCISE PRICE An increase in the exercise price reduces the value of the call. For example, imagine that there are two calls on a stock selling at $60. The first call has an exercise price of $50 and the page 525 second one has an exercise price of $40. Which call would you 944 rather have? Clearly, you would rather have the call with an exercise price of $40, because that one is $20 (= $60 − 40) in the money. In other words, the call with an exercise price of $40 should sell for more than an otherwise identical call with an exercise price of $50. FIGURE 17.7 The Upper and Lower Boundaries of Call Option Values TABLE 17.2 Factors Affecting American Option Values CALL OPTION* PUT OPTION* Value of underlying asset (stock price) + − Exercise price − + Stock volatility + + Interest rate + − Time to expiration date + + INCREASE IN In addition to the preceding, we have presented the following four relationships for American calls: 1. The call price can never be greater than the stock price (upper bound). 2. The call price can never be less than either zero or the difference 945 between the stock price and the exercise price (lower bound). 3. The call is worth zero if the stock is worth zero. 4. When the stock price is much greater than the exercise price, the call price tends toward the difference between the stock price and the present value of the exercise price. * The signs (+, −) indicate the effect of the variables on the value of the option. For example, the two +s for stock volatility Indicate that an increase in volatility will increase both the value of a call and the value of a put. EXPIRATION DATE The value of an American call option must be at least as great as the value of an otherwise identical option with a shorter term to expiration. Consider two American calls: One has a maturity of nine months and the other expires in six months. Obviously, the ninemonth call has the same rights as the six-month call, and it also has an additional three months within which these rights can be exercised. It cannot be worth less and will generally be more valuable.5 STOCK PRICE Other things being equal, the higher the stock price, the more valuable the call option will be. For example, if a stock is worth $80, a call with an exercise price of $100 isn’t worth very much. If the stock soars to $120, the call becomes much more valuable. page 526 FIGURE 17.8 Value of an American Call as a Function of Stock Price 946 Now consider Figure 17.8, which shows the relationship between the call price and the stock price prior to expiration. The curve indicates that the call price increases as the stock price increases. Furthermore, it can be shown that the relationship is represented, not by a straight line, but by a convex curve. That is, the increase in the call price for a given change in the stock price is greater when the stock price is high than when the stock price is low. There are two special items of note regarding the curve in Figure 17.8: 1. The Stock Is Worthless. The call must be worthless if the underlying stock is worthless. That is, if the stock has no chance of attaining any value, it is not worthwhile to pay the exercise price in order to obtain the stock. 2. The Stock Price Is Very High Relative to the Exercise Price. In this situation, the owner of the call knows that he will end up exercising the call. He can view himself as the owner of the stock now, with one difference. He must pay the exercise price at expiration. Thus, the value of his position, i.e., the value of the call, is: These two points are summarized in the bottom half of Table 17.2. THE KEY FACTOR: THE VARIABILITY OF THE UNDERLYING ASSET The greater the variability of the underlying asset, the more valuable the call option will be. Consider the following example. Suppose that just before the call expires, the stock price will be either $100 with probability .5 or $80 with probability .5. What will be the value of a call with an exercise price of $110? Clearly, it will be worthless because no matter what happens to the stock, its price will always be below the exercise price. For an option-oriented site focusing on volatilities, visit www.ivolatility.com. Now let us see what happens if the stock is more variable. Suppose that we add $20 to the best case and take $20 away from the worst case. Now the stock has a one-half chance of being worth $60 and a one-half chance of being worth $120. We have spread the stock returns, but, of course, the expected value of the stock has stayed the same: 947 Notice that the call option has value now because there is a one-half chance that the stock price will be $120, or $10 above the exercise price of $110. This illustrates a very important point. There is a fundamental distinction between holding an option on an underlying asset and holding the underlying asset. If investors in the marketplace are risk-averse, a rise in the variability of the stock will decrease its market value. page 527 However, the holder of a call receives payoffs from the positive tail of the probability distribution. As a consequence, a rise in the variability in the underlying stock increases the market value of the call. FIGURE 17.9 Distribution of Common Stock Price at Expiration for Both Security A and Security B. Options on the Two Securities Have the Same Exercise Price. This result can also be seen in Figure 17.9. Consider two stocks, A and B, each of which is normally distributed. For each security, the figure illustrates the probability of different stock prices on the expiration date.6 As can be seen from the figure, Stock B has more volatility than does Stock A. This means that Stock B has a higher probability of both abnormally high returns and abnormally low returns. Let us assume that options on each of the two securities have the same exercise price. To option holders, a return much below average on Stock B is no worse than a return only moderately below average on Stock A. In either situation, the option expires out of the money. However, to option holders, a return 948 much above average on Stock B is better than a return only moderately above average on Stock A. Because a call’s price at the expiration date is the difference between the stock price and the exercise price, the value of the call on B at expiration will be higher in this case. THE INTEREST RATE Call prices are also a function of the level of interest rates. Buyers of calls do not pay the exercise price until they exercise the option, if they do so at all. The ability to delay payment is more valuable when interest rates are high and less valuable when interest rates are low. Thus, the value of a call is positively related to interest rates. A Quick Discussion of Factors Determining Put Option Values Given our extended discussion of the factors influencing a call’s value, we can examine the effect of these factors on puts very easily. Table 17.2 summarizes the five factors influencing the prices of both American calls and American puts. The effect of three factors on puts is the opposite of the effect of these three factors on calls: 1. Value of Underlying Asset. The put’s market value decreases as the stock price increases because puts are in the money when the stock sells below the exercise price. 2. Exercise Price. The value of a put with a high exercise price is greater than the value of an otherwise identical put with a low exercise price for the reason given in (1). 3. Interest Rate. A high interest rate adversely affects the page 528 value of a put. The ability to sell a stock at a fixed exercise price sometime in the future is worth less if the present value of the exercise price is diminished by a high interest rate. The effect of the other two factors on puts is the same as the effect of these factors on calls: 4. Time to Expiration Date. The value of an American put with a distant expiration date is greater than an otherwise identical put with an earlier expiration.7 The longer time to maturity gives the put holder more flexibility, just as it did in the case of a call. 5. Stock Volatility. Volatility of the underlying stock increases the value of the put. The reasoning is analogous to that for a call. At expiration, 949 a put that is way in the money is more valuable than a put only slightly in the money. However, at expiration, a put way out of the money is worth zero, just as is a put only slightly out of the money. 17.8 AN OPTION PRICING FORMULA We have explained qualitatively that the value of a call option is a function of five variables: ExcelMaster coverage online www.mhhe.com/RossCore5e 1. The current price of the underlying asset, which for stock options is the price of a share of common stock. 2. The exercise price. 3. The time to expiration date. 4. The variance of the underlying asset’s rate of return. 5. The risk-free interest rate. It is time to replace the qualitative model with a precise option valuation model. The model we choose is the famous Black–Scholes option pricing model. You can put numbers into the Black–Scholes model and get values back. The Black–Scholes model is represented by a rather imposing formula. A derivation of the formula is simply not practical in this textbook, as many students will be happy to learn. However, some appreciation for the achievement as well as some intuitive understanding is in order. In the early chapters of this book, we showed how to discount capital budgeting projects using the net present value formula. We also used this approach to value stocks and bonds. Why, students sometimes ask, can’t the same NPV formula be used to value puts and calls? It is a good question because the earliest attempts at valuing options used NPV. Unfortunately, the attempts were simply not successful because no one could determine the appropriate discount rate. An option is generally riskier than the underlying stock, but no one knew exactly how much riskier. Fischer Black and Myron Scholes attacked the problem by pointing out that a strategy of borrowing to finance a stock purchase duplicates the risk of a call. Then, knowing the price of a stock already, one can determine 950 the price of a call such that its return is identical to that of the stock-withborrowing alternative. We illustrate the intuition behind the Black–Scholes approach by considering a simple example where a combination of a call and a stock eliminates all risk. This example works because we let the future stock price be one of only two values. Hence, the example is called a two-state option model. By eliminating the possibility that the stock price can take on other values, we are able to duplicate the call exactly. page 529 A Two-State Option Model Consider the following example. Suppose the current market price of a stock is $50 and the stock will sell for either $60 or $40 at the end of the year. Further, imagine a call option on this stock with a one-year expiration date and a $50 exercise price. Investors can borrow at 10 percent. Our goal is to determine the value of the call. In order to value the call correctly, we need to examine two strategies. The first is to simply buy the call. The second is to: a. Buy one-half a share of stock. b. Borrow $18.18, implying a payment of principal and interest at the end of the year of $20 (= $18.18 × 1.10). As you will see shortly, the cash flows from the second strategy exactly match the cash flows from buying a call. (A little later, we will show how we came up with the exact fraction of a share of stock to buy and the exact borrowing amount.) Because the cash flows match, we say that we are duplicating the call with the second strategy. At the end of the year, the future payoffs are set out as follows: FUTURE PAYOFFS INITIAL TRANSACTIONS $60 − 50 = $10 1. Buy a call 2. Buy IF STOCK PRICE IS $60 × $60 = $30 share of stock Borrow $18.18 at 10% −($18.18 × 1.10) = −$20 951 Total from stock and borrowing strategy $10 Note that the future payoff structure of the “buy-a-call” strategy is duplicated by the strategy of “buy stock” and “borrow.” That is, under either strategy, an investor would end up with $10 if the stock price rose and $0 if the stock price fell. Thus, these two strategies are equivalent as far as traders are concerned. Now, if two strategies always have the same cash flows at the end of the year, how must their initial costs be related? The two strategies must have the same initial cost. Otherwise, there will be an arbitrage possibility. We can easily calculate this cost for our strategy of buying stock and borrowing. This cost is: Buy share of stock Borrow $18.18 × $50 = $25.00 − 18.18 $ 6.82 Because the call option provides the same payoffs at expiration as does the strategy of buying stock and borrowing, the call must be priced at $6.82. This is the value of the call option in a market without arbitrage profits. We left two issues unexplained in the preceding example. DETERMINING THE DELTA How did we know to buy one-half a share of stock in the duplicating strategy? Actually, the answer is easier than it might at first appear. The call price at the end of the year will be either $10 or $0, whereas the stock price will be either $60 or $40. Thus, the call price has a potential swing of $10 (= $10 − 0) next period, whereas the stock price has a potential swing of $20 (= $60 − 40). We can write this in terms of the following ratio: page 530 As indicated, this ratio is called the delta of the call. In words, a $1 swing in the price of the stock gives rise to a $1/2 swing in the price of the call. Because we are trying to duplicate the call with the stock, it seems sensible to buy one-half a share of stock instead of buying one call. In other words, 952 the risk of buying one-half a share of stock should be the same as the risk of buying one call. DETERMINING THE AMOUNT OF BORROWING How did we know how much to borrow? Buying one-half a share of stock brings us either $30 or $20 at expiration, which is exactly $20 more than the payoffs of $10 and $0, respectively, from the call. To duplicate the call through a purchase of stock, we should also borrow enough money so that we have to pay back exactly $20 of interest and principal. This amount of borrowing is merely the present value of $20, which is $18.18 (= $20/1.10). Now that we know how to determine both the delta and the amount of borrowing, we can write the value of the call as: We will find this intuition very useful in explaining the Black–Scholes model. RISK-NEUTRAL VALUATION Before leaving this simple example, we should comment on a remarkable feature. We found the exact value of the option without even knowing the probability that the stock would go up or down! If an optimist thought the probability of an up move was very high and a pessimist thought it was very low, they would still agree on the option value. How could that be? The answer is that the current $50 stock price already balances the views of the optimist and the pessimist. The option reflects that balance because its value depends on the stock price. This insight provides us with another approach to valuing the call. If we don’t need the probabilities of the two states to value the call, perhaps we can select any probabilities we want and still come up with the right answer. Suppose we selected probabilities such that the return on the stock is equal to the risk-free rate of 10 percent. We know that the stock return given a rise is 20 percent (= $60/$50 − 1) and the stock return given a fall is −20 percent (= $40/$50 − 1). Thus, we can solve for the probability of a rise necessary to achieve an expected return of 10 percent as: Solving this formula, we find that the probability of a rise is 3/4 and the probability of a fall is 1/4. If we apply these probabilities to the call, 953 we can value it as: the same value that we got from the duplicating approach. Why did we select probabilities such that the expected return on the stock is 10 percent? We wanted to work with the special case where investors are risk-neutral. This case occurs when the expected return on any asset (including both the stock and the call) is equal to the risk-free rate. In other words, this case occurs when investors demand no additional compensation beyond the risk-free rate, regardless of the risk of the asset in question. What would have happened if we had assumed that the expected return on the stock was greater than the risk-free rate? The value of the call would still be $6.82. However, the calculations would be more difficult. For example, if we assumed that the expected return on the page 531 stock was, say 11 percent, we would have had to derive the expected return on the call. Although the expected return on the call would be higher than 11 percent, it would take a lot of work to determine it precisely. Why do any more work than you have to? Because we can’t think of any good reason, we (and most other financial economists) choose to assume risk-neutrality. Thus, the preceding material allows us to value a call in the following two ways: 1. Determine the cost of a strategy to duplicate the call. This strategy involves an investment in a fractional share of stock financed by partial borrowing. 2. Calculate the probabilities of a rise and a fall under the assumption of risk-neutrality. Use those probabilities, in conjunction with the riskfree rate, to discount the payoffs of the call at expiration. The Black–Scholes Model The preceding example illustrates the duplicating strategy. Unfortunately, a strategy such as this will not work in the real world over, say, a one-year time frame, because there are many more than two possibilities for next year’s stock price. However, the number of possibilities is reduced as the time period is shortened. In fact, the assumption that there are only two 954 possibilities for the stock price over the next infinitesimal instant is quite plausible.8 There’s a Black–Scholes calculator (and a lot more) at www.numa.com. In our opinion, the fundamental insight of Black and Scholes is to shorten the time period. They show that a specific combination of stock and borrowing can indeed duplicate a call over an infinitesimal time horizon. Because the price of the stock will change over the first instant, another combination of stock and borrowing is needed to duplicate the call over the second instant and so on. By adjusting the combination from moment to moment, one can continually duplicate the call. It may boggle the mind that a formula can (1) determine the duplicating combination at any moment and (2) value the option based on this duplicating strategy. Suffice it to say that their dynamic strategy allows one to value a call in the real world, just as we showed how to value a call in the two-state model. This is the basic intuition behind the Black–Scholes model. Because the actual derivation of their formula is, alas, far beyond the scope of this text, we simply present the formula itself. The formula is: where: This formula for the value of a call, C, is one of the most complex in finance. However, it involves only five parameters: 1. 2. 3. 4. 5. S = Current stock price. E = Exercise price of call. R = Annual risk-free rate of return, continuously compounded. σ2 = Variance (per year) of the continuous return on the stock. t = Time (in years) to expiration date. page 532 955 The small e in the formula is the mathematical constant 2.71828. . . . In addition, there is the statistical concept: Rather than discuss the formula in its algebraic state, we illustrate the formula with an example: EXAMPLE 17.4 Black–Scholes Consider Private Equipment Company (PEC). On October 4 of Year 0, the PEC April 49 call option has a closing value of $4. The stock itself is selling at $50. The option has 199 days to expiration (maturity date = April 21, Year 1). The annual risk-free interest rate, continuously compounded, is 7 percent. This information determines three variables directly: 1. The stock price, S, is $50. 2. The exercise price, E, is $49. 3. The risk-free rate, R, is .07. In addition, the time to maturity, t, can be calculated quickly: The formula calls for t to be expressed in years. 4. We express the 199-day interval in years as t = 199/365. In the real world, an option trader would know S and E exactly. Traders generally view U.S. Treasury bills as riskless, so a current quote from The Wall Street Journal or a similar source would be obtained for the interest rate. The trader would also know (or could count) the number of days to expiration exactly. Thus, the fraction of a year to expiration, t, could be calculated quickly. The problem comes in determining the variance of the stock’s return. The formula calls for the variance in effect between the purchase date of October 4 and the expiration date. Unfortunately, this represents the future, so the correct value for variance is simply not available. Instead, traders frequently estimate variance from past data, just as we calculated variance in an earlier chapter. In addition, some traders may use intuition to adjust their estimate. For example, if anticipation of an upcoming event is currently increasing the volatility of the stock, the trader might adjust her estimate of variance upward to reflect this. (This problem was most severe right after the October 19, 1987, crash. The 956 stock market was quite risky in the aftermath, so estimates using precrash data were too low.) The above discussion was intended merely to mention the difficulties in variance estimation, not to present a solution. For our purposes, we assume that a trader has come up with an estimate of variance: 5. The variance of PEC has been estimated to be .09 per year. Using the above five parameters, we calculate the Black–Scholes value of the PEC option in three steps: Step 1: Calculate d1 and d2. These values can be determined by a straightforward, albeit tedious, insertion of our parameters into the basic formula. We have page 533 FIGURE 17.10 Graph of Cumulative Probability Step 2: Calculate N(d1) and N(d2). The values N(d1) and N(d2) can best be understood by examining Figure 17.10. The figure shows the normal distribution with an expected value of 0 and a standard deviation 957 of 1. This is frequently called the standardized normal distribution. We mentioned in an earlier chapter that the probability that a drawing from this distribution will be between −1 and +1 (within one standard deviation of its mean, in other words) is 68.26 percent. Now let us ask a different question. What is the probability that a drawing from the standardized normal distribution will be below a particular value? For example, the probability that a drawing will be below 0 is clearly 50 percent because the normal distribution is symmetric. Using statistical terminology, we say that the cumulative probability of 0 is 50 percent. Statisticians also say that N(0) = 50%. It turns out that The first value means that there is a 64.59 percent probability that a drawing from the standardized normal distribution will be below .3742. The second value means that there is a 56.07 percent probability that a drawing from the standardized normal distribution will be below .1527. More generally, N(d) is the notation that a drawing from the standardized normal distribution will be below d. In other words, N(d) is the cumulative probability of d. Note that d1 and d2 in our example are slightly above zero, so N(d1) and N(d2) are slightly greater than .50. Perhaps the easiest way to determine N(d1) and N(d2) is from the Excel function NORMSDIST. In our example, NORMSDIST(.3742) and NORMSDIST(.1527) are .6459 and .5607, respectively. We can also determine the cumulative probability from Table 17.3. For example, consider d = .37. This can be found in the table as .3 on the vertical and .07 on the horizontal. The value in the table for d = .37 is .1443. This value is not the cumulative probability of .37. One must first make an adjustment to determine cumulative probability. That is: Unfortunately, our table handles only two significant digits, whereas our value of .3742 has four significant digits. Hence, we must interpolate to find N(.3742). Because N(.37) = .6443 and N(.38) = .6480, the difference between the two values is .0037 (= .6480 − .6443). Because .3742 is 42 percent of the way between .37 and .38, we interpolate as:9 TABLE 17.3 Cumulative Probabilities of the Standard Normal Distribution Function 958 d .00 .01 .02 .03 .04 .05 .06 .07 .0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 2.0 .4773 .4778 .4783 .4788 .4793 .4798 .4803 .4808 2.1 .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 2.2 .4861 .4866 .4830 .4871 .4875 .4878 .4881 .4884 2.3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 2.4 .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 2.5 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 2.6 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 2.7 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 2.8 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 959 2.9 .4981 .4982 .4982 .4982 .4984 .4984 .4985 .4985 3.0 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 N(d) represents areas under the standard normal distribution function. Suppose that d1 = .24. This table implies a cumulative probability of .5000 + .0948 = .5948. If d1 is equal to .2452, we must estimate the probability by interpolating between N(.25) and N(.24). page 534 Step 3: Calculate C. We have: The estimated price of $5.85 is greater than the $4 actual price, implying that the call option is underpriced. A trader believing in the Black–Scholes model would buy a call. Of course, the Black–Scholes model is fallible. Perhaps the disparity between the model’s estimate and the market price reflects error in the trader’s estimate of variance. page 535 The previous example stressed the calculations involved in using the Black–Scholes formula. Is there any intuition behind the formula? Yes, and that intuition follows from the stock purchase and borrowing strategy in our binomial example. The Black–Scholes equation is: which is exactly analogous to Equation 17.2: that we presented in the binomial example. It turns out that N(d1) is the delta in the Black–Scholes formula. N(d1) is .6459 in the previous example. In addition, Ee−Rt N(d2) is the amount that an investor must borrow to duplicate a call. In the previous example, this value is $26.45 (= $49 × .9626 × .5607). Thus, the model tells us that we can duplicate the 960 call of the preceding example by both: Another good options calculator can be found at www.margrabe.com/optionpricing.html. 1. Buying .6459 share of stock. 2. Borrowing $26.45. It is no exaggeration to say that the Black–Scholes formula is among the most important contributions in finance. It allows anyone to calculate the value of an option given a few parameters. The attraction of the formula is that four of the parameters are observable: the current price of stock, S, the exercise price, E, the interest rate, R, and the time to expiration date, t. Only one of the parameters must be estimated: the variance of return, σ2. To see how truly attractive this formula is, note what parameters are not needed. First, the investor’s risk aversion does not affect the option value. The formula can be used by anyone, regardless of willingness to bear risk. Second, it does not depend on the expected return on the stock! Investors with different assessments of the stock’s expected return will nevertheless agree on the call price. As in the two-state example, this is because the call depends on the stock price and that price already balances investors’ divergent views. 17.9 STOCKS AND BONDS AS OPTIONS The previous material in this chapter described, explained, and valued publicly traded options. This is important material to any finance student because much trading occurs in these listed options. The study of options has another purpose for the student of corporate finance. You may have heard the one-liner about the elderly gentleman who was surprised to learn that he had been speaking prose all of his life. The same can be said about the corporate finance student and options. Although options were formally defined for the first time in this chapter, we can actually describe the corporation in terms of options. We begin by illustrating the implicit options in stocks and bonds through a simple example. 961 EXAMPLE 17.5 Stocks and Bonds as Options The Popov Company has been awarded the concessions at next year’s Olympic Games in Antarctica. Because the firm’s principals do not live in Antarctica and because there is no other concession business in that continent, their enterprise will disband after the games. The firm has issued debt to help finance this venture. Interest and principal due on the debt next year will be $800, at which time the debt will be paid off in full. The firm’s cash flows next year are forecasted as page 536 POPOV’S CASH FLOW SCHEDULE VERY SUCCESSFUL GAMES MODERATELY SUCCESSFUL GAMES MODERATELY UNSUCCESSFUL GAMES $1,000 $850 $700 800 − 800 $ 200 $ 50 Cash flow before interest and principal − Interest and principal Cash flow to stockholders − − $ As can be seen, the principals forecasted four equally likely scenarios. If either of the first two scenarios occurs, the bondholders will be paid in full. The extra cash flow goes to the stockholders. However, if either of the last two scenarios occurs, the bondholders will not be paid in full. Instead, they will receive the firm’s entire cash flow, leaving the stockholders with nothing. This example is similar to the bankruptcy examples presented in our chapters on capital structure. Our new insight is that the relationship between the common stock and the firm can be expressed in terms of options. We consider call options first because the intuition is easier. The put option scenario is treated next. The Firm Expressed in Terms of Call Options 962 THE STOCKHOLDERS We now show that stock can be viewed as a call option on the firm. To illustrate this, Figure 17.11 graphs the cash flow to the stockholders as a function of the cash flow to the firm. The stockholders receive nothing if the firm’s cash flows are less than $800; here, all of the cash flows go to the bondholders. However, the stockholders earn a dollar for every dollar that the firm receives above $800. The graph looks exactly like the call option graphs that we considered earlier in this chapter. But what is the underlying asset upon which the stock is a call option? The underlying asset is the firm itself. That is, we can view the bondholders as owning the firm. However, the stockholders have a call option on the firm with an exercise price of $800. FIGURE 17.11 Cash Flow to Stockholders of Popov Company as a Function of Cash Flow to Firm If the firm’s cash flow is above $800, the stockholders would choose to exercise this option. In other words, they would buy the firm from the bondholders for $800. Their net cash flow is the difference page 537 between the firm’s cash flow and their $800 payment. This would be $200 (= $1,000 − 800) if the games are very successful and $50 (= $850 − 800) if the games are moderately successful. FIGURE 17.12 Cash Flow to Bondholders of Popov Company as a Function of Cash Flow to Firm 963 Should the value of the firm’s cash flows be less than $800, the stockholders would not choose to exercise their option. Instead, they would walk away from the firm, as any call option holder would do. The bondholders then would receive the firm’s entire cash flow. This view of the firm is a novel one, and students are frequently bothered by it on first exposure. However, we encourage students to keep looking at the firm in this way until the view becomes second nature to them. THE BONDHOLDERS What about the bondholders? Our earlier cash flow schedule showed that they would get the entire cash flow of the firm if the firm generated less cash than $800. Should the firm earn more than $800, the bondholders would receive only $800. That is, they would be entitled only to interest and principal. This schedule is graphed in Figure 17.12. In keeping with our view that the stockholders have a call option on the firm, what does the bondholders’ position consist of? The bondholders’ position can be described by two claims: 1. They own the firm. 2. They have written a call against the firm with an exercise price of $800. As we mentioned before, the stockholders walk away from the firm if cash flows are less than $800. Thus, the bondholders retain ownership in this case. However, if the cash flows are greater than $800, the stockholders exercise their option. They call the stock away from the 964 bondholders for $800. The Firm Expressed in Terms of Put Options The preceding analysis expresses the positions of the stockholders and the bondholders in terms of call options. We can now express the situation in terms of put options. THE STOCKHOLDERS The stockholders’ position can be expressed by three claims: 1. They own the firm. 2. They owe $800 in interest and principal to the bondholders. If the debt were risk-free, these two claims would fully describe the stockholders’ situation. However, because of the possibility of default, we have a third claim as well: page 538 3. The stockholders own a put option on the firm with an exercise price of $800. The group of bondholders is the seller of the put. Now consider two possibilities: CASH FLOW IS LESS THAN $800 Because the put has an exercise price of $800, the put is in the money. The stockholders “put,” that is, sell, the firm to the bondholders. Normally, the holder of a put receives the exercise price when the asset is sold. However, the stockholders already owe $800 to the bondholders. Thus, the debt of $800 is simply canceled— and no money changes hands—when the stock is delivered to the bondholders. Because the stockholders give up the stock in exchange for extinguishing the debt, the stockholders end up with nothing if the cash flow is below $800. CASH FLOW IS GREATER THAN $800 Because the put is out of the money here, the stockholders do not exercise. Thus, the stockholders retain ownership of the firm but pay $800 to the bondholders as interest and principal. THE BONDHOLDERS The bondholders’ position can be described by two claims: 965 1. The bondholders are owed $800. 2. They have sold a put option on the firm to the stockholders with an exercise price of $800. CASH FLOW IS LESS THAN $800 As mentioned before, the stockholders will exercise the put in this case. This means that the bondholders are obligated to pay $800 for the firm. Because they are owed $800, the two obligations offset each other. Thus, the bondholders simply end up with the firm in this case. CASH FLOW IS GREATER THAN $800 Here, the stockholders do not exercise the put. Thus, the bondholders merely receive the $800 that is due them. Expressing the bondholders’ position in this way is illuminating. With a riskless default-free bond, the bondholders are owed $800. Thus, we can express the risky bond in terms of a riskless bond and a put: That is, the value of the risky bond is the value of the default-free bond less the value of the stockholders’ option to sell the company for $800. A Resolution of the Two Views We have argued above that the positions of the stockholders and the bondholders can be viewed either in terms of calls or in terms of puts. These two viewpoints are summarized in Table 17.4: We have found from past experience that it is generally harder for students to think of the firm in terms of puts than in terms of calls. Thus, it would be helpful if there were a way to show that the two viewpoints are equivalent. Fortunately, there is put–call parity. In an earlier section, we presented the put–call parity relationship as Equation 17.1, which we now repeat: page 539 TABLE 17.4 Positions of Stockholders and Bondholders in Popov Company in Terms of Calls and 966 Puts STOCKHOLDERS BONDHOLDERS Positions Viewed in Terms of Call Options 1. Stockholders own a call on the firm with exercise price of $800. 1. Bondholders own the firm. 2. Bondholders have sold a call on the firm to the stockholders. Positions Viewed in Terms of Put Options 1. Stockholders own the firm. 2. Stockholders owe $800 in interest and principal to bondholders. 3. Stockholders own a put option on the firm with exercise price of $800. 1. Bondholders are owed $800 in interest and principal. 2. Bondholders have sold a put on the firm to the stockholders. Using the results of this section, Equation 17.1 can be rewritten as: Going from Equation 17.1 to Equation 17.4 involves a few steps. First, we treat the firm, not the stock, as the underlying asset in this section. (In keeping with common convention, we refer to the value of the firm and the price of the stock.) Second, the exercise price is now $800, the principal and interest on the firm’s debt. Taking the present value of this amount at the riskless rate yields the value of a default-free bond. Third, the order of the terms in Equation 17.1 is rearranged in Equation 17.4. Note that the left-hand side of Equation 17.4 is the stockholders’ position in terms of call options, as shown in Table 17.4. The right-hand side of Equation 17.4 is the stockholders’ position in terms of put options, as shown in the table. Thus, put–call parity shows that viewing the stockholders’ position in terms of call options is equivalent to viewing the stockholders’ position in terms of put options. Now, let’s rearrange the terms in Equation 17.4 to yield: 967 The left-hand side of Equation 17.5 is the bondholders’ position in terms of call options, as shown in Table 17.4. The right-hand side of the equation is the bondholders’ position in terms of put options, as shown in Table 17.4. Thus, put–call parity shows that viewing the bondholders’ position in terms of call options is equivalent to viewing the bondholders’ position in terms of put options. A Note on Loan Guarantees In the Popov example given earlier, the bondholders bore the risk of default. Of course, bondholders generally ask for an interest rate that is enough to compensate them for bearing risk. When firms experience financial distress, they can no longer attract new debt at moderate interest rates. Thus, firms experiencing distress have frequently sought loan guarantees from the government. Our framework can be used to understand these guarantees. page 540 If the firm defaults on a guaranteed loan, the government must make up the difference. In other words, a government guarantee converts a risky bond into a riskless bond. What is the value of this guarantee? Recall that, with option pricing: This equation shows that the government is assuming an obligation that has a cost equal to the value of a put option. This analysis differs from that of either politicians or company spokespeople. They generally say that the guarantee will cost the taxpayer nothing because the guarantee enables the firm to attract debt, thereby staying solvent. However, it should be pointed out that, although solvency may be a strong possibility, it is never a certainty. Thus, at the time when the guarantee is made, the government’s obligation has a cost in terms of present value. To say that a government guarantee costs the government nothing is like saying a put on the stock of Microsoft has no value because the stock is likely to rise in price. 968 Who benefits from a typical loan guarantee? 1. If existing risky bonds are guaranteed, all gains accrue to the existing bondholders. The stockholders gain nothing because the limited liability of corporations absolves the stockholders of any obligation in bankruptcy. 2. If new debt is being issued and guaranteed, the new debtholders do not gain. Rather, in a competitive market, they must accept a low interest rate because of the debt’s low risk. The stockholders gain here because they are able to issue debt at a low interest rate. In addition, some of the gains accrue to the old bondholders because the firm’s value is greater than would otherwise be true. Therefore, if shareholders want all the gains from loan guarantees, they should renegotiate or retire existing bonds before the guarantee is in place. SUMMARY AND CONCLUSIONS This chapter serves as an introduction to options. 1. The most familiar options are puts and calls. These options give the holder the right to sell or buy shares of common stock at a given exercise price. American options can be exercised any time up to and including the expiration date. European options can be exercised only on the expiration date. 2. We showed that a strategy of buying a stock and buying a put is equivalent to a strategy of buying a call and buying a zero coupon bond. From this, the put–call parity relationship was established: 3. The value of an option depends on five factors: a. b. c. d. e. The price of the underlying asset. The exercise price. The expiration date. The variability of the underlying asset’s return. The interest rate on risk-free bonds. The Black–Scholes model can determine the intrinsic price of an option from these five factors. 4. The positions of stockholders and bondholders can be 969 described in terms of calls and puts. page 541 CONCEPT QUESTIONS 1. Options What is a call option? A put option? Under what circumstances might you want to buy each? Which one has greater potential profit? Why? 2. Options Complete the following sentence for each of these investors: a. b. c. d. A buyer of call options A buyer of put options A seller (writer) of call options A seller (writer) of put options “The (buyer/seller) of a (put/call) option (pays/receives) money for the (right/obligation) to (buy/ sell) a specified asset at a fixed price for a fixed length of time.” 3. American and European Options What is the difference between an American option and a European option? 4. Intrinsic Value What is the intrinsic value of a call option? Of a put option? How do we interpret this value? 5. Option Pricing You notice that shares of stock in the Patel Corporation are going for $50 per share. Call options with an exercise price of $35 per share are selling for $10. What’s wrong here? Describe how you can take advantage of this mispricing if the option expires today. 6. Options and Stock Risk If the risk of a stock increases, what is likely to happen to the price of call options on the stock? To the price of put options? Why? 7. Option Rise True or false: The unsystematic risk of a share of stock is irrelevant in valuing the stock because it can be diversified away; therefore, it is also irrelevant for valuing a call option on the stock. Explain. 8. Option Pricing Suppose a certain stock currently sells for $30 per share. If a put option and a call option are available with $30 exercise prices, which do you think will sell for more, the put or the call? 970 Explain. 9. Option Price and Interest Rates Suppose the interest rate on Tbills suddenly and unexpectedly rises. All other things being the same, what is the impact on call option values? On put option values? 10. Contingent Liabilities When you take out an ordinary student loan, it is often the case that whoever holds that loan is given a guarantee by the U.S. government, meaning that the government will make up any payments you skip. This is just one example of the many loan guarantees made by the U.S. government. Such guarantees don’t show up in calculations of government spending or in official deficit figures. Why not? Should they show up? 11. Options and Expiration Dates What is the impact of lengthening the time to expiration on an option’s value? Explain. 12. Options and Stock Price Volatility What is the impact of an increase in the volatility of the underlying stock’s return on an option’s value? Explain. 13. Insurance as an Option An insurance policy is considered analogous to an option. From the policyholder’s point of view, what type of option is an insurance policy? Why? 14. Equity as a Call Option It is said that the equityholders of a levered firm can be thought of as holding a call option on the firm’s assets. Explain what is meant by this statement. 15. Option Valuation and NPV You are CEO of Titan Industries and have just been awarded a large number of employee stock options. The company has two mutually exclusive projects available. The first project has a large NPV and will reduce the total risk of the company. The second project has a small NPV and will increase the total risk of the company. You have decided to accept the first project when you remember your employee stock options. How might this affect your decision? 16. Put–Call Parity You find a put and a call with the same exercise price and maturity. What do you know about the relative prices of the put and call? Prove your answer and provide an intuitive explanation. 17. Put–Call Parity A put and a call have the same maturity page 542 and strike price. If they have the same price, which one is in the money? Prove your answer and provide an intuitive explanation. 18. Put–Call Parity One thing put-call parity tells us is that given any three of a stock, a call, a put, and a T-bill, the fourth can be synthesized or replicated using the other three. For example, how can 971 we replicate a share of stock using a call, a put, and a T-bill? QUESTIONS AND PROBLEMS Basic (Questions 1–14) 1. Understanding Option Quotes Use the option quote information shown here to answer the questions that follow. The stock is currently selling for $82. OPTION AND NY CLOSE CALLS EXPIRATION STRIKE PRICE VOL. Mar 80 230 Apr 80 Jul Oct PUTS LAST VOL. LAST 1.90 160 .80 170 6 127 1.40 80 139 8.05 43 3.90 80 60 10.20 11 3.65 RWJ a. Are the call options in the money? What is the intrinsic value of an RWJ Corp. call option? b. Are the put options in the money? What is the intrinsic value of an RWJ Corp. put option? c. Two of the options are clearly mispriced. Which ones? At a minimum, what should the mispriced options sell for? Explain how you could profit from the mispricing in each case. 2. Calculating Payoffs Use the option quote information shown below to answer the questions that follow. The stock is currently selling for $97. OPTION AND NY CLOSE CALLS PUTS EXPIRATION STRIKE PRICE VOL. LAST VOL. LAST Feb 95 85 2.85 40 1.13 Macrosoft 972 Mar 95 61 5.25 22 4.05 May 95 22 7.78 11 5.94 Aug 95 3 10.20 3 9.12 a. Suppose you buy 10 contracts of the February 95 call option. How much will you pay, ignoring commissions? b. In part (a), suppose that Macrosoft stock is selling for $105 per share on the expiration date. How much is your options investment worth? What if the terminal stock price is $112? Explain. c. Suppose you buy 10 contracts of the August 95 put option. What is your maximum gain? On the expiration date, Macrosoft is selling for $89 per share. How much is your options investment worth? What is your net gain? d. In part (c), suppose you sell 10 of the August 95 put contracts. What is your net gain or loss if Macrosoft is selling for $87 at expiration? For $103? What is the breakeven price, that is, the terminal stock price that results in a zero profit? Put–Call 3. Parity A stock is currently selling for $47 per share. A call option with an exercise price of $50 sells for $2.61 and expires in three months. If the risk-free rate of interest is 3.9 percent per year, compounded continuously, what is the price of a put option with the same exercise price? 4. Put–Call Parity A put option that expires in five page 543 months with an exercise price of $60 sells for $2.87. The stock is currently priced at $58, and the risk-free rate is 4.1 percent per year, compounded continuously. What is the price of a call option with the same exercise price? 5. Put–Call Parity A put option and a call option with an exercise price of $60 and three months to expiration sell for $3.87 and $4.89, respectively. If the risk-free rate is 2.7 percent per year, compounded continuously, what is the current stock price? 6. Put–Call Parity A put option and a call option with an exercise price of $90 expire in four months and sell for $7.05 and $11.74, respectively. If the stock is currently priced at $94.13, what is the annual continuously compounded rate of interest? 973 7. Black–Scholes What are the prices of a call option and a put option with the following characteristics? 8. Black–Scholes What are the prices of a call option and a put option with the following characteristics? Delta 9. What are the deltas of a call option and a put option with the following characteristics? What does the delta of the option tell you? 10. Black–Scholes and Asset Value You own a lot in Key West, Florida, that is currently unused. Similar lots have recently sold for $1.45 million. Over the past five years, the price of land in the area has increased 7 percent per year, with an annual standard deviation of 25 percent. A buyer has recently approached you and wants an option to buy the land in the next 12 months for $1.6 million. The risk-free rate of interest is 5 percent per year, compounded continuously. How much should you charge for the option? 11. Black–Scholes and Asset Value In the previous problem, suppose you wanted the option to sell the land to the buyer in one year. Assuming all the facts are the same, describe the transaction that would occur today. What is the price of the transaction today? 974 12. Time Value of Options You are given the following information concerning options on a particular stock: a. What is the intrinsic value of the call option? Of page 544 the put option? b. What is the time value of the call option? Of the put option? c. Does the call or the put have the larger time value component? Would you expect this to be true in general? 13. Risk-Neutral Valuation A stock is currently priced at $58. The stock will either increase or decrease by 13 percent over the next year. There is a call option on the stock with a strike price of $55 and one year until expiration. If the risk-free rate is 6 percent, what is the risk-neutral value of the call option? 14. Risk-Neutral Valuation In the previous problem, assume the risk-free rate is only 4 percent. What is the risk-neutral value of the option now? What happens to the risk-neutral probabilities of a stock price increase and a stock price decrease? Intermediate (Questions 15–25) 15. Black–Scholes A call option matures in six months. The underlying stock price is $75, and the stock’s return has a standard deviation of 20 percent per year. The risk-free rate is 4 percent per year, compounded continuously. If the exercise price is $0, what is the price of the call option? 16. Black–Scholes A call option has an exercise price of $65 and matures in six months. The current stock price is $68, and the risk-free rate is 4 percent per year, compounded continuously. What is the price of the call if the standard deviation of the stock is 0 percent per year? 17. Black–Scholes A stock is currently priced at $35. A call option with an expiration of one year has an exercise price of $40. The risk-free rate is 3 percent per year, compounded continuously, and the standard deviation of the stock’s return is infinitely large. What is the price of the call option? 18. Equity as an Option Sunburn Sunscreen has a zero coupon 975 bond issue outstanding with a $15,000 face value that matures in one year. The current market value of the firm’s assets is $16,100. The standard deviation of the return on the firm’s assets is 32 percent per year, and the annual risk-free rate is 6 percent per year, compounded continuously. Based on the Black–Scholes model, what is the market value of the firm’s equity and debt? 19. Equity as an Option and NPV Suppose the firm in the previous problem is considering two mutually exclusive investments. Project A has an NPV of $800, and Project B has an NPV of $1,300. As a result of taking Project A, the standard deviation of the return on the firm’s assets will increase to 50 percent per year. If Project B is taken, the standard deviation will fall to 23 percent per year. a. What is the value of the firm’s equity and debt if Project A is undertaken? If Project B is undertaken? b. Which project would the stockholders prefer? Can you reconcile your answer with the NPV rule? c. Suppose the stockholders and bondholders are in fact the same group of investors. Would this affect your answer to (b)? d. What does this problem suggest to you about stockholder incentives? 20. Equity as an Option Frostbite Thermalwear has a zero coupon bond issue outstanding with a face value of $22,000 that matures in one year. The current market value of the firm’s assets is $23,200. The standard deviation of the return on the firm’s assets is 34 percent per year, and the annual risk-free rate is 6 percent per year, compounded continuously. Based on the Black–Scholes model, what is the market value of the firm’s equity and debt? What is the firm’s continuously compounded cost of debt? 21. Equity as an Option and NPV A company has a single zero coupon bond outstanding that matures in 10 years and has a face value of $12 million. The current value of the company’s assets is $10.7 million, and the standard deviation of the return on the firm’s assets is 47 percent per year. The risk-free rate is 6 percent per year, compounded continuously. a. What is the current market value of the company’s equity? b. What is the current market value of the company’s debt? 976 c. What is the company’s continuously compounded cost of debt? d. The company has a new project available. The project has an NPV of $1.3 million. If the company undertakes the project, what will be the new market value of equity? Assume volatility is unchanged. e. Assuming the company undertakes the new page 545 project and does not borrow any additional funds, what is the new continuously compounded cost of debt? What is happening here? 22. Two-State Option Pricing Model Ken is interested in buying a European call option written on Southeastern Airlines, Inc., a non-dividend-paying common stock, with a strike price of $60 and one year until expiration. Currently, the company’s stock sells for $64 per share. In one year, Ken knows that the stock will be trading at either $71 per share or $56 per share. Ken is able to borrow and lend at the risk-free EAR of 5.5 percent. a. What should the call option sell for today? b. If no options currently trade on the stock, is there a way to create a synthetic call option with identical payoffs to the call option described above? If there is, how would you do it? c. How much does the synthetic call option cost? Is this greater than, less than, or equal to what the actual call option costs? Does this make sense? 23. Two-State Option Pricing Model Rob wishes to buy a European put option on BioLabs, Inc., a non-dividend-paying common stock, with a strike price of $75 and six months until expiration. The company’s common stock is currently selling for $73 per share, and Rob expects that the stock price will either rise to $83 or fall to $61 in six months. Rob can borrow and lend at the risk-free EAR of 4.3 percent. a. What should the put option sell for today? b. If no options currently trade on the stock, is there a way to create a synthetic put option with identical payoffs to the put option described above? If there is, how would you do it? c. How much does the synthetic put option cost? Is this greater than, less than, or equal to what the actual put option costs? Does this make sense? 24. Two-State Option Pricing Model Maverick Manufacturing, 977 Inc., must purchase gold in three months for use in its operations. Maverick’s management has estimated that if the price of gold were to rise above $1,310 per ounce, the firm would go bankrupt. The current price of gold is $1,230 per ounce. The firm’s chief financial officer believes that the price of gold will either rise to $1,350 per ounce or fall to $1,120 per ounce over the next three months. Management wishes to eliminate any risk of the firm going bankrupt. The company can borrow and lend at the risk-free EAR of 4 percent. a. Should the company buy a call option or a put option on gold? In order to avoid bankruptcy, what strike price and time to expiration would the company like this option to have? b. How much should such an option sell for in the open market? c. If no options currently trade on gold, is there a way for the company to create a synthetic option with identical payoffs to the option described above? If there is, how would the firm do it? d. How much does the synthetic option cost? Is this greater than, less than, or equal to what the actual option costs? Does this make sense? 25. Black–Scholes and Collar Cost An investor is said to take a position in a “collar” if she buys the asset, buys an out-of-themoney put option on the asset, and sells an out-of-the-money call option on the asset. The two options should have the same time to expiration. Suppose Marie wishes to purchase a collar on Hollywood, Inc., a non-dividend-paying common stock, with six months until expiration. She would like the put to have a strike price of $50 and the call to have a strike price of $60. The current price of the stock is $54 per share. Marie can borrow and lend at the continuously compounded risk-free rate of 4.7 percent per year, and the annual standard deviation of the stock’s return is 45 percent. Use the Black–Scholes model to calculate the total cost of the collar that Marie is interested in buying. What is the effect of the collar? Challenge (Questions 26–35) 26. Mergers and Equity as an Option Suppose Sunburn Sunscreen (Problem 18) and Frostbite Thermalwear (Problem 20) have decided to merge. Since the two companies have seasonal sales, the combined firm’s return on assets will have a standard deviation of 16 percent per year. 978 a. What is the combined value of equity in the two existing companies? Value of debt? b. What is the value of the new firm’s equity? Value of debt? c. What was the gain or loss for shareholders? For page 546 bondholders? d. What happened to shareholder value here? 27. Debt Valuation and Time to Maturity Eagle Industries has a zero coupon bond issue that matures in two years with a face value of $75,000. The current value of the company’s assets is $58,000, and the standard deviation of the return on assets is 60 percent per year. a. Assume the risk-free rate is 5 percent per year, compounded continuously. What is the value of a risk-free bond with the same face value and maturity as the company’s bond? b. What price would the bondholders have to pay for a put option on the firm’s assets with a strike price equal to the face value of the debt? c. Using the answers from (a) and (b), what is the value of the firm’s debt? What is the continuously compounded yield on the company’s debt? d. From an examination of the value of the assets of the company, and the fact that the debt must be repaid in two years, it seems likely that the company will default on its debt. Management has approached bondholders and proposed a plan whereby the company would repay the same face value of debt, but the repayment would not occur for five years. What is the value of the debt under the proposed plan? What is the new continuously compounded yield on the debt? Explain why this occurs. 28. Debt Valuation and Asset Variance Watson Corp. has a zero coupon bond that matures in five years with a face value of $75,000. The current value of the company’s assets is $61,000, and the standard deviation of its return on assets is 50 percent per year. The risk-free rate is 6 percent per year, compounded continuously. a. What is the value of a risk-free bond with the same face value and maturity as the current bond? b. What is the value of a put option on the firm’s assets with a strike price equal to the face value of the debt? 979 c. Using the answers from (a) and (b), what is the value of the firm’s debt? What is the continuously compounded yield on the company’s debt? d. Assume the company can restructure its assets so that the standard deviation of its return on assets increases to 60 percent per year. What happens to the value of the debt? What is the new continuously compounded yield on the debt? Reconcile your answers in (c) and (d). e. What happens to bondholders if the company restructures its assets? What happens to shareholders? How does this create an agency problem? 29. Two-State Option Pricing and Corporate Valuation Masters Real Estate, Inc., a construction firm financed by both debt and equity, is undertaking a new project. If the project is successful, the value of the firm in one year will be $125 million, but if the project is a failure, the firm will only be worth $85 million. The company’s current value is $103 million, a figure that includes the prospects for the new project. The company has outstanding zero coupon bonds due in one year with a face value of $95 million. Treasury bills that mature in one year yield 7 percent EAR. The company pays no dividends. a. Use the two-state option pricing model to find the current value of the company’s debt and equity. b. Suppose the company has 300,000 shares of common stock outstanding. What is the price per share of the firm’s equity? c. Compare the market value of the company’s debt to the present value of an equal amount of debt that is riskless with one year until maturity. Is the firm’s debt worth more than, less than, or the same as the riskless debt? Does this make sense? What factors might cause these two values to be different? d. Suppose that in place of the project described above, the company’s management decides to undertake a project that is even more risky. The value of the firm will either increase to $135 million or decrease to $70 million by the end of the year. Use the two-state option pricing model to determine the value of the firm’s debt and equity if the firm plans on undertaking this new project. What is the stock price if the firm undertakes this project? Which project do bondholders prefer? 30. Black–Scholes and Dividends In addition to the 980 page 547 five factors discussed in the chapter, dividends also affect the price of an option. The Black–Scholes option pricing model with dividends is All of the variables are the same as the Black–Scholes model without dividends except for the variable d, which is the continuously compounded dividend yield on the stock. a. What effect do you think the dividend yield will have on the price of a call option? Explain. b. A stock is currently priced at $87 per share, the standard deviation of its return is 50 percent per year, and the risk-free rate is 5 percent per year, compounded continuously. What is the price of a call option with a strike price of $80 and a maturity of 4 months if the stock has a dividend yield of 2 percent per year? 31. Put–Call Parity and Dividends The put–call parity condition is altered when dividends are paid. The dividend adjusted put– call parity formula is where d is again the continuously compounded dividend yield. a. What effect do you think the dividend yield will have on the price of a put option? Explain. b. From the previous question, what is the price of a put option with the same strike price and time to expiration as the call option? 32. Put Delta The delta for a put option is N(d1) − 1. Is this the same thing as −N(−d1)? (Hint: Yes, but why?) 33. Black–Scholes Put Pricing Model Use the Black–Scholes model for pricing a call, put–call parity, and the previous question to show that the Black–Scholes model for directly pricing a put can be written as 34. Black–Scholes A stock is currently priced at $50. The stock will never pay a dividend. The risk-free rate is 12 percent per year, compounded continuously, and the standard deviation of 981 the stock’s return is 60 percent. A European call option on the stock has a strike price of $100 and no expiration date, meaning that it has an infinite life. Based on Black–Scholes, what is the value of the call option? Do you see a paradox here? Do you see a way out of the paradox? 35. Delta You purchase one call and sell one put with the same strike price and expiration date. What is the delta of your portfolio? Why? page 548 WHAT’S ON THE WEB? 1. Black-Scholes Go to www.option-price.com. Find the stock option calculator. The call option and put option on a stock expire in 30 days. The strike price is $50 and the current stock price is $51.20. The standard deviation of the stock is 60 percent per year, and the riskfree rate is 4.8 percent per year, compounded continuously. What are the prices of the call and the put? What are the deltas? 2. Black-Scholes Go to www.cboe.com, and find the options calculator. A stock is currently priced at $93 per share, and its return has a standard deviation of 48 percent per year. Options are available with an exercise price of $90, and the risk-free rate is 5.2 percent per year, compounded continuously. What are the prices of the call and the put that expire next month? What are the deltas? How do your answers change for an exercise price of $95? 3. Black-Scholes with Dividends Recalculate the first two problems assuming a dividend yield of 2 percent per year. How does this change your answers? Can you explain why dividends have the effect they do? EXCEL MASTER IT! PROBLEM In addition to spinners and scroll bars, there are numerous other controls in Excel. For this assignment, you need to build a Black– Scholes option pricing model spreadsheet using several of these controls. a. Buttons are always used in sets. Using buttons permits you to check an option, and the spreadsheet will use that input. In this case, you need to create two buttons, one for a call option and 982 one for a put option. When using the spreadsheet, if you click the call option, the spreadsheet will calculate a call price, and if you click the put option, it will calculate the price of a put. Notice on the next spreadsheet that cell B20 is empty. This cell should change names. The names should be “Call option price” and “Put option price.” In the price cell, only the price for the call option or put option is displayed depending on which button is selected. For the button, use the button under Form Controls. b. A combo box uses a drop-down menu with values entered by the spreadsheet developer. One advantage of a combo box is that the user can either choose values from the drop-down menu or enter another value. In this case, you want to create one combo box for the stock price and a separate combo box for the strike price. On the right-hand side of the spreadsheet, we have values for the dropdown menu. These values should be created in an array before the combo box is inserted. To create an ActiveX combo box, go to Developer, Insert, and select Combo Box from the ActiveX Controls menu. After you draw the combo box, right-click on the box, select Properties, and enter the LinkedCell, which is the cell where you want the output displayed, and the ListFillRange, which is the range that contains the list of values you want displayed in the drop-down menu. c. In contrast to a combo box, a list box permits the user to scroll through a list of possible values that are predetermined by the spreadsheet developer. No other values can be entered. You need to create a list box for the interest rate using the interest rate array on the right-hand side of the spreadsheet. To insert a list box, go to Developer, Insert, and choose the List Box from the ActiveX Controls. To enter the linked cell and array of values, right-click on List Box and select Properties from the menu. We should note here that to edit both the combo box and list box you will need to make sure that Design Mode is checked on the Developer tab. CLOSING CASE EXOTIC CUISINES EMPLOYEE STOCK OPTIONS 983 As a newly minted MBA, you’ve taken a management position with Exotic Cuisines, Inc., a restaurant chain that just went public last year. The company’s restaurants specialize in exotic main dishes, using ingredients such as alligator, buffalo, and ostrich. A concern you had going in was that the restaurant business is very risky. However, after some due diligence, you discovered a common misperception about the restaurant industry. It is widely thought that 90 percent of new restaurants close within three years; however, recent evidence suggests the failure rate is closer to 60 percent over three years. So, it is a risky business, although not as risky as you originally thought. During your interview process, one of the benefits mentioned was employee stock options. Upon signing your employment contract, you received options with a strike price of $45 for 10,000 shares of company stock. As is fairly common, your stock options have a three-year vesting period and a 10-year expiration, meaning that you cannot exercise the options for a period of three years, and you lose them if you leave before they vest. After the three-year vesting period, you can exercise the options at any time. Thus, the employee stock options are European (and subject to forfeit) for the first three years and American afterward. Of course, you cannot sell the options, nor can you enter into any sort of hedging agreement. If you leave the company after the options vest, you must exercise within 90 days or forfeit. Exotic Cuisines stock is currently trading at $17.89 per share, a slight increase from the initial offering price last year. There are no markettraded options on the company’s stock. Because the company has only been traded for about a year, you are reluctant to use the page 549 historical returns to estimate the standard deviation of the stock’s return. However, you have estimated that the average annual standard deviation for restaurant company stocks is about 55 percent. Since Exotic Cuisines is a newer restaurant chain, you decide to use a 65 percent standard deviation in your calculations. The company is relatively young, and you expect that all earnings will be reinvested back into the company for the near future. Therefore, you expect no dividends will be paid for at least the next 10 years. A three-year Treasury note currently has a yield of 1.5 percent, and a 10-year Treasury note has a yield of 2.6 percent. 1. You’re trying to value your options. What minimum value would you assign? What is the maximum value you would assign? 2. Suppose that, in three years, the company’s stock is trading at $60. At that time, should you keep the options or exercise them immediately? What are some of the important determinants in making such a decision? 984 3. Your options, like most employee stock options, are not transferable or tradeable. Does this have a significant effect on the value of the options? Why? 4. Why do you suppose employee stock options usually have a vesting provision? Why must they be exercised shortly after you depart the company even after they vest? 5. A controversial practice with employee stock options is repricing. What happens is that a company experiences a stock price decrease, which leaves employee stock options far out of the money or “underwater.” In such cases, many companies have “repriced” or “restruck” the options, meaning that the company leaves the original terms of the option intact, but lowers the strike price. Proponents of repricing argue that since the option is very unlikely to end in the money because of the stock price decline, the motivational force is lost. Opponents argue that repricing is in essence a reward for failure. How do you evaluate this argument? How does the possibility of repricing affect the value of an employee stock option at the time it is granted? 6. As we have seen, much of the volatility in a company’s stock price is due to systematic or marketwide risks. Such risks are beyond the control of a company and its employees. What are the implications for employee stock options? In light of your answer, can you recommend an improvement over traditional employee stock options? 1 We use buyer, owner, and holder interchangeably. 2 This example assumes that the call lets the holder purchase one share of stock at $100. In reality, one call option contract would let the holder purchase 100 shares. The profit would then equal $3,000 = [($130 – 100) × 100]. 3 Actually, because of differing exercise prices, the two graphs are not quite mirror images of each other. The exercise price In Figure 17.1 is $100 and the exercise price in Figure 17.3 is $50. 4 Our discussion in this section is of American options because they are more commonly traded in the real world. As necessary, we will indicate differences for European options. 5 This relationship need not hold for a European call option. Consider a firm with two otherwise identical European call options, one expiring at the end of May and the other expiring a few months later. Further assume that a huge dividend is paid in early June. If the first call is exercised at the end of May, its holder will receive the underlying stock. If he does not sell the stock, he will receive the large dividend shortly thereafter. However, the holder of the second call will receive the stock through exercise after the dividend is paid. Because the market knows that the 985 holder of this option will miss the dividend, the value of the second call option could be less than the value of the first. 6 This graph assumes that, for each security, the exercise price is equal to the expected stock price. This assumption is employed merely to facilitate the discussion. It is not needed to show the relationship between a call’s value and the volatility of the underlying stock. 7 Though this result must hold in the case of an American put, it need not hold for a European put. 8 A full treatment of this assumption can be found in John C. Hull, Options, Futures and Other Derivatives, 9th ed. (Upper Saddle River, NJ: Pearson, 2014). 9 This method is called linear interpolation. It is only one of a number of possible methods of interpolation. 986 page 550 Short-Term Finance and Planning 18 OPENING CASE Most often, when news breaks about a firm’s c...
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