Chapter 8
Financial Options and Their Valuation

8-1 a. An option is a contract which gives its holder the right to buy or sell an asset at some predetermined price within a specified period of time. A call option allows the holder to buy the asset, while a put option allows the holder to sell the asset.

b. A simple measure of an option’s value is its exercise value. The exercise value is equal to the current price of the stock (underlying the option) less the striking price of the option. The strike price is the price stated in the option contract at which the security can be bought (or sold). For example, if the underlying stock sells for \$50 and the striking price is \$20, the exercise value of the option would be \$30.

c. The Black-Scholes Option Pricing Model is widely used by option traders to value options. It is derived from the concept of a riskless hedge. By buying shares of a stock and simultaneously selling call options on that stock, the investor will create a risk-free investment position. This riskless return must equal the risk-free rate or an arbitrage opportunity would exist. People would take advantage of this opportunity until the equilibrium level estimated by the Black-Scholes model was reached.

8-2 The market value of an option is typically higher than its exercise value due to the speculative nature of the investment. Options allow investors to gain a high degree of personal leverage when buying securities. The option allows the investor to limit his or her loss but amplify his or her return. The exact amount this protection is worth is the premium over the exercise value.

8-3 (1) An increase in stock price causes an increase in the value of a call option. (2) An increase in exercise price causes a decrease in the value of a call option. (3) An increase in the time to expiration causes an increase in the value of a call option. (4) An increase in the risk-free rate causes an increase in the value of a call option. (1) An increase in the variance of stock return causes an increase in the value of a call option.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS

8-1 P = \$15; X = \$15; t = 0.5; rRF = 0.06; s2 = 0.12; d1 = 0.24495;
d2 = 0.0000; N(d1) = 0.59675; N(d2) = 0.500000; V = ?

Using the Black-Scholes Option Pricing Model, you calculate the option’s value as:

V = P[N(d1)] - [N(d2)]
= \$15(0.59675) - \$15e(-0.10)(0.5)(0.50000)
= \$8.95128 - \$15(0.9512)(0.50000)
= \$1.6729 » \$1.67.

8-2 Option’s exercise price = \$15; Exercise value = \$22; Premium value = \$5;
V = ? P0 = ?

Premium = Market price of option - Exercise value
\$5 = V - \$22
V = \$27.

Exercise value = P0 - Exercise price
\$22 = P0 - \$15
P0 = \$37.

8-3

d2 = d1 – s (t)0.5 = -0.3319 – 0.5(0.33333)0.5 = -0.6206.

N(d1) = 0.3700 (from Excel NORMSDIST function).

N(d2) = 0.2674 (from Excel NORMSDIST function).

V = P[N(d1)] - [N(d2)]
= \$30(0.3700) - \$35e(-0.05)(0.33333)(0.2674)
= \$11.1000 - \$9.2043
= \$1.8957 » \$1.90.

8-4 The stock’s range of payoffs in one year is \$26 - \$16 = \$10. At expiration, the option will be worth \$26 - \$21 = \$5 if the stock price is \$26, and zero if the stock price \$16. The range of payoffs for the stock option is \$5 – 0 = \$5.

Equalize the range to find the number of shares of stock: Option range / Stock range = \$5/\$10 = 0.5.

With 0.5 shares, the stock’s payoff will be either \$13 or \$8. The portfolio’s payoff will be \$13 - \$5 = \$8, or \$8 – 0 = \$8.

The present value of \$8 at the daily compounded risk-free rate is: PV = \$8 / (1+ (0.05/365))365 = \$7.610.

The option price is the current value of the stock in the portfolio minus the PV of the payoff:

V = 0.5(\$20) - \$7.610 = \$2.39.

8-5 The stock’s range of payoffs in six months is \$18 - \$13 = \$5. At expiration, the option will be worth \$18 - \$14 = \$4 if the stock price is \$18, and zero if the stock price \$13. The range of payoffs for the stock option is \$4 – 0 = \$5.

Equalize the range to find the number of shares of stock: Option range / Stock range = \$4/\$5 = 0.8.

With 0.8 shares, the stock’s payoff will be either 0.8(\$18) = \$14.40 or 0.8(\$13) = \$10.40. The portfolio’s payoff will be \$14.4 - \$4 = \$10.40, or \$10.40 – 0 = \$10.40.

The present value of \$10.40 at the daily compounded risk-free rate is: PV = \$10.40 / (1+ (0.06/365))365/2 = \$10.093.

The option price is the current value of the stock in the portfolio minus the PV of the payoff:

V = 0.8(\$15) - \$10.093 = \$1.907 ».\$1.91.

8-6 Put = V – P + X exp(-rRF t)
= \$6.56 - \$33 + \$32 e-0.06(1)
= \$6.56 - \$33 + \$30.136 = \$3.696 » \$3.70.

8-7 The detailed solution for the problem is available both on the instructor’s resource CD-ROM (in the file Solution for FM 11 Ch 08 P07 Build a Model.xls) and on the instructor’s side of the textbook’s web site, http://brigham.swcollege.com.

MINI CASE

Assume that you have just been hired as a financial analyst by Triple Trice Inc., a mid-sized California company that specializes in creating exotic clothing. Since no one at Triple Trice is familiar with the basics of financial options, you have been asked to prepare a brief report that the firm's executives could use to gain at least a cursory understanding of the topic.
To begin, you gathered some outside materials the subject and used these materials to draft a list of pertinent questions that need to be answered. In fact, one possible approach to the paper is to use a question-and-answer format. Now that the questions have been drafted, you have to develop the answers.

a. What is a financial option? What is the single most important characteristic of an option?

Answer: A financial option is a contract which gives its holder the right to buy (or sell) an asset at a predetermined price within a specified period of time. An option’s most important characteristic is that it does not obligate its owner to take any action; it merely gives the owner the right to buy or sell an asset.

b. Options have a unique set of terminology. Define the following terms: (1) call option; (2) put option; (3) exercise price; (4) striking, or strike, price; (5) option price; (6) expiration date; (7) exercise value; (8) covered option; (9) naked option; (10) in-the-money call; (11) out-of-the-money call; and (12) LEAPS.

Answer: 1. A call option is an option to buy a specified number of shares of a security within some future period.

2. A put option is an option to sell a specified number of shares of a security within some future period.

3. Exercise price is another name for strike price, the price stated in the option contract at which the security can be bought (or sold).

4. The strike price is the price stated in the option contract at which the security can be bought (or sold).

5. The option price is the market price of the option contract.

6. The expiration date is the date the option matures.

7. The exercise value is the value of a call option if it were exercised today, and it is equal to the current stock price minus the strike price. Note: the exercise value is zero if the stock price is less than the strike price.

8. A covered option is a call option written against stock held in an investor's portfolio.

9. A naked option is an option sold without the stock to back it up.

10. An in-the-money call is a call option whose exercise price is less than the current price of the underlying stock.

11. An out-of-the-money call is a call option whose exercise price exceeds the current stock price.

12. LEAPS stands for long-term equity anticipation securities. They are similar to conventional options except they are long-term options with maturities of up to 2? years.

c. Consider Triple Trice’s call option with a \$25 strike price. The following table contains historical values for this option at different stock prices:

Stock Price Call Option Price
\$25 \$ 3.00
30 7.50
35 12.00
40 16.50
45 21.00
50 25.50

1. Create a table which shows (a) stock price, (b) strike price, (c) exercise value, (d) option price, and (e) the premium of option price over exercise value.

Stock Price Of Option Of Option (D) - (C) =
(A) (B) (A) - (B) = (C) (D) (E)
\$25.00 \$25.00 \$ 0.00 \$ 3.00 \$3.00
30.00 25.00 5.00 7.50 2.50
35.00 25.00 10.00 12.00 2.00
40.00 25.00 15.00 16.50 1.50
45.00 25.00 20.00 21.00 1.00
50.00 25.00 25.00 25.50 0.50

c. 2. What happens to the premium of option price over exercise value as the stock price rises? Why?

Answer: As the table shows, the premium of the option price over the exercise value declines as the stock price increases. This is due to the declining degree of leverage provided by options as the underlying stock prices increase, and to the greater loss potential of options at higher option prices.

d. In 1973, Fischer Black and Myron Scholes developed the Black-Scholes Option Pricing Model (OPM).

1. What assumptions underlie the OPM?

Answer: The assumptions which underlie the OPM are as follows:

The stock underlying the call option provides no dividends during the life of the option.

No transactions costs are involved with the sale or purchase of either the stock or the option.

The short-term, risk-free interest rate is known and is constant during the life of the option.

Security buyers may borrow any fraction of the purchase price at the short-term, risk-free rate.

Short-term selling is permitted without penalty, and sellers receive immediately the full cash proceeds at today's price for securities sold short.
The call option can be exercised only on its expiration date.

Security trading takes place in continuous time, and stock prices move randomly in continuous time.

d. 2. Write out the three equations that constitute the model.

Answer: The OPM consists of the following three equations:

V = P[N(d1) - [N(d2)].

d1 = .

d2 = d1 - .

Here,

V = current value of a call option with time t until expiration.
P = current price of the underlying stock.
N(di) = probability that a deviation less than di will occur in a standard normal distribution. Thus, N(d1) and N(d2) represent areas under a standard normal distribution function.
X = exercise, or strike, price of the option.
e » 2.7183.
rRF = risk-free interest rate.
t = time until the option expires (the option period).
ln(P/X) = natural logarithm of P/X.
s2 = variance of the rate of return on the stock.

d. 3. What is the value of the following call option according to the OPM?

Stock Price = \$27.00.
Exercise Price = \$25.00
Time To Expiration = 6 Months.
Risk-Free Rate = 6.0%.
Stock Return Variance = 0.11.

P = \$27.00; X = \$25.00; rRF = 6.0%; t = 6 months = 0.5 years; and s2 = 0.11.

Now, we proceed to use the OPM:

V = \$27[N(d1)] - \$25e-(0.06)(0.5)[N(d2)].

d1 =
= = 0.5736.

d2 = d1 - (0.3317)(0.7071) = d1 - 0.2345
= 0.5736 - 0.2345 = 0.3391.

N(d1) = N(0.5736) = 0.5000 + 0.2168 = 0.7168.

N(d2) = N(0.3391) = 0.5000 + 0.1327 = 0.6327.

Therefore,

V = \$27(0.7168) - \$25e-0.03(0.6327) = \$19.3536 - \$25(0.97045)(0.6327)
= \$19.3536 - \$15.3500 = \$4.0036 » \$4.00.

Thus, under the OPM, the value of the call option is about \$4.00.

e. What impact does each of the following call option parameters have on the value of a call option?

1. Current Stock Price
2. Exercise Price
3. Option’s Term To Maturity
4. Risk-Free Rate
5. Variability Of The Stock Price

Answer: 1. The value of a call option increases (decreases) as the current stock price increases (decreases).

2. As the exercise price of the option increases (decreases), the value of the option decreases (increases).

3. As the expiration date of the option is lengthened, the value of the option increases. This is because the value of the option depends on the chance of a stock price increase, and the longer the option period, the higher the stock price can climb.

4. As the risk-free rate increases, the value of the option tends to increase as well. Since increases in the risk-free rate tend to decrease the present value of the option's exercise price, they also tend to increase the current value of the option.

5. The greater the variance in the underlying stock price, the greater the possibility that the stock's price will exceed the exercise price of the option; thus, the more valuable the option will be.

f. What is put-call parity?

Answer: Put-call parity specifies the relationship between puts, calls, and the underlying stock price that must hold to prevent arbitrage:

Put + Stock = Call + PV Of Exercise Price