## CHAPTER 21

### Valuing Options

1.                  a.

b.

2.                  a.         Let p equal the probability of a rise in the stock price.  Then, if investors are risk-neutral:

(p ΄ 0.15) + (1 - p)΄(-0.13) = 0.10

p = 0.821

The possible stock prices next period are:

\$60 ΄ 1.15 = \$69.00

\$60 ΄ 0.87 = \$52.20

Let X equal the break-even exercise price.  Then the following must be true:

X  60 = (p)(\$0) + [(1  p)(X  52.20)]/1.10

That is, the value of the put if exercised immediately equals the value of the put if it is held to next period.  Solving for X, we find that the break-even exercise price is \$61.52.

b.                  If the interest rate is increased, the value of the put option decreases.

3.

 If there is an increase in: The change in the put option price is: Stock price (P) Negative Exercise price (EX) Positive Interest rate (rf) Negative Time to expiration (t) Positive Volatility of stock price (s) Positive

Consider the following base case assumptions:

P = 100, EX = 100, rf = 5%, t = 1, s = 50%

Then, using the Black-Scholes model, the value of the put is \$16.98

The base case value along with values computed for various changes in the assumed values of the variables are shown in the table below:

 Black-Scholes put value: Base case 16.98 P = 120 11.04 EX = 120 29.03 rf = 10% 14.63 t = 2 21.94 s = 100% 35.04

4.                  a.         The future stock prices of Matterhorn Mining are:

With dividend

Ex-dividend

Let p equal the probability of a rise in the stock price.  Then, if investors are risk-neutral:

(p ΄ 0.25) + (1 - p)΄(-0.20) = 0.10

p = 0.67

Now, calculate the expected value of the call in month 6.

If stock price decreases to SFr80 in month 6, then the call is worthless.  If stock price increases to SFr125, then, if it is exercised at that time, it has a value of (125  80) = SFr45.  If the call is not exercised, then its value is:

Therefore, it is preferable to exercise the call.

The value of the call in month 0 is:

b.                  The future stock prices of Matterhorn Mining are:

With dividend

Ex-dividend

Let p equal the probability of a rise in the price of the stock.  Then, if investors are risk-neutral:

(p ΄ 0.25) + (1 - p)΄(-0.20) = 0.10

p = 0.67

Now, calculate the expected value of the call in month 6.

If stock price decreases to SFr80 in month 6, then the call is worthless.  If stock price increases to SFr125, then, if it is exercised at that time, it has a value of (125  80) = SFr45.  If the call is not exercised, then its value is:

Therefore, it is preferable to exercise the call.

The value of the call in month 0 is:

5.                  a.         The possible prices of Buffelhead stock and the associated call option values (shown in parentheses) are:

Let p equal the probability of a rise in the stock price.  Then, if investors are risk-neutral:

p (1.00) + (1 - p)(-0.50) = 0.10

p = 0.4

If the stock price in month 6 is \$110, then the option will not be exercised so that it will be worth:

[(0.4 ΄ 55) + (0.6 ΄ 0)]/1.10 = \$20

Similarly, if the stock price is \$440 in month 6, then, if it is exercised, it will be worth (\$440 - \$165) = \$275.  If the option is not exercised, it will be worth:

[(0.4 ΄ 715) + (0.6 ΄ 55)]/1.10 = \$290

Therefore, the call option will not be exercised, so that its value today is:

[(0.4 ΄ 290) + (0.6 ΄ 20)]/1.10 = \$116.36

b.         (i)         If the price rises to \$440:

(ii)        If the price falls to \$110:

c.                  The option delta is 1.0 when the call is certain to be exercised and is zero when it is certain not to be exercised.  If the call is certain to be exercised, it is equivalent to buying the stock with a partly deferred payment.  So a one-dollar change in the stock price must be matched by a one-dollar change in the option price.  At the other extreme, when the call is certain not to be exercised, it is valueless, regardless of the change in the stock price.

d.                  If the stock price is \$110 at 6 months, the option delta is 0.33.  Therefore, in order to replicate the stock, we buy three calls and lend, as follows:

 Initial Stock Stock Outlay Price = 55 Price = 220 Buy 3 calls -60 0 165 Lend PV(55) -50 +55 +55 -110 +55 +220 This strategy Is equivalent to: Buy stock -110 +55 +220

6.                  a.         Yes, it is rational to consider the early exercise of an American put option.

b.                  The possible prices of Buffelhead stock and the associated American put option values (shown in parentheses) are:

Let p equal the probability of a rise in the stock price.  Then, if investors are risk-neutral:

p (1.00) + (1 - p)(-0.50) = 0.10

p = 0.4

If the stock price in month 6 is \$110, and if the American put option is not exercised, it will be worth:

[(0.4 ΄ 0) + (0.6 ΄ 165)]/1.10 = \$90

On the other hand, if it is exercised after 6 months, it is worth \$110.  Thus, the investor should exercise the put early.

Similarly, if the stock price in month 6 is \$440, and if the American put option is not exercised, it will be worth:

[(0.4 ΄ 0) + (0.6 ΄ 0)]/1.10 = \$0

On the other hand, if it is exercised after 6 months, it will cost the investor \$220.  The investor should not exercise early.

Finally, the value today of the American put option is:

[(0.4 ΄ 0) + (0.6 ΄ 110)]/1.10 = \$60

c.                  Unlike the American put in part (b), the European put can not be exercised prior to expiration.  We noted in part (b) that, If the stock price in month 6 is \$110, the American put would be exercised because its value if exercised (i.e., \$110) is greater than its value if not exercised (i.e., \$90).  For the European put, however, the value at that point is \$90 because the European put can not be exercised early.  Therefore, the value of the European put is:

[(0.4 ΄ 0) + (0.6 ΄ 90)]/1.10 = \$49.09

7.                  The following tree shows stock prices, with option values in parentheses:

With dividend

Ex-dividend

We calculate the option value as follows:

1.                  The option values in month 6, if the option is not exercised, are computed as follows:

If the stock price in month 6 is \$110, then it would not pay to exercise the option.  If the stock price in month 6 is \$440, then the call is worth:

(440 - 165) = 275.  Therefore, the option would be exercised at that time.

2.                  Working back to month 0, we find the option value as follows:

b.                  If the option were European, it would not be possible to exercise early.  Therefore, if the price rises to \$440 at month 6, the value of the option is \$265, not \$275 as is the case for the American option.  Therefore, in this case, the value of the European option is less than the value of the American option.  The value of the European option is computed as follows:

8.                  The following tree (see Practice Question 5) shows stock prices, with the values for the one-year option values in parentheses:

The put option is worth \$55 in month 6 if the stock price falls and \$0 if the stock price rises.  Thus, with a 6-month stock price of \$110, it pays to exercise the put (value = \$55).  With a price in month 6 of \$440, the investor would not exercise the put since it would cost \$275 to exercise.  The value of the option in month 6, if it is not exercised, is determined as follows:

Therefore, the month 0 value of the option is:

9.                  a.         The following tree shows stock prices (with put option values in parentheses):

Let p equal the probability that the stock price will rise.  Then, for a risk-neutral investor:

(p ΄ 0.111) + (1 - p)΄(-0.10) = 0.05

p = 0.71

If the stock price in month 6 is C\$111.1, then the value of the European put is:

If the stock price in month 6 is C\$90.0, then the value of the put is:

Since this is a European put, it can not be exercised at month 6.

The value of the put at month 0 is:

b.                  Since the American put can be exercised at month 6, then, if the stock price is C\$90.0, the put is worth (102  90) = \$12 if exercised, compared to \$7.15 if not exercised.  Thus, the value of the American put in month 0 is:

10.             a.         P = 200          EX = 180        s = 0.223       t = 1.0             rf = 0.21

N(d1) = N(1.4388) = 0.9249

N(d2) = N(1.2158) = 0.8880

Call value = [N(d1) ΄ P]  [N(d2) ΄ PV(EX)]

= [0.9249 ΄ 200]  [0.8880 ΄ (180/1.21)] = \$52.88

b.

Let p equal the probability that the stock price will rise.  Then, for a risk-neutral investor:

(p ΄ 0.25) + (1 - p)΄(-0.20) = 0.21

p = 0.91

In one year, the stock price will be either \$250 or \$160, and the option values will be \$70 or \$0, respectively.  Therefore, the value of the option is:

c.

Let p equal the probability that the stock price will rise.  Then, for a risk-neutral investor:

(p ΄ 0.171) + (1 - p)΄(-0.146) = 0.10

p = 0.776

The following tree gives stock prices, with option values in parentheses:

Option values are calculated as follows:

1.

2.

3.

d.                  (i)

To replicate a call, buy 0.89 shares and borrow:

[(0.89 ΄ 200) - 52.63] = \$125.37

(ii)

To replicate a call, buy one share and borrow:

[(1.0 ΄ 200) - 70.53] = \$129.47

(iii)

To replicate a call, buy 0.37 shares and borrow:

[(0.37 ΄ 200) - 14.11] = \$59.89

11.             To hold time to expiration constant, we will look at a simple one-period binomial problem with different starting stock prices.  Here are the possible stock prices:

Now consider the effect on option delta:

 Current Stock Price Option Deltas 100 110 In-the-money (EX =   60) 140/150 = 0.93 160/165 = 0.97 At-the-money (EX = 100) 100/150 = 0.67 120/165 = 0.73 Out-of-the-money (EX = 140) 60/150 = 0.40 80/165 = 0.48

Note that, for a given difference in stock price, out-of-the-money options result in a larger change in the option delta.  If you want to minimize the number of times you rebalance an option hedge, use in-the-money options.

12.             a.         The call option.  (You would delay the exercise of the put until after the dividend has been paid and the stock price has dropped.)

b.                  The put option.  (You never exercise a call if the stock price is below exercise price.)

c.         The put when the interest rate is high.  (You can invest the exercise price.)

13.             a.         When you exercise a call, you purchase the stock for the exercise price.  Naturally, you want to maximize what you receive for this price, and so you would exercise on the with-dividend date in order to capture the dividend.

b.                  When you exercise a put, your gain is the difference between the price of the stock and the amount you receive upon exercise, i.e., the exercise price.  Therefore, in order to maximize your profit, you want to minimize the price of the stock and so you would exercise on the ex-dividend date.

14.             [Note: the answer to this question is based on the assumption that the stock price is known.]

We can value the call by using the put-call parity relationship:

Value of put = value of call  share price + present value of exercise price

Then we must purchase two items of information [value of European put and PV(Exercise price)] and, hence, will spend \$20.

If we use the Black-Scholes model, we must also purchase two items [standard deviation times square root of time to maturity and PV(exercise price)] and, hence, will spend \$20.

15.             Internet exercise; answers will vary.

Challenge Questions

1.                  For the one-period binomial model, assume that the exercise price of the options (EX) is between u and d.  Then, the spread of possible option prices is:

For the call:    [(u  EX)  0]

For the put:     [(d  EX)  0]

The option deltas are:

Option delta(call) = [(u  EX)  0]/(u  d) = (u  EX)/(u  d)

Option delta(put) = [(d  EX)  0]/(u  d) = (d  EX)/(u  d)

Therefore:

[Option delta(call)  1] = [(u  EX)/(u  d)]  1

= [(u  EX)]/(u  d)]  [(u  d)/(u  d)]

= [(u  EX)  (u  d)]/(u  d)

= [d  EX]/(u  d) = Option delta(put)

2.                  If the exercise price of a call is zero, then the option is equivalent to the stock, so that, in order to replicate the stock, you would buy one call option.  Therefore, if the exercise price is zero, the option delta is one.  If the exercise price of a call is indefinitely large, then the option value remains low even if there is a large percentage change in the price of the stock.  Therefore, the dollar change in the value of the option will be much smaller than the dollar change in the price of the stock, so that the option delta is close to zero.  Between these two extreme cases, the option delta varies between zero and one.

4.                  Both of these announcements may convey information about company prospects, and thereby affect the price of the stock.  But, when the dividend is paid, stock price decreases by an amount approximately equal to the amount of the dividend.  This price decrease reduces the value of the option.  On the other hand, a stock repurchase at the market price does not affect the price of the stock.  Therefore, you should hope that the board will decide to announce a stock repurchase program.

5.                  a.         Assume the following:

1.                  The annual market standard deviation is 21 percent.

2.                  The risk-free interest rate is 3 percent.

3.                  Dividends equal 2 percent of the index value, and grow by 9 percent per year to give a total return of 11 percent.

4.                  The yield on a 4-year bond is 5.5 percent.

Then:

Therefore, the value of the call option = 172.78

b.                  Salomon Brothers has sold a four-year call option on the market.  To hedge this position, Salomon needs to replicate the purchase of an equivalent option.  It could do this by a series of levered investments in a diversified stock portfolio.  (A more practical alternative would be to use index futures, rather than the underlying stocks; these are discussed in Chapter 27.)

6.                  a.         As the life of the call option increases, the present value of the exercise price becomes infinitesimal.  Thus the only difference between the call option and the stock is that the option holder misses out on any dividends.  If dividends are negligible, the value of the option approaches its upper bound, i.e., the stock price.

b.                  While it is true that the value of an option approaches the upper bound as maturity increases and dividend payments on the stock decrease, a stock that never pays dividends is valueless.