# OPTION VALUATION. Topics in Corporate Finance P A R T 8. ON JULY 7, 2008, the closing stock prices for LEARNING OBJECTIVES

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1 LEARNING OBJECTIVES After studying this chapter, you should understand: LO1 The relationship between stock prices, call prices, and put prices using put call parity. LO2 The famous Black Scholes option pricing model and its uses. LO3 How the five factors in the Black Scholes formula affect the value of an option. LO4 How the Black Scholes model can be used to value the debt and equity of a firm. LO5 How option valuation can result in some surprising conclusions regarding mergers and capital budgeting decisions. OPTION VALUATION 25 ON JULY 7, 2008, the closing stock prices for Costco, biotech company Genzyme, and Amazon. com were \$72.87, \$72.79, and \$72.49, respectively. Each company had a call option trading on the Chicago Board Options Exchange with a \$70 strike price and an expiration date of October 18, 103 days away. You might expect that the prices on these call options would be similar, but they weren t. The Costco options sold for \$4.80, Genzyme options traded at \$7 and Amazon.com options traded at \$9.70. 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? If you go back to our earlier chapter about options, the volatility of the underlying stock is an important determinant of an option s value, and, in fact, these three stocks have very different volatilities. In this chapter, we explore this issue and many others in much greater depth using the Nobel Prize-winning Black Scholes Option Pricing Model. Master the ability to solve problems in this chapter by using a spreadsheet. Access Excel Master on the student Web site In an earlier chapter, we explored the basics of options, but we didn t discuss how to value them in much detail. Our goal in this chapter is to take this next step and examine how to actually estimate what an option is worth. To do this, we will explore two very famous results: the put call parity condition and the Black Scholes Option Pricing Model. An understanding of option valuation lets us illustrate and explore some very important ideas in corporate finance. For example, we will show why certain types of mergers are a bad idea. We will also examine some conflicts between bondholder and stockholder interests. We will even provide some examples under which companies have an incentive to take negative NPV projects. In each case, option-related effects underlie the issue. Topics in Corporate Finance P A R T ros46128_ch25.indd 799 1/16/09 10:32:02 AM

3 CHAPTER 25 Option Valuation 801 Stock Price in One Year Value of Call Option (Strike Price \$105) Value of Risk-Free Asset Combined Value Total Gain or Loss (Combined Value Less \$115) \$125 \$20 \$105 \$125 \$ TABLE 25.2 Gains and Losses in One Year. Original investment: purchase a one-year call option with a strike price of \$105 for \$15. Invest \$100 in a risk-free asset paying 5 percent. Total cost is \$115. What does this strategy accomplish? Once again, we will create a table to illustrate your gains and losses. Notice that in Table 25.2 your \$100 grows to \$105 based on a 5 percent interest rate. If you compare Table 25.2 to Table 25.1, you will make an interesting discovery. No matter what the stock price is one year from now, the two strategies always have the same value in one year! The fact that the two strategies always have exactly the same value in one year explains why they have the same cost today. If one of these strategies were cheaper than the other today, there would be an arbitrage opportunity involving buying the one that s cheaper and simultaneously selling the one that s more expensive. THE RESULT Our example illustrates a very important pricing relationship. What it shows is that a protective put strategy can be exactly duplicated by a combination of a call option (with the same strike price as the put option) and a riskless investment. In our example, notice that the investment in the riskless asset, \$100, is exactly equal to the present value of the strike price on the option calculated at the risk-free rate: \$105/1.05 \$100. Putting it all together, what we have discovered is the put call parity (PCP) condition. It says that: Share of stock A put option Present value of strike price A call option In symbols, we can write: [25.1] S P PV(E ) C [25.2] where S and P are stock and put values, and PV( E ) and C are the present value of the exercise price and the value of the call option. Because the present value of the exercise price is calculated using the risk-free rate, you can think of it as the price of a risk-free, pure discount instrument (such as a T-bill) with a face value equal to the strike price. In our experience, the easiest way to remember the PCP condition is to remember that stock plus put equals T-bill plus call. The PCP condition is an algebraic expression, meaning that it can be rearranged. For example, suppose we know that the risk-free rate is.5 percent per month. A call with a strike price of \$40 sells for \$4, and a put with the same strike price sells for \$3. Both have a three-month maturity. What s the stock price? put call parity (PCP) The relationship between the prices of the underlying stock, a call option, a put option, and a riskless asset. ros46128_ch25.indd 801 1/16/09 10:32:04 AM

4 802 PART 8 Topics in Corporate Finance To answer, we just use the PCP condition to solve for the stock price: S PV(E) C P [25.3] \$ \$40.41 The PCP condition really says that between a riskless asset (like a T-bill), a call option, a put option, and a share of stock, we can always figure out the price of any one of the four given the prices of the other three. EXAMPLE 25.1 Put Call Parity Suppose a share of stock sells for \$60. A six-month call option with a \$70 strike price sells for \$2. The risk-free rate is.4 percent per month. What s the price of a six-month put option with a \$70 strike? If we just use the PCP condition to solve for the put price, we get: P PV(E ) C S \$ \$60 \$10.34 Notice that, in this example, the put option is worth a lot more than the call. Why? EXAMPLE 25.2 More Parity Suppose a share of stock sells for \$110. A one-year, at-the-money call option sells for \$15. An at-the-money put with the same maturity sells for \$5. Can you create a risk-free investment by combining these three instruments? How? What s the risk-free rate? Here, we can use the PCP condition to solve for the present value of the strike price: PV(E) S P C \$ \$100 The present value of the strike price is thus \$100. Notice that because the options are at the money, the strike is the same as the stock price, \$110. So, if you put \$100 in a riskless investment today and receive \$110 in one year, the implied risk-free rate is obviously 10 percent. CONTINUOUS COMPOUNDING: A REFRESHER COURSE Back in Chapter 6, we saw that the effective annual interest rate (EAR) on an investment depends on compounding frequency. We also saw that, in the extreme, compounding can occur every instant, or continuously. So, as a quick refresher, suppose you invest \$100 at a rate of 6 percent per year compounded continuously. How much will you have in one year? How about in two years? In Chapter 6, we saw that the EAR with continuous compounding is: EAR e q 1 ros46128_ch25.indd 802 1/16/09 10:32:05 AM

5 CHAPTER 25 Option Valuation 803 where q is the quoted rate (6 percent, or.06, in this case) and e is the number , the base of the natural logarithms. Plugging in the numbers, we get: EAR e q or about 6.2 percent. Notice that most calculators have a key labeled e x, so doing this calculation is simply a matter of entering.06 and then pressing this key. With an EAR of percent, your \$100 investment will grow to \$ in one year. In two years, it will grow to: Future value \$ \$ \$ When we move into option valuation, continuous compounding shows up quite a bit, and it helps to have some shortcuts. In our examples here, we first converted the continuously compounded rate to an EAR and then did our calculations. It turns out that we don t need to do the conversion at all. Instead, we can calculate present and future values directly. In particular, the future value of \$1 for t periods at a continuously compounded rate of R per period is simply: Future value \$1 e Rt For example, looking back at the problem we just solved, the future value of \$100 in two years at a continuously compounded rate of 6 percent is: Future value \$100 e.06(2) \$ \$ \$ which is exactly what we had before. Similarly, we can calculate the present value of \$1 to be received in t periods at a continuously compounded rate of R per period as follows: Present value \$1 e Rt So, if we want the present value of \$15,000 to be received in five years at 8 percent compounded continuously, we would calculate: Present value \$15,000 e.08(5) \$15, \$15, \$10, Continuous Compounding EXAMPLE 25.3 What is the present value of \$500 to be received in six months if the discount rate is 9 percent per year, compounded continuously? In this case, notice that the number of periods is equal to one-half because six months is half of a year. Thus, the present value is: Present value \$500 e.09(1/2) \$ \$ \$478 ros46128_ch25.indd 803 1/16/09 10:32:05 AM

6 804 PART 8 Topics in Corporate Finance Looking back at our PCP condition, we wrote: S P PV(E ) C If we assume that R is the continuously compounded risk-free rate per year, then we could write this as: S P E e Rt C [25.4] where t is the time to maturity (in years) on the options. Finally, suppose we are given an EAR and we need to convert it to a continuously compounded rate. For example, if the risk-free rate is 8 percent per year compounded annually, what s the continuously compounded risk-free rate? Going back to our first formula, we had that: EAR e q 1 Now, we need to solve for q, the continuously compounded rate. Plugging in the numbers, we have:.08 e q 1 e q 1.08 We need to take the natural logarithm (ln) of both sides to solve for q : ln(e q ) ln(1.08) q or about 7.7 percent. Notice that most calculators have a button labeled ln, so doing this calculation involves entering 1.08 and then pressing this key. EXAMPLE 25.4 Even More Parity Suppose a share of stock sells for \$30. A three-month call option with a \$25 strike sells for \$7. A three-month put with the same strike price sells for \$1. What s the continuously compounded risk-free rate? We need to plug the relevant numbers into the PCP condition: S P E e Rt C \$30 1 \$25 e R(1/4) 7 Notice that we used one-fourth for the number of years because three months is a quarter of a year. We now need to solve for R : \$24 \$25 e R(1/4).96 e R(1/4) ln(.96) ln(e R(1/4) ).0408 R(1 4) R.1632 This is about percent, which would be a very high risk-free rate! Concept Questions 25.1a What is a protective put strategy? 25.1b What strategy exactly duplicates a protective put? ros46128_ch25.indd 804 1/16/09 10:32:06 AM

7 CHAPTER 25 Option Valuation 805 The Black Scholes Option Pricing Model We re now in a position to discuss one of the most celebrated results in modern finance, the Black Scholes Option Pricing Model (OPM). The OPM is a sufficiently important discovery that it was the basis for the Nobel Prize in Economics in The underlying development of the Black Scholes OPM is fairly complex, so we will focus only on the main result and how to use it. THE CALL OPTION PRICING FORMULA Black and Scholes showed that the value of a European-style call option on a non dividendpaying stock, C, can be written as follows: C S N(d 1 ) E e Rt N(d 2 ) [25.5] where S, E, and e Rt are as we previously defined them and N( d 1 ) and N( d 2 ) are probabilities that must be calculated. More specifically, N( d 1 ) is the probability that a standardized, normally distributed random variable (widely known as a z variable) is less than or equal to d 1, and N(d 2 ) is the probability of a value less than or equal to d 2. Determining these probabilities requires a table such as Table To illustrate, suppose we are given the following information: S \$100 E \$90 R f 4% per year, continuously compounded d 1.60 d 2.30 t 9 months Based on this information, what is the value of the call option, C? To answer, we need to determine N( d 1 ) and N(d 2 ). In Table 25.3, we first find the row corresponding to a d of.60. The corresponding probability N( d ) is.7258, so this is N( d 1 ). For d 2, the associated probability N( d 2 ) is Using the Black Scholes OPM, we calculate that the value of the call option is: C S N(d 1 ) E e Rt N(d 2 ) \$ \$90 e.04(3 4).6179 \$18.61 Notice that t, the time to expiration, is 9 months, which is of one year. As this example illustrates, if we are given values for d 1 and d 2 (and the table), then using the Black Scholes model is not difficult. Generally, however, we would not be given the values of d 1 and d 2, and we must calculate them. This requires a little extra effort. The values for d 1 and d 2 for the Black Scholes OPM are given by: d 1 [ln(s E) (R 2 2) t] ( _ t ) _ [25.6] d 2 d 1 t In these expressions, is the standard deviation of the rate of return on the underlying asset. Also, ln( S E ) is the natural logarithm of the current stock price divided by the exercise price There s a Black Scholes calculator (and a lot more) at www. numa.com. ros46128_ch25.indd 805 1/16/09 10:32:06 AM

8 806 PART 8 Topics in Corporate Finance TABLE 25.3 Cumulative Normal Distribution d N(d ) d N(d ) d N(d ) d N(d ) d N(d ) d N(d ) This table shows the probability [N(d )] of observing a value less than or equal to d. For example, as illustrated, if d is.24, then N(d ) is ros46128_ch25.indd 806 1/16/09 10:32:07 AM

9 CHAPTER 25 Option Valuation 807 The formula for d 1 looks a little intimidating, but it is mostly a matter of plug and chug with a calculator. To illustrate, suppose we have the following: S \$70 E \$80 R 4% per year, continuously compounded 60% per year t 3 months With these numbers, d 1 is: d 1 [ln(s E) (R 2 2) t] ( _ t ) [ln(70 80) ( ) ¼] (.6 ¼ ).26 d 2 d 1 _ t.26.6 ¼.56 Referring to Table 25.3, the values of N( d 1 ) and N( d 2 ) are.3974 and.2877, respectively. Plugging all the numbers in: C S N(d 1 ) E e Rt N(d 2 ) \$ \$80 e.04(1 4).2877 \$5.03 If you take a look at the Black Scholes formula and our examples, you will see that the price of a call option depends on five, and only five, factors. These are the same factors that we identified earlier: namely, the stock price, the strike price, the time to maturity, the riskfree rate, and the standard deviation of the return on the stock. Call Option Pricing EXAMPLE 25.5 Suppose you are given the following: S \$40 E \$36 R 4% per year, continuously compounded 70% per year t 3 months What s the value of a call option on the stock? We need to use the Black Scholes OPM. So, we first need to calculate d 1 and d 2 : d 1 [ln(s E) (R 2 2) t] ( _ t ) [ln(40 36) ( ) ¼] (.7 ¼ ).50 d 2 d 1 _ t.50.7 ¼.15 (continued ) ros46128_ch25.indd 807 1/16/09 10:32:10 AM

10 808 PART 8 Topics in Corporate Finance Referring to Table 25.3, the values of N( d 1 ) and N( d 2 ) are.6915 and.5597, respectively. To get the second of these, we averaged the two numbers on each side, ( ) Plugging all the numbers in: C S N(d 1 ) E e Rt N(d 2 ) \$ \$36 e.04(1/4).5597 \$7.71 A question that sometimes comes up concerns the probabilities N( d 1 ) and N( d 2 ). Just what are they the probabilities of? In other words, how do we interpret them? The answer is that they don t really correspond to anything in the real world. We mention this because there is a common misconception about N( d 2 ) in particular. It is frequently thought to be the probability that the stock price will exceed the strike price on the expiration day, which is also the probability that a call option will finish in the money. Unfortunately, that s not correct at least not unless the expected return on the stock equals the risk-free rate. Tables such as Table 25.3 are the traditional means of looking up z values, but they have been mostly replaced by computers. They are not as accurate because of rounding, and they also have only a limited number of values. Our nearby Spreadsheet Strategies box shows how to calculate Black Scholes call option prices using a spreadsheet. Because this is so much easier and more accurate, we will do all the calculations in the rest of this chapter using computers instead of tables. SPREADSHEET STRATEGIES A B C D E F G H I J K Using a spreadsheet to calculate Black Scholes option prices XYZ stock has a price of \$65 and an annual return standard deviation of 50%. The riskless interest rate is 5%. Calculate call and put option prices with a strike of \$60 and a 3-month time to expiration. Stock = 65 d1 =.4952 N(d1) =.6898 Strike = 60 Sigma =.5 d2 =.2452 N(d2) =.5968 Time =.25 Rate =.05 Call = Stock x N(d1) Strike x exp( Rate x Time) x N(d2) = \$9.47 Put = Strike x exp( Rate x Time) + Call Stock = \$3.72 Formula entered in E8 is =(LN(B8/B9)+(B12+.5*B10^2)*B11)/(B10*SQRT(B11)) Formula entered in E10 is =E8 B10*SQRT(B11) Formula entered in H8 is =NORMSDIST(E8) Formula entered in H10 is =NORMSDIST(E10) Formula entered in K14 is =B8*H8 B9*EXP( B12*B11)*H10 Formula entered in K16 is =B9*EXP( B12*B11)+K14 B8 PUT OPTION VALUATION Our examples thus far have focused only on call options. Just a little extra work is needed to value put options. Basically, we pretend that a put option is a call option and use the Black Scholes formula to value it. We then use the put call parity (PCP) condition to solve ros46128_ch25.indd 808 1/16/09 8:13:44 PM

11 CHAPTER 25 Option Valuation 809 for the put value. To see how this works, suppose we have the following: S \$40 E \$40 R 4% per year, continuously compounded 80% per year t 4 months What s the value of a put option on the stock? For practice, calculate the Black Scholes call option price and see if you agree that a call option would be worth about \$7.52. Now, recall the PCP condition: S P E e Rt C which we can rearrange to solve for the put price: P E e Rt C S Plugging in the relevant numbers, we get: P \$40 e.04(1 3) \$6.99 Thus, the value of a put option is \$6.99. So, once we know how to value call options, we also know how to value put options. A CAUTIONARY NOTE For practice, let s consider another put option value. Suppose we have the following: S \$70 E \$90 R 8% per year, continuously compounded 20% per year t 12 months What s the value of a put option on the stock? For practice, calculate the call option s value and see if you get \$1.65. Once again, we use PCP to solve for the put price: P E e Rt C S The put value we get is: P \$90 e.08(1) \$14.73 Does something about our put option value seem odd? The answer is yes. Because the stock price is \$70 and the strike price is \$90, you could get \$20 by exercising the put immediately; so it looks like we have an arbitrage possibility. Unfortunately, we don t. This example illustrates that we have to be careful with assumptions. The Black Scholes formula is for European- style options (remember that European-style options can be exercised only on the final day, whereas American-style options can be exercised anytime). In fact, our PCP condition holds only for European-style options. What our example shows is that an American-style put option is worth more than a European-style put. The reason is not hard to understand. Suppose you buy a put with a ros46128_ch25.indd 809 1/16/09 10:32:12 AM

12 810 PART 8 Topics in Corporate Finance strike price of \$80. The very best thing that can happen is for the stock price to fall to zero. If the stock price did fall to zero, no further gain on your option would be possible, so you would want to exercise it immediately rather than wait. If the option is American style, you can, but not if it is European style. More generally, it often pays to exercise a put option once it is well into the money because any additional potential gains are limited, so American-style exercise is valuable. What about call options? Here the answer is a little more encouraging. As long as we stick to non dividend-paying stocks, it will never be optimal to exercise a call option early. Again, the reason is not complicated. A call option is worth more alive than dead, meaning you would always be better off selling the option than exercising it. In other words, for a call option, the exercise style is irrelevant. Here is a challenge for the more mathematically inclined among you. We have a formula for a European-style put option. What about for an American-style put? Despite a great deal of effort, this problem has never been solved, so no formula is known. Just to be clear, we have numerical procedures for valuing put options, but no explicit formula. Call us if you figure one out. Concept Questions 25.2a What are the five factors that determine an option s value? 25.2b Which is worth more, an American-style put or a European-style put? Why? 25.3 More about Black Scholes In this section, we take a closer look at the inputs into the option pricing formula and their effects on option values. Table 25.4 summarizes the inputs and their impacts (positive or negative) on option values. In the table, a plus sign means that increasing the input increases the option s value and vice versa. Table 25.4 also indicates that four of the five effects have common names. For fairly obvious reasons given their names, these effects are collectively called the greeks. We discuss them in the next several sections. In some cases, the calculations can be fairly involved; but the good news is that options calculators are widely available on the Web. See our nearby Work the Web box for one example. VARYING THE STOCK PRICE The effect that the stock price has on put and call values is pretty obvious. Increasing the stock price increases call values and decreases put values. However, the strength of the effect varies depending on the moneyness of the option (how far in or out of the money it is). TABLE 25.4 Five Inputs Determining the Value of an American Option on a Non Dividend-Paying Stock NOTE: The effect of increasing the time to maturity is positive for an American put option, but the impact is ambiguous for a European put. Impact on Option Price from an Increase in Input Input Call Options Put Options Common Name Stock price (S) Delta Strike price (E) Time to expiration (t) Theta Standard deviation of return on stock ( ) Vega Risk-free rate (R) Rho ros46128_ch25.indd 810 1/16/09 10:32:13 AM

13 CHAPTER 25 Option Valuation 811 WORK THE WEB The Black Scholes OPM is a wonderful tool; but as we have seen, the calculations can get somewhat tedious. One way to fi nd the price of an option without the effort is to work the Web. We went to the options calculator at and entered MSFT, the ticker symbol for Microsoft. As shown, the current stock price is \$25.98, the standard deviation of the stock s return is percent per year, and the risk-free rate is 2.46 percent. Here is what we get: As you can see, a call option on MSFT with a strike price of \$23.50 should sell for \$ and a put option should sell for \$ Now that s easy! Notice that the greeks are also calculated. What does gamma tell you? Visit the site to learn more. Questions 1. Go to and find current option prices for Microsoft. Compare the prices of calls and puts with the same strike prices for the closest maturities and the most distant maturities. What relationship do you see in comparing these prices? 2. Go to and find current option prices for ebay with strike prices closest to the current stock price. Compare the deltas of the calls and puts with the closest maturities and the most distant maturities. What relationship do you see in the deltas? For a given set of input values, we illustrate the relationship between call and put option prices and the underlying stock price in Figure In the figure, stock prices are measured on the horizontal axis and option prices are measured on the vertical axis. Notice that the lines for put and call values are bowed. The reason is that the value of an option that is far out of the money is not as sensitive to a change in the underlying stock price as an inthe-money option. The sensitivity of an option s value to small changes in the price of the underlying stock is called the option s delta. For European options, we can directly measure the deltas as follows: Call option delta N(d 1 ) Put option delta N(d 1 ) 1 Another good options calculator can be found at com/optionpricing.html. delta Measures the effect on an option s value of a small change in the value of the underlying stock. ros46128_ch25.indd 811 1/16/09 8:07:23 PM

14 812 PART 8 Topics in Corporate Finance FIGURE 25.1 Put and Call Option Prices 25 Option price (\$) Put price Call price Input values: E \$100 t 3 months R 5% 25% Stock price (\$) The N(d 1 ) that we need to calculate these deltas is the same one we used to calculate option values, so we already know how to do it. Remember that N( d 1 ) is a probability, so its value ranges somewhere between zero and 1. For a small change in the stock price, the change in an option s price is approximately equal to its delta multiplied by the change in the stock price: Change in option value Delta Change in stock value To illustrate this, suppose we are given the following: S \$120 E \$100 R 8% per year, continuously compounded 80% per year t 6 months Using the Black Scholes formula, the value of a call option is \$ The delta (N( d 1 )) is.75, which tells us that if the stock price changes by, say, \$1, the option s value will change in the same direction by \$.75. We can check this directly by changing the stock price to \$121 and recalculating the option value. If we do this, the new value of the call is \$38.55, an increase of \$.75; so the approximation is pretty accurate (it is off in the third decimal point). If we price a put option using these same inputs, the value is \$ The delta is.75 1, or.25. If we increase the stock price to \$121, the new put value is 13.63, a change of.24; so, again, the approximation is fairly accurate as long as we stick to relatively small changes. Looking back at our graph in Figure 25.1, we now see why the lines get progressively steeper as the stock price rises for calls and falls for puts. The delta for a deeply in-themoney option is close to 1, whereas the delta for a deeply out-of-the-money option is close to zero. ros46128_ch25.indd 812 1/16/09 10:32:16 AM

15 CHAPTER 25 Option Valuation 813 Delta EXAMPLE 25.6 Suppose you are given the following: S \$40 E \$30 R 6% per year, continuously compounded 90% per year t 3 months What s the delta for a call option? A put option? Which one is more sensitive to a change in the stock price? Why? We need to calculate N( d 1 ). See if you agree that it s.815, which is the delta for the call. The delta for the put is , which is much smaller (in absolute value). The reason is that the call option is well in the money and the put is out of the money. VARYING THE TIME TO EXPIRATION The impact of changing the time to maturity on American-style options is also fairly obvious. Because an American-style option can be exercised any time, increasing the option s time to expiration can t possibly hurt and (especially for out-of-the-money options) might help. Thus, for both puts and calls, increasing the time to expiration has a positive effect. For a European-style call option, increasing the time to expiration also never hurts because, as we discussed earlier, the option is always worth more alive than dead, and any extra time to expiration only adds to its alive value. With a European-style put, however, increasing the time to expiration may or may not increase the value of the option. As we have discussed, for a deep in-the-money put, immediate exercise is often desirable, so increasing the time to expiration only reduces the value of the option. If a put is out of the money, then increasing the time to expiration will probably increase its value. Figure 25.2 shows the effect of increasing the time to expiration on a put and a call. The options are exactly at the money. In the figure, notice that once time to maturity reaches FIGURE 25.2 Option Prices and Time to Expiration Call price 25 Option price (\$) Put price Input values: S \$100 E \$100 R 5% 25% Time to expiration (months) ros46128_ch25.indd 813 1/16/09 10:32:17 AM

17 CHAPTER 25 Option Valuation 815 FIGURE 25.3 Option Prices and Sigma Input values: S \$100 E \$100 t 3 months R 5% Option price (\$) Call price Put price Sigma (%) The sensitivity of an option s value to the volatility of the underlying asset is called its vega. 2 Once again, the formula is somewhat complicated, so we will omit it. The main thing to understand from Figure 25.3 is that option values are very sensitive to standard deviations, and changes in the volatility of the underlying asset s return can have a strong impact on option values. VARYING THE RISK-FREE RATE We illustrate the effect of changing the risk-free rate on option values in Figure As shown, increasing the risk-free rate has a positive impact on call values and a negative impact on put values. Notice, however, that for realistic changes in interest rates, option values don t change a lot. In other words, option values are not as sensitive to changes in interest rates as they are to, say, changes in volatilities. An option s sensitivity to interest rate changes is called its rho. There are a few other greeks in addition to the ones we have discussed, but we will end our Greek lesson here. What we now discuss is a very important use of Black Scholes: the calculation of implied volatilities. IMPLIED STANDARD DEVIATIONS Thus far, we have focused on using the Black Scholes OPM to calculate option values; but there is another very important use. Of the five factors that determine an option s value, four can be directly observed: the stock price, the strike price, the risk-free rate, and the life of the option. Only the standard deviation must be estimated. The standard deviation we use in the OPM is actually a prediction of what the standard deviation of the underlying asset s return is going to be over the life of the option. Often, we already know the value of an option because we observe its price in the financial markets. In such cases, we can use the value of the option, along with the four observable vega Measures the sensitivity of an option s value to a change in the standard deviation of the return on the underlying asset. rho Measures the sensitivity of an option s value to a change in the risk-free rate. For an optionoriented Web site focusing on volatilities, visit 2 The Greek scholars among you will recognize that vega is not a Greek letter. (It is a star in the constellation Lyra and also a particularly forgettable automobile manufactured by Chevrolet in the 1960s and 1970s.) ros46128_ch25.indd 815 1/16/09 10:32:18 AM

18 816 PART 8 Topics in Corporate Finance FIGURE 25.4 Options Prices and Interest Rates Call price Option price (\$) Put price Input values: S \$100 E \$100 t 3 months 25% Interest rate (%) implied standard deviation An estimate of the future standard deviation of the return on an asset obtained from the Black Scholes OPM. inputs, to back out a value for the standard deviation. When we solve for the standard deviation this way, the result is called the implied standard deviation (ISD, which some people pronounce as iz-dee ), also known as the implied volatility. To illustrate this calculation, suppose we are given the following: S \$12 E \$8 R 5% per year, continuously compounded t 6 months We also know that the call option sells for \$4.59. Based on this information, how volatile is the stock expected to be over the next three months? If we plug all this information into the Black Scholes formula, we are left with one unknown: the standard deviation ( ). However, it s not possible to directly solve for, so trial and error must be used. In other words, we just start plugging in values for until we find one that produces the call price of \$4.59. For a stock option,.50 is a good place to start. If you plug this in, you will see that the calculated call value is \$4.38, which is too low. Recall that option values increase as we increase, so we might try.60. Now the option value is \$4.52, so we re getting close, but we re still low. At.65, the calculated value is \$4.61, which is just a little too high. After a little more work, we discover that the implied volatility is.64, or 64 percent. EXAMPLE 25.8 ISD Here is an actual example. On July 7, 2008, common stock in network hardware manufacturer Cisco Systems closed at \$ A call option expiring on October 18, 2008, with a strike price of \$22.00 traded for \$2.03. Treasury bills maturing on October 18 were paying percent. Based on this information, how volatile is the return on Cisco predicted to be? (continued) ros46128_ch25.indd 816 1/16/09 10:32:20 AM

19 CHAPTER 25 Option Valuation 817 To summarize, the relevant numbers we have are: S \$22.57 E \$22.00 R 3.125% per year, compounded annually? t 103 days C \$2.03 From here, it s plug and chug. As you have probably figured out by now, it s easier to use an options calculator to solve this problem. That s what we did; the implied standard deviation is about 35 percent. Our nearby Work the Web box shows you how to do this. In principle, to solve this problem, we need to convert the interest rate of percent to a continuously compounded rate. If we do this, we get percent. However, we ve seen that option values are not very sensitive to small changes in interest rates; and in this case, it actually makes almost no difference. For this reason, in practice, the continuous compounding issue is often ignored, especially when rates are low. WORK THE WEB From our discussion of implied standard deviation, you can see that solving for ISD when you know the option price is simply trial and error. Fortunately, most option calculators will do the work for you. To illustrate, we found the ISD for the Cisco call option we discussed in Example Just to refresh your memory, Cisco stock closed at \$ A call option with a strike price of \$22.00 and a maturity of 103 days was selling for \$2.03. Treasury bills with the same maturity had a yield of percent. What is the ISD of Cisco stock? We went to com and used the options calculator at the site. After entering all the information, this is what we found: Notice the calculator changes the days to maturity to.282, which is of a year. So, Cisco stock has an ISD of percent per year. Questions 1. Go to finance.yahoo.com and find option quotes for IBM. Pick a maturity and find the lowest strike price call, the call with a strike price nearest the current stock price, and the call with the highest strike price. Calculate the implied standard deviation for each using the calculator at What do you observe? 2. Go to finance.yahoo.com and find option quotes for Dell. Pick three call options with a strike price near the current stock price with different expiration months. Calculate the implied standard deviations using the calculator at What do you observe? ros46128_ch25.indd 817 1/16/09 10:32:22 AM

20 818 PART 8 Topics in Corporate Finance Concept Questions 25.3a What are an option s delta, rho, theta, and vega? 25.3b What is an ISD? 25.4 Valuation of Equity and Debt in a Leveraged Firm In our earlier chapter about options, we pointed out that the equity in a leveraged corporation (a corporation that has borrowed money) can be viewed as a call option on the assets of the business. The reason is that when a debt comes due, the stockholders have the option to pay off the debt, and thereby acquire the assets free and clear, or else default. The act of paying off the debt amounts to exercising an in-the-money call option to acquire the assets. Defaulting amounts to letting an out-of-the-money call option expire. In this section, we expand on the idea of equity as a call option in several ways. VALUING THE EQUITY IN A LEVERAGED FIRM Consider a firm that has a single zero coupon bond issue outstanding with a face value of \$10 million. It matures in six years. The firm s assets have a current market value of \$12 million. The volatility (standard deviation) of the return on the firm s assets is 40 percent per year. The continuously compounded risk-free rate is 6 percent. What is the current market value of the firm s equity? Its debt? What is its continuously compounded cost of debt? What this case amounts to is that the stockholders have the right, but not the obligation, to pay \$10 million in six years. If they do, they get the assets of the firm. If they don t, they default and get nothing. So, the equity in the firm is a call option with a strike price of \$10 million. Using the Black Scholes formula in this case can be a little confusing because now we are solving for the stock price. The symbol C is the value of the stock, and the symbol S is the value of the firm s assets. With this in mind, we can value the equity of the firm by plugging the numbers into the Black Scholes OPM with S \$12 million and E \$10 million. When we do so, we get \$6.554 million as the value of the equity, with a delta of.852. Now that we know the value of the equity, we can calculate the value of the debt using the standard balance sheet identity. The firm s assets are worth \$12 million and the equity is worth \$6.554 million, so the debt is worth \$12 million \$6.554 million \$5.446 million. To calculate the firm s continuously compounded cost of debt, we observe that the present value is \$5.446 million and the future value in six years is the \$10 million face value. We need to solve for a continuously compounded rate, R D, as follows: \$5.446 \$10 e R D (6).5446 e R D (6) R D 1 6 ln(.5446).10 So, the firm s cost of debt is 10 percent, compared to a risk-free rate of 6 percent. The extra 4 percent is the default risk premium: the extra compensation the bondholders demand because of the risk that the firm will default and bondholders will receive assets worth less than \$10 million. We also have that the delta of the option here is.852. How do we interpret this? In the context of valuing equity as a call option, the delta tells us what happens to the value of the equity when the value of the firm s assets changes. This is an important consideration. For ros46128_ch25.indd 818 1/16/09 10:32:23 AM

21 CHAPTER 25 Option Valuation 819 example, suppose the firm undertakes a project with an NPV of \$100,000, meaning that the value of the firm s assets will rise by \$100,000. We now see that the value of the stock will rise (approximately) by only.852 \$100,000 \$85, Why? The reason is that the firm has made its assets more valuable, which means default is less likely to occur in the future. As a result, the bonds gain value, too. How much do they gain? The answer is \$100,000 85,200 \$14,800 in other words, whatever value the stockholders don t get. Equity as a Call Option EXAMPLE 25.9 Consider a firm that has a single zero-coupon bond issue outstanding with a face value of \$40 million. It matures in five years. The risk-free rate is 4 percent. The firm s assets have a current market value of \$35 million, and the firm s equity is worth \$15 million. If the firm takes a project with a \$200,000 NPV, approximately how much will the stockholders gain? To answer this question, we need to know the delta, so we need to calculate N( d 1 ). To do this, we need to know the relevant standard deviation, which we don t have. We do have the value of the option (\$15 million), though, so we can calculate the ISD. If we use C \$15 million, S \$35 million, and E \$40 million along with the risk-free rate of 4 percent and time to expiration of five years, we find that the ISD is 48.1 percent. With this value, the delta is.725; so if \$200,000 in value is created, the stockholders will get 72.5 percent of it, or \$145,000. OPTIONS AND THE VALUATION OF RISKY BONDS Let s continue with the case we just examined of a firm with \$12 million in assets and a six-year, zero-coupon bond with a face value of \$10 million. Given the other numbers, we showed that the bonds were worth \$5.446 million. Suppose that the holders of these bonds wish to eliminate the risk of default. In other words, the holders want to turn their risky bonds into risk-free bonds. How can they do this? The answer is that the bondholders can do a protective put along the lines we described earlier in the chapter. In this case, the bondholders want to make sure their bonds will never be worth less than their face value of \$10 million, so the bondholders need to purchase a put option with a six-year life and a \$10 million face value. The put option is an option to sell the assets of the firm for \$10 million. Remember that if the assets of the firm are worth more than \$10 million in six years, the shareholders will pay the \$10 million. If the assets are worth less than \$10 million, the stockholders will default, and the bondholders will receive the assets of the firm. At that point, however, the bondholders will exercise their put and sell the assets for \$10 million. Either way, the bondholders get their \$10 million. What we have discovered is that a risk-free bond is the same thing as a combination of a risky bond and a put option on the assets of the firm with a matching maturity and a strike price equal to the face value of the bond: Value of risky bond Put option Value of risk-free bond [25.7] In our example, the face value of the debt is \$10 million, and the risk-free rate is 6 percent, so the value of the bonds if they were risk-free is: Value of risk-free bonds \$10 million e.06(6) \$6.977 million 3 Delta is used to evaluate the effect of a small change in the underlying asset s value, so it might look like we shouldn t use it to evaluate a shift of \$100,000. Small is relative, however, and \$100,000 is small relative to the \$12 million total asset value. ros46128_ch25.indd 819 1/16/09 10:32:24 AM

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