Chapter 8
Financial Options and Applications in Corporate Finance
ANSWERS TO END-OF-CHAPTER QUESTIONS
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 options time value, which is the difference between the option’s price and its exercise value.
8-3 (1) An increase in stock price causes an increase in the value of a call option. (2) An increase in strike 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 standard deviation of stock return causes an increase in the value of a call option.
SOLUTIONS TO END-OF-CHAPTER PROBLEMS
8-1 Exercise value = Current stock price – strike price
= $30 - $25 = $5.
Time value = Option price – Exercise value
= $7 - $5 = $2.
8-2 Option’s strike price = $15; Exercise value = $22; Time value = $5;
V = ? P0 = ?
Time Value = Market price of option - Exercise value
$5 = V - $22
V = $27.
Exercise value = P0 - Strike price
$22 = P0 - $15
P0 = $37.
8-3 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(-0.06)(0.5)(0.50000)
= $8.95128 - $15(0.9512)(0.50000)
= $1.6729 » $1.67.
8-4 Put = V – P + X exp(-rRF t)
= $6.56 - $33 + $32 e-0.06(1)
= $6.56 - $33 + $30.136 = $3.696 » $3.70.
8-5
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.
8-6 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.
8-7 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.
SOLUTION TO SPREADSHEET PROBLEMS
8-8 The detailed solution for the problem is available in the file Ch08 P08 Build a Model Solution.xls at the textbook’s web site.
Answers and Solutions: 8 - 5
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MINI CASE
Assume that you have just been hired as a financial analyst by Triple Play Inc., a mid-sized California company that specializes in creating high-fashion clothing. Since no one at Triple Play 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) strike price; (4) expiration date; (5) exercise value (6) option price; (7) time 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. The strike price is the price stated in the option contract at which the security can be bought (or sold).
4. The expiration date is the last date the option can be exercised.
5. 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.
6. The option price is the market price of the option contract.
7. The time value is the difference between the option price and the exercise value.
8. For every new option, there is an investor who “writes” the option. A writer creates the contract, sells it to another investor, and must fulfill the option contract if it is exercised. For example, the writer of a call must be prepared to sell a share of stock to the investor who owns the call.
9. A covered option is a call option written against stock held in an investor's portfolio.
10. A naked option is an option sold without the stock to back it up.
11. An in-the-money call is a call option whose strike price is less than the current price of the underlying stock.
12. An out-of-the-money call is a call option whose strike price exceeds the current stock price.
13. 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 2½ years.
c. Consider Triple Play’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 time value, which is the option’s price less its exercise value.
Answer: Price Of Strike Exercise Value Market Price Time 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 option’s time value as the stock price rises? Why?
Answer: As the table shows, the option’s time 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. Consider a stock with a current price of P = $27. Suppose that over the next 6 months the stock price will either go up by a factor of 1.41 or down by a factor of 0.71. Consider a call option on the stock with a strike price of $25 which expires in 6 months. The risk-free rate is 6%.
1. Using the binomial model, what are the ending values of the stock price? What are the payoffs of the call option?
Answer: The assumptions which underlie the OPM are as follows:
Strike price: X = / $25.00Current stock price: P = / $27.00
Up factor for stock price: u = / 1.41
Down factor for stock price: d = / 0.71
Up option payoff: Cu = MAX[0,P(u)-X] = / $13.07
Down option payoff: Cd =MAX[0,P(d)-X] = / $0.00
Ending "up" stock price = P (u) = / $38.07
/ Option payoff: Cu = MAX[0,P(u)-X] = / $13.07
Current
stock price
P = / $27
Ending "down" stock price = P (d) = / $19.17
Option payoff: Cd =MAX[0,P(d)-X] = / $0.00
d. 2. Suppose you write 1 call option and buy Ns shares of stock. How many shares must you buy to create a portfolio with a riskless payoff (which is called a hedge portfolio)? What is the payoff of the portfolio?
Answer:
Ns = / Cu - Cd / = / 0.69153P(u - d)
:
/ Stock price = P (u) = / $38.07Portfolio’s stock payoff: = P(u)(Ns) = / $26.33
Subtract option's payoff: Cu = / $13.07
Portfolio’s net payoff = P(u)Ns - Cu = / $13.26
P = $27 /
Stock price = P (d) = / $19.17
Portfolio’s stock payoff: = P(d)(Ns) = / $13.26
Subtract option's payoff: Cd = / $0.00
Portfolio’s net payoff = P(d)Ns - Cd = / $13.26
d. 3. What is the present value of the hedge portfolio’s riskless payoff? What is the value of the call option?
Answer:
PV of payoff = / Payoff / = / $13.2567 / = / $12.865(1 + rRF/365)365*(t) / 1.03045
VC = / Ns (P) - Present value of riskless payoff
VC = / $5.81
d. 4. What is a replicating portfolio? What is arbitrage?
Answer: If you borrow an amount equal to the present value of the hedge portfolio’s riskless payoff and purchase Ns shares of stock, the portfolio’s payoff’s will replicate the call option’s payoffs.
The option’s value must be the same as the portfolio’s cost, otherwise you would have an opportunity for arbitrage, which is a situation in which you have none of your own money invested, you have no risk, yet you have a positive cash flow. Arbitrage opportunities can’t exist long in a well functioning economy, so the option’s price will be driven towards the cost of the replicating portfolio.
e. In 1973, Fischer Black and Myron Scholes developed the Black-Scholes Option Pricing Model (OPM).