Use the Black-Scholes Model to find the price for a call option with the following inputs:

(1) current stock price is $30, (2) strike price is $35, (3) time to expiration is 4 months,

(4) annualized risk-free rate is 5%, and (5) variance of stock return is 0.25.The attached file is the book, and the “Black-Scholes Model” is on the page 16.CHAPTER

© Adalberto Rios Szalay/Sexto Sol/Getty Images

I

8

Financial Options and

Applications in Corporate

Finance

n 2012, Cisco had over 621 million outstanding employee stock options and

about 5.4 billion outstanding shares of stock. If all these options are exercised,

then the option holders will own about 10% of Cisco’s stock: 0.621/(5.4 +

0.621) = 0.10. Many of these options may never be exercised, but any way you look

at it, 621 million is a lot of options. Cisco isn’t the only company with mega-grants:

Pfizer, Time Warner, Ford, and Bank of America are among the many companies

that have granted to their employees options to buy more than 100 million shares.

Whether your next job is with a high-tech firm, a financial services company, or a

manufacturer, you will probably receive stock options, so it’s important that you

understand them.

In a typical grant, you receive options allowing you to purchase shares of

stock at a fixed price, called the strike price or exercise price, on or before a stated

expiration date. Most plans have a vesting period, during which you can’t exercise

the options. For example, suppose you are granted 1,000 options with a strike

price of $50, an expiration date 10 years from now, and a vesting period of

3 years. Even if the stock price rises above $50 during the first 3 years, you can’t

exercise the options because of the vesting requirement. After 3 years, if you are

still with the company, you have the right to exercise the options. For example, if

the stock goes up to $110, you could pay the company $50(1,000) = $50,000

and receive 1,000 shares of stock worth $110,000. However, if you don’t

exercise the options within 10 years, they will expire and thus be worthless.

Even though the vesting requirement prevents you from exercising the options

the moment they are granted to you, the options clearly have some immediate

value. Therefore, if you are choosing between different job offers where options are

involved, you will need a way to determine the value of the alternative options.

This chapter explains how to value options, so read on.

325

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The Intrinsic Value of Stock Options

© Rob Webb/Getty Images

In previous chapters we showed that the intrinsic value

of an asset is the present value of its cash flows. This

time value of money approach works well for stocks

and bonds, but we must use another approach for

options and derivatives. If we can find a portfolio of

stocks and risk-free bonds that replicates an option’s

cash flows, then the intrinsic value of the option must

be identical to the value of the replicating portfolio.

Cost of

equity (rs)

Stock price =

Risk-free bond

Dividends (Dt)

D1

(1 + rs)1

+

D2

(1 + rs)2

Portfolio of stock and

risk-free bond that

replicates cash flows

of the option

+ …+

D∞

(1 + rs)∞

Value of option must

be the same as the

replicating portfolio

© Cengage Learning 2014

resource

The textbook’s Web site

contains an Excel file that

will guide you through the

chapter’s calculations. The

file for this chapter is

Ch08 Tool Kit.xls, and we

encourage you to open the

file and follow along as you

read the chapter.

There are two fundamental approaches to valuing assets. The first is the discounted cash flow

(DCF) approach, which we covered in previous chapters: An asset’s value is the present value

of its cash flows. The second is the option pricing approach. It is important that every manager

understands the basic principles of option pricing for the following reasons. First, many

projects allow managers to make strategic or tactical changes in plans as market conditions

change. The existence of these “embedded options” often means the difference between a

successful project and a failure. Understanding basic financial options can help you manage

the value inherent in these real options. Second, many companies use derivatives to manage

risk; many derivatives are types of financial options, so an understanding of basic financial

options is necessary before tackling derivatives. Third, option pricing theory provides insights

into the optimal debt/equity choice, especially when convertible securities are involved. And

fourth, knowing about financial options will help you understand any employee stock options

that you receive.

8-1 Overview of Financial Options

In general, an option is a contract that gives its owner the right to buy (or sell) an asset at

some predetermined price within a specified period of time. However, there are many

types of options and option markets.1 Consider the options reported in Table 8-1, which

1

For an in-depth treatment of options, see Don M. Chance and Robert Brooks, An Introduction to Derivatives

and Risk Management, 8th ed. (Mason, OH: South-Western, Cengage Learning, 2010), or John C. Hull, Options,

Futures, and Other Derivatives, 8th ed. (Upper Saddle River, NJ: Prentice-Hall, 2012).

Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

TABLE 8-1

Listed Options Quotations for January 7, 2013

CALLS—LAST QUOTE

Closing Price

Strike Price

PUTS—LAST QUOTE

February

March

May

February

March

May

General Computer Corporation (GCC)

53.50

50

4.25

4.75

5.50

0.65

1.40

2.20

53.50

55

1.30

2.05

3.15

2.65

r

4.50

53.50

60

0.30

0.70

1.50

6.65

r

8.00

Note: r means not traded on January 7.

© Cengage Learning 2014

is an extract from a Listed Options Quotations table as it might appear on a Web site or in

a daily newspaper. The first column reports the closing stock price. For example, the table

shows that General Computer Corporation’s (GCC) stock price closed at $53.50 on

January 7, 2013.

A call option gives its owner the right to buy a share of stock at a fixed price, which is

called the strike price (sometimes called the exercise price because it is the price at which

you exercise the option). A put option gives its owner the right to sell a share of stock at a

fixed strike price. For example, the first row in Table 8-1 is for GCC’s options that have a

$50 strike price. Observe that the table has columns for call options and for put options

with this strike price.

Each option has an expiration date, after which the option may not be exercised. Table 8-1

reports data for options that expire in February, March, and May.2 If the option can be

exercised any time before the expiration, it is called an American option; if it can be

exercised only on its expiration date, it is a European option. All of GCC’s options are

American options. The first row shows that GCC has a call option with a strike price of

$50 that expires on May 17 (the third Saturday in May 2013 is the 18th). The quoted

price for this option is $5.50.3

When the current stock price is greater than the strike price, the option is

in-the-money. For example, GCC’s $50 (strike) May call option is in-the-money by

$53.50 − $50 = $3.50. Thus, if the option were immediately exercised, it would have a

payoff of $3.50. On the other hand, GCC’s $55 (strike) May call is out-of-the-money

because the current $53.50 stock price is below the $55 strike price. Obviously, you

currently would not want to exercise this option by paying the $55 strike price for a

share of stock selling for $53.50. Therefore, the exercise value, which is any profit

from immediately exercising an option, is4

2

At its Web site, www.cboe.com/learncenter/glossary.aspx, the CBOE defines the expiration date as follows:

“The day on which an option contract becomes void. The expiration date for listed stock options is the Saturday

after the third Friday of the expiration month. Holders of options should indicate their desire to exercise, if they

wish to do so, by this date.” The CBOE also defines the expiration time as: “The time of day by which all exercise

notices must be received on the expiration date. Technically, the expiration time is currently 5:00PM on the

expiration date, but public holders of option contracts must indicate their desire to exercise no later than 5:30PM

on the business day preceding the expiration date. The times are Eastern Time.”

3

Option contracts are generally written in 100-share multiples, but to reduce confusion we focus on the cost and

payoffs of a single option.

4

MAX means choose the maximum. For example, MAX[15, 0] = 15 and MAX[−10, 0] = 0.

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Exercise value ¼ MAX½Current price of the stock − Strike price; 0#

(8-1)

An American option’s price always will be greater than (or equal to) its exercise value.

If the option’s price were less, you could buy the option and immediately exercise it,

reaping a sure gain. For example, GCC’s May call with a $50 strike price sells for $5.50,

which is greater than its exercise value of $3.50. Also, GCC’s out-of-the-money May call

with a strike price of $55 sells for $3.15 even though it would be worthless if it had to be

exercised immediately. An option always will be worth more than zero as long as there is

still any chance it will end up in-the-money: Where there is life, there is hope! The

difference between the option’s price and its exercise value is called the time value because

it represents the extra amount over the option’s immediate exercise value that a purchaser

will pay for the chance the stock price will appreciate over time.5 For example, GCC’s May

call with a $50 strike price sells for $5.50 and has an exercise value of $3.50, so its time

value is $5.50 − $3.50 = $2.00.

Suppose you bought GCC’s $50 (strike) May call option for $5.50 and then the stock

price increased to $60. If you exercised the option by purchasing the stock for

the $50 strike price, you could immediately sell the share of stock at its market price

of $60, resulting in a payoff of $60 − $50 = $10. Notice that the stock itself had a return of

12.1% = ($60 − $53.50)/$53.50, but the option’s return was 81.8% = ($10 − $5.50)/$5.50.

Thus, the option offers the possibility of a higher return.

However, if the stock price fell to $50 and stayed there until the option expired, the

stock would have a return of −6.5% = ($50.00 − $53.50)/$53.50, but the option would

have a 100% loss (it would expire worthless). As this example shows, call options are a lot

riskier than stocks. This works to your advantage if the stock price goes up but to your

disadvantage if the stock price falls.

Suppose you bought GCC’s May put option (with a strike price of $50) for $2.20 and

then the stock price fell to $45. You could buy a share of stock for $45 and exercise the put

option, which would allow you to sell the share of stock at its strike price of $50. Your

payoff from exercising the put would be $5 = $50 − $45. Stockholders would lose money

because the stock price fell, but a put holder would make money. In this example, your

rate of return would be 127.3% = ($5 − $2.20)/$2.20. So if you think a stock price is going

to fall, you can make money by purchasing a put option. On the other hand, if the stock

price doesn’t fall below the strike price of $50 before the put expires, you would lose 100%

of your investment in the put option.6

Options are traded on a number of exchanges, with the Chicago Board Options

Exchange (CBOE) being the oldest and the largest. Existing options can be traded in

the secondary market in much the same way that existing shares of stock are traded in

secondary markets. But unlike new shares of stock that are issued by corporations, new

options can be “issued” by investors. This is called writing an option.

For example, you could write a call option and sell it to some other investor. You

would receive cash from the option buyer at the time you wrote the option, but you

would be obligated to sell a share of stock at the strike price if the option buyer later

5

Among traders, an option’s market price is also called its “premium.” This is particularly confusing because for

all other securities the word premium means the excess of the market price over some base price. To avoid

confusion, we will not use the word premium to refer to the option price.

6

Most investors don’t actually exercise an option prior to expiration. If they want to cash in the option’s profit

or cut its losses, they sell the option to some other investor. As you will see later in the chapter, the cash flow

from selling an American option before its expiration is always greater than (or equal to) the profit from

exercising the option.

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

WWW

The Chicago Board Options

Exchange provides

20-minute delayed quotes

for equity, index, and

LEAPS options at

www.cboe.com.

decided to exercise the option.7 Thus, each option has two parties, the writer and the

buyer, with the CBOE (or some other exchange) acting as an intermediary. Other

than commissions, the writer’s profits are exactly opposite those of the buyer. An

investor who writes call options against stock held in his or her portfolio is said to be

selling covered options. Options sold without the stock to back them up are called

naked options.

In addition to options on individual stocks, options are also available on several

stock indexes such as the NYSE Index and the S&P 100 Index. Index options permit

one to hedge (or bet) on a rise or fall in the general market as well as on individual

stocks.

The leverage involved in option trading makes it possible for speculators with just a

few dollars to make a fortune almost overnight. Also, investors with sizable portfolios

can sell options against their stocks and earn the value of the option (less brokerage

commissions) even if the stock’s price remains constant. Most important, though,

options can be used to create hedges that protect the value of an individual stock or

portfolio.8

Conventional options are generally written for 6 months or less, but a type of option

called a Long-Term Equity AnticiPation Security (LEAPS) is different. Like conventional options, LEAPS are listed on exchanges and are available on both individual

stocks and stock indexes. The major difference is that LEAPS are long-term options,

having maturities of up to almost 3 years. One-year LEAPS cost about twice as much as

the matching 3-month option, but because of their much longer time to expiration,

LEAPS provide buyers with more potential for gains and offer better long-term protection for a portfolio.

Corporations on whose stocks the options are written have nothing to do with

the option market. Corporations do not raise money in the option market, nor do

they have any direct transactions in it. Moreover, option holders do not vote for

corporate directors or receive dividends. There have been studies by the SEC and

others as to whether option trading stabilizes or destabilizes the stock market and

whether this activity helps or hinders corporations seeking to raise new capital. The

studies have not been conclusive, but research on the impact of option trading is

ongoing.

SELF-TEST

What is an option? A call option? A put option?

Define a call option’s exercise value. Why is the market price of a call option usually

above its exercise value?

Brighton Memory’s stock is currently trading at $50 a share. A call option on the

stock with a $35 strike price currently sells for $21. What is the exercise value of the

call option? ($15.00) What is the time value? ($6.00)

7

Your broker would require collateral to ensure that you kept this obligation.

Insiders who trade illegally generally buy options rather than stock because the leverage inherent in options

increases the profit potential. However, it is illegal to use insider information for personal gain, and an insider

using such information would be taking advantage of the option seller. Insider trading, in addition to being

unfair and essentially equivalent to stealing, hurts the economy: Investors lose confidence in the capital markets

and raise their required returns because of an increased element of risk, and this raises the cost of capital and

thus reduces the level of real investment.

8

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Financial Reporting for Employee

Stock Options

When granted to executives and other employees, options

are a “hybrid” form of compensation. At some companies,

especially small ones, option grants may be a substitute

for cash wages: Employees are willing to take lower cash

salaries if they have options. Options also provide an

incentive for employees to work harder. Whether issued

to motivate employees or to conserve cash, options clearly

have value at the time they are granted, and they transfer

wealth from existing shareholders to employees to the

extent that they do not reduce cash expenditures or

increase employee productivity enough to offset their

value at the time of issue.

Companies like the fact that an option grant requires

no immediate cash expenditure, although it might dilute

shareholder wealth if it is exercised later. Employees,

and especially CEOs, like the potential wealth they

receive when they are granted options. When option

grants were relatively small, they didn’t show up on

investors’ radar screens. However, as the high-tech

sector began making mega-grants in the 1990s, and as

other industries followed suit, stockholders began to

© Rob Webb/Getty Images

realize that large grants were making some CEOs filthy

rich at the stockholders’ expense.

Before 2005, option grants were barely visible in

companies’ financial reports. Even though such grants

are clearly a wealth transfer to employees, companies

were required only to footnote the grants and could

ignore them when reporting their income statements

and balance sheets. The Financial Accounting Standards

Board now requires companies to show option grants as

an expense on the income statement. To do this, the value

of the options is estimated at the time of the grant and

then expensed during the vesting period, which is the

amount of time the employee must wait before being

allowed to exercise the options. For example, if the initial

value is $100 million and the vesting period is 2 years, the

company would report a $50 million expense for each of

the next 2 years. This approach isn’t perfect, because the

grant is not a cash expense; nor does the approach take

into account changes in the option’s value after the initial

grant. However, it does make the option grant more

visible to investors, which is a good thing.

8-2 The Single-Period Binomial Option

Pricing Approach

We can use a model like the Capital Asset Pricing Model (CAPM) to calculate the

required return on a stock and then use that required return to discount its expected

future cash flows to find its value. No such model exists for the required return on

options, so we must use a different approach to find an option’s value. In Section 8-5

we describe the Black-Scholes option pricing model, but in this section we explain the

binomial option pricing model. The idea behind this model is different from that of

the DCF model used for stock valuation. Instead of discounting cash flows at a

required return to obtain a price, as we did with the stock valuation model, we will

use the option, shares of stock, and the risk-free rate to construct a portfolio whose

value we already know and then deduce the option’s price from this portfolio’s value.

The following sections describe and apply the binomial option pricing model to

Western Cellular, a manufacturer of cell phones. Call options exist that permit the

holder to buy 1 share of Western at a strike price, X, of $35. Western’s options will

expire at the end of 6 months (t is the number of years until expiration, so t = 0.5 for

Western’s options). Western’s stock price, P, is currently $40 per share. Given this

background information, we will use the binomial model to determine the call option’s

value. The first step is to determine the option’s possible payoffs, as described in the

next section.

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

8-2a Payoffs in a Single-Period Binomial Model

resource

See Ch08 Tool Kit.xls on

the textbook’s Web site.

In general, the time until expiration can be divided into many periods, with n denoting the

number of periods. But in a single-period model, which we describe in this section, there is only

one period. We assume that, at the end of the period, the stock’s price can take on only one of

two possible values, so this is called the binomial approach. For this example, Western’s stock

will either go up (u) by a factor of 1.25 or go down (d) by a factor of 0.80. If we were considering

a riskier stock, then we would have assumed a wider range of ending prices; we will show how

to estimate this range later in the chapter. If we let u = 1.25 and d = 0.80, then the ending stock

price will be either P(u) = $40(1.25) = $50 or P(d) = $40(0.80) = $32. Figure 8-1 illustrates the

stock’s possible price paths and contains additional information about the call option that is

explained in the text that follows.

When the option expires at the end of the year, Western’s stock will sell for either $50

or $32. As shown in Figure 8-1, if the stock goes up to $50 then the option will have a

payoff, Cu, of $15 at expiration because the option is in-the-money: $50 − $35 = $15. If

the stock price goes down to $32, then the option’s payoff, Cd, will be zero because the

option is out-of-the-money.

8-2b The Hedge Portfolio Approach

Suppose we created a portfolio by writing 1 call option and purchasing 1 share of stock. As

Figure 8-1 shows, if the stock price goes up then our portfolio’s stock will be worth $50 but we

will owe $15 on the option, so our portfolio’s net payoff is $35 = $50 − $15. If the stock price

goes down then our portfolio’s stock will be worth only $32, but the amount we owe on the

written option also will fall to zero, leaving the portfolio’s net payoff at $32. The portfolio’s

end-of-period price range is smaller than if we had just owned the stock, so writing the call

FIGURE 8-1

Binomial Payoffs from Holding Western Cellular’s Stock or Call Option

Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

option reduces the portfolio’s price risk. Taking this further: Is it possible for us to choose the

number of shares held by our portfolio so that it will have the same net payoff whether the

stock goes up or down? If so, then our portfolio is hedged and will have a riskless payoff when

the option expires. Therefore, it is called a hedge portfolio.

We are not really interested in investing in the hedge portfolio, but we want to use it to

help us determine the value of the option. Notice that if the hedge portfolio has a riskless

net payoff when the option expires, then we can find the present value of this payoff by

discounting it at the risk-free rate. Our current portfolio value must equal this present

value, which allows us to determine the option’s value. The following example illustrates

the steps in this approach.

1. FIND NS,

THE

NUMBER

OF

SHARES

OF

STOCK

IN THE

HEDGE PORTFOLIO

We want the portfolio’s payoff to be the same whether the stock goes up or down. If we write

1 call option and buy Ns shares of stock, then the portfolio’s stock will be worth Ns(P)(u)

should the stock price go up, so its net payoff will be Ns(P)(u) − Cu. The portfolio’s stock will

be worth Ns(P)(d) if the stock price goes down, so its net payoff will be Ns(P)(d) − Cd. Setting

these portfolio payoffs equal to one another and then solving for Ns yields

Ns ¼

Cu − Cd

Cu − Cd

¼

PðuÞ − PðdÞ

Pðu − dÞ

(8-2)

For Western, the hedge portfolio has 0.83333 share of stock:9

Cu − Cd

$15 − $0

¼

Ns ¼

¼ 0:83333

PðuÞ − PðdÞ $50 − $32

2. FIND

THE

HEDGE PORTFOLIO’S PAYOFF

Our next step is to find the hedge portfolio’s payoff when the stock price goes up (you will

get the same result if instead you find the portfolio’s payoff when the stock goes down).

Recall that the hedge portfolio has Ns shares of stock and that we have written the call

option, so the call option’s payoff must be subtracted:

Hedge portfolio’s payoff if stock is up ¼ Ns PðuÞ − Cu

¼ 0:83333ð$50Þ − $15

¼ $26:6665

Hedge portfolio’s payoff if stock is down ¼ Ns PðdÞ − Cd

¼ 0:83333ð$32Þ − $0

¼ $26:6665

Figure 8-2 illustrates the payoffs of the hedge portfolio.

3. FIND

THE

PRESENT VALUE

OF THE

HEDGE PORTFOLIO’S PAYOFF

Because the hedge portfolio’s payoff is riskless, the current value of the hedge portfolio

must be equal to the present value of its riskless payoff. Suppose the nominal annual riskfree rate, rRF, is 8%. What is the present value of the hedge portfolio’s riskless payoff of

9

An easy way to remember this formula is to notice that Ns is equal to the range in possible option payoffs

divided by the range in possible stock prices.

Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

FIGURE 8-2

Hedge Portfolio with Riskless Payoffs

180

181

182

183

184

185

186

187

188

189

190

191

A

B

E

Strike price: X =

Current stock price: P =

Up factor for stock price: u =

Down factor for stock price: d =

Up option payoff: Cu = MAX[0,P(u)‐X] =

Down option payoff: Cd =MAX[0,P(d)‐ X] =

Number of shares of stock in portfolio: Ns = (Cu ‐ Cd) / P(u ‐ d) =

F

$35.00

$40.00

1.25

0.80

$15.00

$0.00

0.83333

Stock price = P (u) = $50.00

Portfolio’s stock payoff: = P(u)(Ns) =

Subtract option’s payoff: Cu =

$41.67

$15.00

Portfolio’s net payoff = P(u)Ns ‐ Cu =

$26.67

Stock price = P (d) = $32.00

Portfolio’s stock payoff: = P(d)(Ns) =

Subtract option’s payoff: Cd =

$26.67

$0.00

Portfolio’s net payoff = P(d)Ns ‐ Cd =

$26.67

P,

current

stock price

192

193

194

195

196

197

198

D

$40

199

200

resource

See Ch08 Tool Kit.xls on

the textbook’s Web site.

C

$26.6665 in 6 months? Recall from Chapter 4 that the present value depends on how

frequently interest is compounded. Let’s assume that interest is compounded daily.10 We

can use a financial calculator to find the present value of the hedge portfolio’s payoff by

entering N = 0.5(365), because there are 365 days in a year and the contract expires in half

a year; I/YR = 8/365, because we want a daily interest rate; PMT = 0; and FV = −$26.6665,

because we want to know the amount we would take today in exchange for giving up the

payoff when the option expires. Using these inputs, we solve for PV = $25.6210, which is

the present value of the hedge portfolio’s payoff.11

4. FIND

THE

OPTION’S VALUE

The current value of the hedge portfolio is the value of the stock, Ns(P), less the value of

the call option we wrote:

Current value of hedge portfolio ¼ Ns ðPÞ − VC

Because the payoff is riskless, the current value of the hedge portfolio must also equal the

present value of the riskless payoff:

Current value of hedge portfolio ¼ Present value of riskless payoff

10

Option pricing models usually assume continuous compounding, which we discuss in Web Extension 4C on the

textbook’s Web site, but daily compounding works well. We will apply continuous compounding in Sections 8-3 and 8-4.

11

We could also solve for the present value using the present value equation with the daily periodic interest rate

and the number of daily periods: PV = $26.6665/(1 + 0.08/365)0.5(365) = $25.6210.

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Substituting for the current value of the hedge portfolio, we get:

Ns ðPÞ − VC ¼

Present value of

riskless payoff

Solving for the call option’s value, we get

VC ¼ Ns ðPÞ −

Present value of

riskless payoff

For Western’s option, this is

VC ¼ 0:83333ð$40Þ − $25:621

¼ $7:71

8-2c Hedge Portfolios and Replicating Portfolios

In our previous derivation of the call option’s value, we combined an investment in the stock

with writing a call option to create a risk-free investment. We can modify this approach and

create a portfolio that replicates the call option’s payoffs. For example, suppose we formed a

portfolio by purchasing 0.83333 shares of Western’s stock and borrowing $25.621 at the risk-free

rate (this is equivalent to selling a T-bill short). In 6 months, we would repay the future value of a

$25.621, compounded daily at the risk-free rate. Using a financial calculator, input N = 0.5(365),

I/YR = 8/365, PV = −$25.621, and solve for FV = $26.6665.12 If the stock goes up, our net payoff

would be 0.83333($50) − $26.6665 = $15.00. If the stock goes down, our net payoff would be

0.83333($32) − $26.6665 = $0. The portfolio’s payoffs are exactly equal to the option’s payoffs as

shown in Figure 8-1, so our portfolio of 0.83333 shares of stock and the $25.621 that we

borrowed would exactly replicate the option’s payoffs. Therefore, this is called a replicating

portfolio. Our cost to create this portfolio is the cost of the stock less the amount we borrowed:

Cost of replicating portfolio ¼ 0:83333ð$40Þ − $25:621 ¼ $7:71

If the call option did not sell for exactly $7.71, then a clever investor could make a sure

profit. For example, suppose the option sold for $8. The investor would write an option,

which would provide $8 of cash now but would obligate the investor to pay either $15 or

$0 in 6 months when the option expires. However, the investor could use the $8 to create

the replicating portfolio, leaving the investor with $8 − $7.71 = $0.29. In 6 months, the

replicating portfolio will pay either $15 or $0. Thus, the investor isn’t exposed to any

risk—the payoffs received from the replicating portfolio exactly offset the payoffs owed on

the option. The investor uses none of his own money, has no risk, has no net future

obligations, but has $0.29 in cash. This is arbitrage, and if such an arbitrage opportunity

existed then the investor would scale it up by writing thousands of options.13

Such arbitrage opportunities don’t persist for long in a reasonably efficient economy

because other investors will also see the opportunity and will try to do the same thing.

With so many investors trying to write (i.e., sell) the option, its price will fall; with so

many investors trying to purchase the stock, its price will increase. This will continue until

the option and replicating portfolio have identical prices. And because our financial

markets are really quite efficient, you would never observe the derivative security and

the replicating portfolio trading for different prices—they would always have the same

price and there would be no arbitrage opportunities. What this means is that, by finding

12

Alternatively, use the present value equation with daily compounding: $25.621(1 + 0.08/365)365(0.5/1) =

$26.6665.

13

If the option sold for less than the replicating portfolio, the investor would raise cash by shorting the portfolio

and use the cash to purchase the option, again resulting in arbitrage profits.

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the price of a portfolio that replicates a derivative security, we have also found the price of

the derivative security itself!

SELF-TEST

Describe how a risk-free hedge portfolio can be created using stocks and options.

How can such a portfolio be used to help estimate a call option’s value?

What is a replicating portfolio, and how is it used to find the value of a derivative

security?

What is arbitrage?

Lett Incorporated’s stock price is now $50, but it is expected either to rise by a factor

of 1.5 or fall by a factor of 0.7 by the end of the year. There is a call option on Lett’s

stock with a strike price of $55 and an expiration date 1 year from now. What are the

stock’s possible prices at the end of the year? ($75 or $35) What is the call option’s

payoff if the stock price goes up? ($20) If the stock price goes down? ($0) If we sell 1

call option, how many shares of Lett’s stock must we buy to create a riskless

hedged portfolio consisting of the option position and the stock? (0.5) What is the

payoff of this portfolio? ($17.50) If the annual risk-free rate is 6%, then how much is

the riskless portfolio worth today (assuming daily compounding)? ($16.48) What is

the current value of the call option? ($8.52)

8-3 The Single-Period Binomial Option

Pricing Formula14

The hedge portfolio approach works well if you only want to find the value of one type of

option with one period until expiration. But in all other situations, the step-by-step

approach becomes tedious very quickly. The following sections describe a formula that

replaces the step-by-step approach.

8-3a The Binomial Option Pricing Formula

With a little (or a lot!) of algebra, we can derive a single formula for a call option. Instead

of using daily compounding, we use continuous compounding to make the binomial

formula consistent with the Black-Scholes formula in Section 8-5.15 Here is the resulting

binomial option pricing formula:

Cu

VC ¼

!

!

”

”

erRF ðt = n Þ − d

u − erRF ðt = n Þ

þ Cd

u−d

u−d

r

ðt

=

n

Þ

RF

e

(8-3)

After programming it into Excel, which we did for this chapter’s Tool Kit, it is easy to

change inputs and determine the new value of a call option. Here is the binomial option

pricing formula:

14

The material in this section is relatively technical, and some instructors may choose to skip it with no loss in

continuity.

15

With continuous compounding, the present value is equal to the future value divided by (1 + rRF/365)365(0.5/1).

With continuous compounding, the present value is e−rRF(t/n). See Web Extension 4C on the textbook’s Web site

for more discussion of continuous compounding.

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We can apply this formula to Western’s call option:

!

”

”

e0:08ð0:5 = 1 Þ − 0:80

1:25 − e0:08ð0:5 = 1 Þ

$15

þ $0

1:25 − 0:80

1:25 − 0:80

VC ¼

e0:08ð0:5 = 1 Þ

!

¼

$15ð0:5351Þ þ $0ð0:4649Þ

¼ $7:71

1:04081

Notice that this is the same value that resulted from the step-by-step process shown earlier.

The binomial option pricing formula in Equation 8-3 does not include the actual

probabilities that the stock will go up or down, nor does it include the expected stock return,

which is not what one might expect. After all, the higher the stock’s expected return, the

greater the chance that the call will be in-the-money at expiration. Note, however, that the

stock’s expected return is already indirectly incorporated into the stock price.

8-3b Primitive Securities and the Binomial Option

Pricing Formula

If we want to value other Western call options or puts that expire in 6 months, then we

can use Equation 8-3, but there is a time-saving approach. Notice that for options with the

same time left until expiration, Cu and Cd are the only variables that depend on the option

itself. The other variables depend only on the stock process (u and d), the risk-free rate,

the time until expiration, and the number of periods until expiration. If we group these

variables together, we can then define πu and πd as

!

”

(8-4)

!

”

(8-5)

erRF ðt=nÞ −d

u−d

πu ¼

r

e RF ðt=nÞ

and

u−erRF ðt=nÞ

u−d

πd ¼

erRF ðt=nÞ

By substituting these values into Equation 8-3, we obtain an option pricing model that

can be applied to all of Western’s 6-month options:

VC ¼ Cu πu þ Cd πd

In this example, πu and πd are

(8-6)

!

”

e0:08ð0:5=1Þ −0:80

1:25−0:80

πu ¼

¼ 0:5142

e0:08ð0:5=1Þ

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and

!

”

1:25−e0:08ð0:5=1Þ

1:25−0:80

¼ 0:4466

πd ¼

e0:08ð0:5=1Þ

Using Equation 8-6, the value of Western’s 6-month call option with a strike price

of $35 is

Vc ¼ Cu πu þ Cd πd

¼ $15ð0:5142Þ þ $0ð0:4466Þ

¼ $7:71

Sometimes these π’s are called primitive securities because πu is the price of a simple

security that pays $1 if the stock goes up and nothing if it goes down; πd is the opposite.

This means that we can use these π’s to find the price of any 6-month option on Western.

For example, suppose we want to find the value of a 6-month call option on Western but

with a strike price of $30. Rather than reinvent the wheel, all we have to do is find the

payoffs of this option and use the same values of πu and πd in Equation 8-6. If the stock

goes up to $50, the option will pay $50 − $30 = $20; if the stock falls to $32, the option will

pay $32 − $30 = $2. The value of the call option is:

Value of 6-month call with $30 strike price ¼ Cu πu þ Cd πd

¼ $20ð0:5141Þ þ $2ð0:4466Þ

¼ $11:18

It is a bit tedious initially to calculate πu and πd, but once you save them, it is easy to

find the value of any 6-month call or put option on the stock. In fact, you can use these π’s

to find the value of any security with payoffs that depend on Western’s 6-month stock

prices, which makes them a very powerful tool.

SELF-TEST

Yegi’s Fine Phones has a current stock price of $30. You need to find the value

of a call option with a strike price of $32 that expires in 3 months. Use the

binomial model with one period until expiration. The factor for an increase in

stock price is u = 1.15; the factor for a downward movement is d = 0.85. What

are the possible stock prices at expiration? ($34.50 or $25.50) What are the

option’s possible payoffs at expiration? ($2.50 or $0) What are πu and πd?

(0.5422 and 0.4429) What is the current value of the option (assume each month

is 1/12 of a year)? ($1.36)

8-4 The Multi-Period Binomial Option

Pricing Model16

Clearly, the one-period example is simplified. Although you could duplicate buying

0.8333 share and writing one option by buying 8,333 shares and writing 10,000

options, the stock price assumptions are unrealistic—Western’s stock price could be

16

The material in this section is relatively technical, and some instructors may choose to skip it with no loss in

continuity.

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almost anything after 6 months, not just $50 or $32. However, if we allowed the

stock to move up or down more often, then a more realistic range of ending prices

would result. In other words, dividing the time until expiration into more periods

would improve the realism of the resulting prices at expiration. The key to implementing a multi-period binomial model is to keep the stock return’s annual standard

deviation the same no matter how many periods you have during a year. In fact,

analysts typically begin with an estimate of the annual standard deviation and use it

to determine u and d. The derivation is beyond the scope of a financial management

textbook, but the appropriate equations are

u ¼ eσ

d¼

1

u

pﬃﬃﬃﬃﬃ

t=n

(8-7)

(8-8)

where σ is the annualized standard deviation of the stock’s return, t is the time in

years until expiration, and n is the number of periods until expiration.

The standard deviation of Western’s stock returns is 31.5573%, and application of

Equations 8-7 and 8-8 confirms the values of u and d that we used previously:

pﬃﬃﬃﬃﬃﬃﬃﬃ

1

and

d¼

¼ 0:80

u ¼ e0:315573 0_5=1 ¼ 1:25

1:25

Now suppose we allow stock prices to change every 3 months (which is 0.25 years).

Using Equations 8-7 and 8-8, we estimate u and d to be

pﬃﬃﬃﬃﬃﬃﬃﬃ

1

and

d¼

¼ 0:8540

u ¼ e0:31573 0_5=2 ¼1:1709

1:1709

At the end of the first 3 months, Western’s price would either rise to $40(1.1709) = $46.84

or fall to $40(0.8540) = $34.16. If the price rises in the first 3 months to $46.84, then it

would either go up to $46.84(1.1709) = $54.84 or go down to $46.84(0.8540) = $40 at

expiration. If instead the price initially falls to $40(0.8540) = $34.16 during the first 3

months, then it would either go up to $34.16(1.1709) = $40 or go down to $34.16(0.8540) =

$29.17 by expiration. This pattern of stock price movements is called a binomial lattice and

is shown in Figure 8-3.

Because the interest rate and the volatility (as defined by u and d) are constant for each

period, we can calculate πu and πd for any period and apply these same values for each

period:17

” 0:08 ð 0:5 = 2 Þ

#

e

− 0:80

1:25 − 0:80

πu ¼

¼ 0:51400

e0:08 ð 0:5 = 2 Þ

”

#

1:25 − e0:08 ð 0:5 = 1 Þ

1:25 − 0:80

¼ 0:46620

πd ¼

e0:08 ð 0:5 = 1 Þ

These values are shown in Figure 8-3.

17

These values were calculated in Excel, so there may be small differences due to rounding in intermediate steps.

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FIGURE 8-3

Two-Period Binomial Lattice and Option Valuation

resource

See Ch08 Tool Kit.xls on

the textbook’s Web site.

The lattice shows the possible stock prices at the option’s expiration and we know the

strike price, so we can calculate the option payoffs at expiration. Figure 8-3 also shows the

option payoffs at expiration. If we focus only on the upper right portion of the lattice shown

inside the dotted lines, then it is similar to the single-period problem we solved in Section 8-3.

In fact, we can use the binomial option pricing model from Equation 8-6 to determine the

value of the option in 3 months given that the stock price increased to $46.84. As shown in

Figure 8-3, the option will be worth $12.53 in 3 months if the stock price goes up to $46.84.

We can repeat this procedure on the lower right portion of Figure 8-3 to determine the call

option’s value in 3 months if the stock price falls to $34.16; in this case, the call’s value would

be $2.57. Finally, we can use Equation 8-6 and the 3-month option values just calculated to

determine the current price of the option, which is $7.64. Thus, we are able to find the current

option price by solving three simple binomial problems.

If we broke the year into smaller periods and allowed the stock price to move up or down

more often, then the lattice would have an even more realistic range of possible ending stock

prices. Of course, estimating the current option price would require solving lots of binomial

problems within the lattice, but each problem is simple and computers can solve them

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rapidly. With more outcomes, the resulting estimated option price is more accurate. For

example, if we divide the year into 15 periods then the estimated price is $7.42. With 50

periods, the price is $7.39. With 100 periods it is still $7.39, which shows that the solution

converges to its final value within a relatively small number of steps. In fact, as we break the

time to expiration into smaller and smaller periods, the solution for the binomial approach

converges to the Black-Scholes solution, which is described in the next section.

The binomial approach is widely used to value options with more complicated payoffs

than the call option in our example, such as employee stock options. This is beyond the

scope of a financial management textbook, but if you are interested in learning more

about the binomial approach, you should take a look at the textbooks by Don Chance and

John Hull cited in footnote 1.

SELF-TEST

Ringling Cycle’s stock price is now $20. You need to find the value of a call option

with a strike price of $22 that expires in 2 months. You want to use the binomial

model with 2 periods (each period is a month). Your assistant has calculated that

u = 1.1553, d = 0.8656, πu = 0.4838, and πd = 0.5095. Draw the binomial lattice for

stock prices. What are the possible prices after 1 month? ($23.11 or $17.31) After 2

months? ($26.69, $20, or $14.99) What are the option’s possible payoffs at

expiration? ($4.69, $0, or $0) What will the option’s value be in 1 month if the stock

goes up? ($2.27) What will the option’s value be in 1 month if the stock price goes

down? ($0) What is the current value of the option (assume each month is 1/12 of a

year)? ($1.10)

8-5 The Black-Scholes Option Pricing

Model (OPM)

The Black-Scholes option pricing model (OPM), developed in 1973, helped give rise to the

rapid growth in options trading. This model has been programmed into many handheld and

Web-based calculators, and it is widely used by option traders.

8-5a OPM Assumptions and Results

WWW

For a Web-based option

calculator, see www.cboe

.com/LearnCenter/

OptionCalculator.aspx.

In deriving their model to value call options, Fischer Black and Myron Scholes made the

following assumptions.

1. The stock underlying the call option provides no dividends or other distributions

during the life of the option.

2. There are no transaction costs for buying or selling either the stock or the option.

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

4. Any purchaser of a security may borrow any fraction of the purchase price at the

short-term, risk-free interest rate.

5. Short selling is permitted, and the short seller will receive immediately the full cash

proceeds of today’s price for a security sold short.

6. The call option can be exercised only on its expiration date.

7. Trading in all securities takes place continuously, and the stock price moves randomly.

The derivation of the Black-Scholes model rests on the same concepts as the binomial

model, except time is divided into such small increments that stock prices change

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continuously. The Black-Scholes model for call options consists of the following three

equations:

VC ¼ P½Nðd1 Þ%−Xe−rRF t ½Nðd2 Þ%

d1 ¼

lnðP=XÞ þ ½rRF þðσ2 =2Þ%t

pﬃ

σ t

pﬃ

d2 ¼ d1 −σ t

(8-9)

(8-10)

(8-11)

The variables used in the Black-Scholes model are explained below.

VC = Current value of the call option.

WWW

P = Current price of the underlying stock.

Robert’s Online Option

Pricer can be accessed at

www.intrepid.com/robertl/

index.html. The site

provides a financial service

over the Internet to small

investors for option pricing,

giving anyone a means to

price option trades without

having to buy expensive

software and hardware.

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 = Strike price of the option.

e ≈ 2.7183.

rRF = Risk-free interest rate.18

t = Time until the option expires (the option period).

ln(P/X) = Natural logarithm of P/X.

σ = Standard deviation of the rate of return on the stock.

The value of the option is a function of five variables: (1) P, the stock’s price; (2)

t, the option’s time to expiration; (3) X, the strike price; (4) σ, the standard

deviation of the underlying stock; and (5) rRF, the risk-free rate. We do not derive

the Black-Scholes model—the derivation involves some extremely complicated

mathematics that go far beyond the scope of this text. However, it is not difficult

to use the model. Under the assumptions set forth previously, if the option price is

different from the one found by Equation 8-9, then this would provide the opportunity for arbitrage profits, which would force the option price back to the value

indicated by the model. As we noted earlier, the Black-Scholes model is widely used

by traders because actual option prices conform reasonably well to values derived

from the model.

18

The correct process to estimate the risk-free rate for use in the Black-Scholes model for an option with 6

months to expiration is to find the annual nominal rate (compounded continuously) that has the same effective

annual rate as a 6-month T-bill. For example, suppose a 6-month T-bill is yielding a 6-month periodic rate of

4.081%. The risk-free rate to use in the Black-Scholes model is rRF = ln(1 + 0.0408)/0.5 = 8%. Under continuous

compounding, a nominal rate of 8% produces an effective rate of yields e0.08 – 1 = 8.33%. This is the same

effective rate yielded by the T-bill: (1+0.0408)2 – 1 = 8.33%. The same approach can be applied for options with

different expiration periods. We will provide the appropriate risk-free rate for all problems and examples.

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8-5b Application of the Black-Scholes Option Pricing

Model to a Call Option

The current stock price (P), the exercise price (X), and the time to maturity (t) can all be

obtained from a newspaper, such as The Wall Street Journal, or from the Internet, such as the

CBOE’s Web site. The risk-free rate (rRF) is the yield on a Treasury bill with a maturity equal

to the option expiration date. The annualized standard deviation of stock returns (σ) can be

estimated from daily stock prices. First, find the stock return for each trading day for a sample

period, such as each trading day of the past year. Second, estimate the variance of the daily

stock returns. Third, multiply this estimated daily variance by the number of trading days in a

year, which is approximately 250.19 Take the square root of the annualized variance, and the

result is an estimate of the annualized standard deviation.

We will use the Black-Scholes model to estimate Western’s call option that we

discussed previously. Here are the inputs:

resource

P ¼ $40

X ¼ $35

t ¼ 6 months ð0:5 yearsÞ

rRF ¼ 8:0% ¼ 0:080

σ ¼ 31:557% ¼ 0:31557

Given this information, we first estimate d1 and d2 from Equations 8-10 and 8-11:

See Ch08 Tool Kit.xls on

the textbook’s Web site for

all calculations.

d1 ¼

lnð$40=$35Þ þ ½0:08 þ ðð0:315572 Þ=2Þ&ð0:5Þ

pﬃﬃﬃﬃﬃﬃ

0:31557 0:5

0:13353 þ 0:064896

¼ 0:8892

0:22314

pﬃﬃﬃﬃﬃﬃ

d2 ¼ d1 − 0:31557 0:5¼0:6661

¼

Note that N(d1) and N(d2) represent areas under a standard normal distribution

function. The easiest way to calculate this value is with Excel. For example, we can use

the function =NORMSDIST(0.8892), which returns a value of N(d1) = N(0.8892) = 0.8131.

Similarly, the NORMSDIST function returns a value of N(d2) = 0.7473.20 We can use those

values to solve Equation 6-9:

VC ¼ $40½Nð0:8892Þ& $35e− ð 0:08 Þð 0:5 Þ ½Nð0:6661Þ&

¼ $7:39

Thus, the value of the option is $7.39. This is the same value we found using the binomial

approach with 100 periods in the year.

19

If stocks traded every day of the year, then each return covers a 24-hour period; you would simply estimate the

variance of the 1-day returns with your sample of daily returns and then multiply this estimate by 365 for an estimate of

the annual variance. However, stocks don’t trade every day because of weekends and holidays. If you measure returns

from the close of one trading day until the close of the next trading day (called “trading-day returns”), then some returns

are for 1 day (such as Thursday close to Friday close) and some are for longer periods, like the 3-day return from Friday

close to Monday close. It might seem reasonable that the 3-day returns have 3 times the variance of a 1-day return and

should be treated differently when estimating the daily return variance, but that is not the case. It turns out that the 3day return over a weekend has only slightly higher variance than a 1-day return (perhaps because of less new

information on non-weekdays), and so it is reasonable to treat all of the trading-day returns the same. With roughly

250 trading days in a year, most analysts take the estimate of the variance of daily returns and multiply by 250 (or 252,

depending on the year, to be more precise) to obtain an estimate of the annual variance.

20

If you do not have access to Excel, then you can use the table in Appendix A. For example, the table shows that the

value for d1 = 0.88 is 0.5000 + 0.3106 = 0.8106 and that the value for d1 = 0.89 is 0.5000 + 0.3133 = 0.8133, so N(0.8892)

lies between 0.8106 and 0.8133. You could interpolate to find a closer value, but we suggest using Excel instead.

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FIGURE 8-4

Western Cellular’s Call Options with a Strike Price of $35

30

$

t=1

t = 0.5

t = 0.25

25

20

15

10

Exercise Value

5

0

0

5

10

15

20

25

30

35

40

45

50

55

60

Stock Price ($)

© Cengage Learning 2014

8-5c The Five Factors That Affect Call Option Prices

resource

See Ch08 Tool Kit.xls on

the textbook’s Web site.

The Black-Scholes model has five inputs, so there are five factors that affect call option prices.

As we will see in the next section, these five inputs also affect put option prices. Figure 8-4

shows how three of Western Cellular’s call options are affected by Western’s stock price

(all three options have a strike price of $35). The three options expire in 1 year, in 6 months

(0.5 years, like the option in our example), and in 3 months (or 0.25 years), respectively.

Figure 8-4 offers several insights regarding option valuation. Notice that for all stock

prices in the Figure, the call option prices are always above the exercise value. If this were not

true, then an investor could purchase the call and immediately exercise it for a quick profit.21

Also, when the stock price falls far below the strike price, call option prices fall toward

zero. In other words, calls lose value as they become more and more out-of-the-money.

When the stock price greatly exceeds the strike price, call option prices fall toward the

exercise value. Thus, for very high stock prices, call options tend to move up and down by

about the same amount as does the stock price.

Call option prices increase if the stock price increases. This is because the strike price is

fixed, so an increase in stock price increases the chance that the option will be in-themoney at expiration. Although we don’t show it in the figure, an increase in the strike

price would obviously cause a decrease in the call option’s value because higher strike

prices mean a lower chance of being in-the-money at expiration.

21

More precisely, this statement is true for all American call options (which can be exercised before expiration)

and for European call options written on stocks that pay no dividends. Although European options may not be

exercised prior to expiration, investors could earn a riskless profit if the call price were less than the exercise value

by selling the stock short, purchasing the call, and investing at the risk-free rate an amount equal to the present

value of the strike price. The vast majority of call options are American options, so the call price is almost always

above the exercise value.

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resource

See Ch08 Tool Kit.xls for all

calculations.

The 1-year call option always has a greater value than the 6-month call option, which

always has a greater value than the 3-month call option; thus, the longer a call option has

until expiration, the greater its value. Here is the intuition for that result. With a long time

until expiration, the stock price has a chance to increase well above the strike price by the

expiration date. Of course, with a long time until expiration, there is also a chance that the

stock price will fall far below the strike price by expiration. But there is a big difference in

payoffs for being well in-the-money versus far out-of-the-money. Every dollar that the

stock price is above the strike price means an extra dollar of payoff, but no matter how far

the stock price is below the strike price, the payoff is zero. When it comes to a call option,

the gain in value due to the chance of finishing well in-the-money with a big payoff more

than compensates for the loss in value due to the chance of being far out-of-the money.

How does volatility affect call options? Following are the Black-Scholes model prices

for Western’s call option with the original inputs except for different standard deviations:

Standard Deviation (s)

Call Option Price

0.001%

$ 6.37

10.000

6.38

31.557

7.39

40.000

8.07

60.000

9.87

90.000

12.70

The first row shows the option price if there is very little stock volatility.22 Notice

that as volatility increases, so does the option price. Therefore, the riskier the underlying security, the more valuable the option. To see why this makes sense, suppose you

bought a call option with a strike price equal to the current stock price. If the stock had

no risk (which means σ = 0), then there would be a zero probability of the stock going

up, hence a zero probability of making money on the option. On the other hand, if you

bought a call option on a higher-volatility stock, there would be a higher probability

that the stock would increase well above the strike price by the expiration date. Of

course, with higher volatility there also would be a higher probability that the stock

price would fall far below the strike price. But as we previously explained, an increase

in the price of the stock helps call option holders more than a decrease hurts them: The

greater the stock’s volatility, the greater the value of the option. This makes options on

risky stocks more valuable than those on safer, low-risk stocks. For example, an option

on Cisco should have a greater value than an otherwise identical option on Kroger, the

grocery store chain.

22

With such a low standard deviation, the current stock price of $40 is unlikely to change very much before

expiration, so the option will be in-the-money at expiration and the owner will certainly pay the strike price and

exercise the option at that time. This means that the present value of the strike price is the cost of exercising

expressed in today’s dollars. The present value of a stock’s expected cash flows is equal to the current stock price.

So the value of the option today is approximately equal to the current stock price of $40 less the present value of

the strike price that must be paid when the stock is exercised at expiration. If we assume daily compounding,

then the current option price should be:

VC ðfor s ¼ 0:001%Þ $ $40 % !

1þ

$35

¼ $6:37

”

0:08 365ð0:5Þ

365

Observe that this is the same value given by the Black-Scholes model, even though we calculated it more directly.

This approach only works if the volatility is almost zero.

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Taxes and Stock Options

© Rob Webb/Getty Images

If an employee stock option grant meets certain conditions,

it is called a “tax-qualifying grant” or sometimes an

“Incentive Stock Option”; otherwise, it is a “nonqualifying

grant.” For example, suppose you receive a grant of 1,000

options with an exercise price of $50. If the stock price goes

to $110 and you exercise the options, you must pay $50

(1,000) = $50,000 for stock that is worth $110,000, which is a

sweet deal. But what is your tax liability? If you receive a

nonqualifying grant, then you are liable for ordinary income

taxes on 1,000($110 − $50) = $60,000 when you exercise the

option. But if it is a tax-qualified grant, you owe no regular

taxes when exercised. By waiting at least a year and then

selling the stock for, say, $150, you would have a long-term

capital gain of 1,000($150 − $50) = $100,000, which would

be taxed at the lower capital gains rate.

Before you gloat over your newfound wealth, you had

better consult your accountant. Your “profit” when you

exercise the tax-qualified options isn’t taxable under the

regular tax code, but it is under the Alternative Minimum

Tax (AMT) code. With an AMT tax rate of up to 28%, you

might owe as much as 0.28($110 − $50)(1,000) = $16,800.

Here’s where people get into trouble. The AMT tax isn’t due

until the following April, so you might think about waiting

until then to sell some stock to pay your AMT tax (so that the

sale will qualify as a long-term capital gain).

resource

See Ch08 Tool Kit.xls for all

calculations.

But what happens if the stock price falls to $5 by

next April? You can sell your stock, which raises only

$5(1,000) = $5,000 in cash. Without going into the

details, you will have a long-term capital loss of 1,000

($50 − $5) = $45,000 but IRS regulations limit your net

capital loss in a single year to $3,000. In other words,

the cash from the sale and the tax benefit from the

capital loss aren’t nearly enough to cover the AMT

tax. You may be able to reduce your taxes in future

years because of the AMT tax you pay this year and the

carryforward of the remaining long-term capital loss,

but that doesn’t help right now. You lost $45,000 of

your original $50,000 investment, you now have very

little cash, and—adding insult to injury—the IRS will

insist that you also pay the $16,800 AMT tax.

This is exactly what happened to many people

who made paper fortunes in the dot-com boom only

to see them evaporate in the ensuing bust. They were

left with worthless stock but multimillion-dollar AMT

tax obligations. In fact, many still have IRS liens

garnishing their wages until they eventually pay their

AMT tax. So if you receive stock options, we

congratulate you. But unless you want to be the next

poster child for poor financial planning, we advise you

to settle your AMT tax when you incur it.

The risk-free rate also has a relatively small impact on option prices. Shown below are

the prices for Western’s call option with the original inputs except for the risk-free rate,

which is allowed to vary.

Risk-free rate (rRF)

0%

Call option price

$6.41

4

6.89

8

7.39

12

7.90

20

8.93

As the risk-free rate increases, the value of the option increases. The principal effect of an

increase in rRF is to reduce the present value of the exercise price, which increases the

current value of the option. Option prices in general are not very sensitive to interest rate

changes, at least not to changes within the ranges normally encountered.

Myron Scholes and Robert Merton (who also was a pioneer in the field of options)

were awarded the 1997 Nobel Prize in Economics, and Fischer Black would have been a

co-recipient had he still been living. Their work provided analytical tools and methodologies that are widely used to solve many types of financial problems, not just option

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pricing. Indeed, the entire field of modern risk management is based primarily on their

contributions. Although the Black-Scholes model was derived for a European option that

can be exercised only on its maturity date, it also applies to American options that don’t

pay any dividends prior to expiration. The textbooks by Don Chance and John Hull (cited

in footnote 1) show adjusted models for dividend-paying stocks.

SELF-TEST

What is the purpose of the Black-Scholes option pricing model?

Explain what a “riskless hedge” is and how the riskless hedge concept is used in

the Black-Scholes OPM.

Describe the effect of a change in each of the following factors on the value of a call

option: (1) stock price, (2) exercise price, (3) option life, (4) risk-free rate, and (5)

stock return standard deviation (i.e., risk of stock).

Using an Excel worksheet, what is the value of a call option with these data: P = $35,

X = $25, rRF = 6%, t = 0.5 (6 months), and s = 0.6? ($12.05)

8-6 The Valuation of Put Options

A put option gives its owner the right to sell a share of stock. Suppose a stock pays no

dividends and a put option written on the stock can be exercised only upon its expiration

date. What is the put’s value? Rather than reinventing the wheel, we can establish the

price of a put relative to the price of a call.

8-6a Put-Call Parity

Consider the payoffs for two portfolios at expiration date T, as shown in Table 8-2. The first

portfolio consists of a put option and a share of stock; the second has a call option (with the same

strike price and expiration date as the put option) and some cash. The amount of cash is equal to

the present value of the strike price discounted at the continuously compounded risk-free rate,

which is Xe−rRFt. At expiration, the value of this cash will equal the strike price, X.

TABLE 8-2

Portfolio Payoffs

Payoff at Expiration If:

Put

Stock

Portfolio 1:

Call

Cash

Portfolio 2:

PT < X
PT ≥ X
X − PT
0
PT
PT
X
PT
0
PT − X
X
X
X
PT
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If PT, the stock price at expiration date T, is less than X, the strike price, when the
option expires, then the value of the put option at expiration is X − PT. Therefore, the
value of Portfolio 1, which contains the put and the stock, is equal to X minus PT plus PT,
or just X. For Portfolio 2, the value of the call is zero at expiration (because the call option
is out-of-the-money), and the value of the cash is X, for a total value of X. Notice that
both portfolios have the same payoffs if the stock price is less than the strike price.
What if the stock price is greater than the strike price at expiration? In this case, the
put is worth nothing, so the payoff of Portfolio 1 is equal to PT, the stock price at
expiration. The call option is worth PT − X, and the cash is worth X, so the payoff of
Portfolio 2 is PT. Hence the payoffs of the two portfolios are equal regardless of whether
the stock price is below or above the strike price.
If the two portfolios have identical payoffs, then they must have identical values. This is
known as the put–call parity relationship:
Put option þ Stock ¼ Call option þ PV of exercise price:
If VC is the Black-Scholes value of the call option, then the value of a put is23
Put option ¼ VC − P þ Xe−rRF t
(8-12)
For example, consider a put option written on the stock discussed in the previous
section. If the put option has the same exercise price and expiration date as the call, then
its price is
Put option ¼ $7:39 − $40 þ $35 e−0:08 ð 0:5 Þ
¼ $7:39 − $40 þ $33:63 ¼ $1:02
It is also possible to modify the Black-Scholes call option formula to obtain a put
option formula:
Put option ¼ P½Nðd1 Þ−1& − Xe−rRF t ½Nðd2 Þ−1&
(8-13)
The only difference between this formula for puts and the formula for calls is the
subtraction of 1 from N(d1) and N(d2) in the call option formula.
8-6b The Five Factors That Affect Put Option Prices
Just like with call options, the exercise price, the underlying stock price, the time to
expiration, the stock’s standard deviation, and the risk-free rate affect the price of a put
option. Because a put pays off when the stock price declines below the exercise price, the
impact of the underlying stock price and exercise price and risk-free rate on the put are
opposite that of the call option. That is, put prices are higher when the stock price is lower
and when the exercise price is higher. Put prices are also lower when the risk-free rate is
higher, mostly because a higher risk-free rate reduces the present value of the exercise
price, which for a put is a payout to the option holder when the option is exercised.
23
This model cannot be applied to an American put option or to a European option on a stock that pays a
dividend prior to expiration. For an explanation of valuation approaches in these situations, see the books by
Chance and Hull cited in footnote 1.
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On the other hand, put options are affected by the stock’s standard deviation just like
call options. Both put and call option prices are higher when the stock’s standard
deviation is higher. This is true for put options because the higher the standard deviation,
the bigger the chance of a large stock price decline and a large put payoff. The effect of the
time to maturity on the put option price is indeterminate. A call option is more valuable
the longer the maturity, but some puts are more valuable the longer to maturity, and some
are less valuable. For example, consider an in-the-money put option (the stock price is
below the exercise price) on a stock with a low standard deviation. In this case a longer
maturity put option is less valuable than a shorter maturity put option because the longer
the time to maturity, the more likely the stock is to grow and erode the put’s payoff. But if
the stock’s standard deviation is high, then the longer maturity put option will be more
valuable because the likelihood of the stock declining even more and resulting in a high
payoff to the put is greater.
SELF-TEST
In words, what is put–call parity?
A put option written on the stock of Taylor Enterprises (TE) has an exercise price of
$25 and 6 months remaining until expiration. The risk-free rate is 6%. A call option
written on TE has the same exercise price and expiration date as the put option.
TE’s stock price is $35. If the call option has a price of $12.05, then what is the price
(i.e., value) of the put option? ($1.31)
Explain why both put and call options are worth more if the stock return standard
deviation is higher, but put and call options are affected oppositely by the stock
price.
8-7 Applications of Option Pricing
in Corporate Finance
Option pricing is used in four major areas of corporate finance: (1) real options analysis
for project evaluation and strategic decisions, (2) risk management, (3) capital structure
decisions, and (4) compensation plans.
8-7a Real Options
Suppose a company has a 1-year proprietary license to develop a software application for
use in a new generation of wireless cellular telephones. Hiring programmers and marketing consultants to complete the project will cost $30 million. The good news is that if
consumers love the new cell phones, there will be a tremendous demand for the software.
The bad news is that if sales of the new cell phones are low, the software project will be a
disaster. Should the company spend the $30 million and develop the software?
Because the company has a license, it has the option of waiting for a year, at which time
it might have a much better insight into market demand for the new cell phones. If demand
is high in a year, then the company can spend the $30 million and develop the software.
If demand is low, it can avoid losing the $30 million development cost by simply letting the
license expire. Notice that the license is analogous to a call option: It gives the company the
right to buy something (in this case, software for the new cell phones) at a fixed price
($30 million) at any time during the next year. The license gives the company a real option,
because the underlying asset (the software) is a real asset and not a financial asset.
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There are many other types of real options, including the option to increase capacity at
a plant, to expand into new geographical regions, to introduce new products, to switch
inputs (such as gas versus oil), to switch outputs (such as producing sedans versus SUVs),
and to abandon a project. Many companies now evaluate real options with techniques
that are similar to those described earlier in the chapter for pricing financial options.
8-7b Risk Management
Suppose a company plans to issue $400 million of bonds in 6 months to pay for a new
plant now under construction. The plant will be profitable if interest rates remain at
current levels, but if rates rise then it will be unprofitable. To hedge against rising rates,
the company could purchase a put option on Treasury bonds. If interest rates go up then
the company would “lose” because its bonds would carry a high interest rate, but it would
have an offsetting gain on its put options. Conversely, if rates fall then the company would
“win” when it issues its own low-rate bonds, but it would lose on the put options. By
purchasing puts, the company has hedged the risk due to possible interest rate changes
that it would otherwise face.
Another example of risk management is a firm that bids on a foreign contract. For
example, suppose a winning bid means that the firm will receive a payment of 12 million
euros in 9 months. At a current exchange rate of $1.57 per euro, the project would be
profitable. But if the exchange rate falls to $1.10 per euro, the project would be a loser. To
avoid exchange rate risk, the firm could take a short position in a forward contract that
allows it to convert 12 million euros into dollars at a fixed rate of $1.50 per euro in
9 months, which would still ensure a profitable project. This eliminates exchange rate risk
if the firm wins the contract, but what if the firm loses the contract? It would still be
obligated to sell 12 million euros at a price of $1.50 per euro, which could be a disaster.
For example, if the exchange rate rises to $1.75 per euro, then the firm would have to
spend $21 million to purchase 12 million euros at a price of $1.75/€ and then sell the
euros for $18 million = ($1.50/€)(€12 million), a loss of $3 million.
To eliminate this risk, the firm could instead purchase a currency put option that
allows it to sell 12 million euros in 9 months at a fixed price of $1.50 per euro. If the
company wins the bid, it will exercise the put option and sell the 12 million euros for
$1.50 per euro if the exchange rate has declined. If the exchange rate hasn’t declined, then
it will sell the euros on the open market for more than $1.50 and let the option expire. On
the other hand, if the firm loses the bid, it has no reason to sell euros and could let the
option contract expire. Note, however, that even if the firm doesn’t win the contract, it
still is gambling on the exchange rate because it owns the put; if the price of euros declines
below $1.50, the firm will still make some money on the option. Thus, the company can
lock in the future exchange rate if it wins the bid and can avoid any net payment at all if it
loses the bid. The total cost in either scenario is equal to the initial cost of the option. In
other words, the cost of the option is like insurance that guarantees the exchange rate if
the company wins the bid and guarantees no net obligations if it loses the bid.
Many other applications of risk management involve futures contracts and other
complex derivatives rather than calls and puts. However, the principles used in pricing
derivatives are similar to those used earlier in this chapter for pricing options. Thus,
financial options and their valuation techniques play key roles in risk management.
8-7c Capital Structure Decisions
Decisions regarding the mix of debt and equity used to finance operations are quite
important. One interesting aspect of the capital structure decision is based on option
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pricing. For example, consider a firm with debt requiring a final principal payment of $60
million in 1 year. If the company’s value 1 year from now is $61 million, then it can pay
off the debt and have $1 million left for stockholders. If the firm’s value is less than $60
million, then it may well file for bankruptcy and turn over its assets to creditors, resulting
in stockholders’ equity of zero. In other words, the value of the stockholders’ equity is
analogous to a call option: The equity holders have the right to buy the assets for $60
million (which is the face value of the debt) in 1 year (when the debt matures).
Suppose the firm’s owner-managers are considering two projects. One project has very
little risk, and it will result in an asset value of either $59 million or $61 million. The other
has high risk, and it will result in an asset value of either $20 million or $100 million.
Notice that the equity will be worth zero if the assets are worth less than $60 million, so
the stockholders will be no worse off if the assets end up at $20 million than if they end up
at $59 million. On the other hand, the stockholders would benefit much more if the assets
were worth $100 million rather than $61 million. Thus, the owner-managers have an
incentive to choose risky projects, which is consistent with an option’s value rising with
the risk of the underlying asset. Potential lenders recognize this situation, so they build
covenants into loan agreements that restrict managers from making excessively risky
investments.
Not only does option pricing theory help explain why managers might want to choose
risky projects (consider, for example, the cases of Enron, Lehman Brothers, and AIG) and
why debtholders might want restrictive covenants, but options also play a direct role in
capital structure choices. For example, a firm could choose to issue convertible debt,
which gives bondholders the option to convert their debt into stock if the value of the
company turns out to be higher than expected. In exchange for this option, bondholders
charge a lower interest rate than for nonconvertible debt. Because owner-managers must
share the wealth with convertible-bond holders, they have a smaller incentive to gamble
with high-risk projects.
8-7d Compensation Plans
Many companies use stock options as a part of their compensation plans. It is important
for boards of directors to understand the value of these options before they grant them to
employees. We discuss compensation issues associated with stock options in more detail
in Chapter 13.
SELF-TEST
Describe four ways that option pricing is used in corporate finance.
SUMMARY
In this chapter we discussed option pricing topics, which included the following.
•
•
Financial options are instruments that (1) are created by exchanges rather than
firms, (2) are bought and sold primarily by investors, and (3) are of importance to
both investors and financial managers.
The two primary types of financial options are (1) call options, which give the holder
the right to purchase a specified asset at a given price (the exercise, or strike, price)
for a given period of time, and (2) put options, which give the holder the right to sell
an asset at a given price for a given period of time.
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•
•
•
•
•
A call option’s exercise value is defined as the maximum of zero or the current price
of the stock less the strike price.
The Black-Scholes option pricing model (OPM) or the binomial model can be used
to estimate the value of a call option.
The five inputs to the Black-Scholes model are (1) P, the current stock price; (2) X,
the strike price; (3) rRF, the risk-free interest rate; (4) t, the remaining time until
expiration; and (5) σ, the standard deviation of the stock’s rate of return.
A call option’s value increases if P increases, X decreases, rRF increases, t increases, or
σ increases.
The put–call parity relationship states that
Put option + Stock = Call option + PV of exercise price.
Questions
(8-1)
Define each of the following terms:
a. Option; call option; put option
b. Exercise value; strike price
c. Black-Scholes option pricing model
(8-2)
Why do options sell at prices higher than their exercise values?
(8-3)
Describe the effect on a call option’s price that results from an increase in each of the
following factors: (1) stock price, (2) strike price, (3) time to expiration, (4) risk-free rate,
and (5) standard deviation of stock return.
SELF-TEST PROBLEMS
(ST-1)
Binomial Option
Pricing
(ST-2)
Black-Scholes Model
PROBLEMS
Solutions Appear in Appendix A
The current price of a stock is $40. In 1 year, the price will be either $60 or $30. The
annual risk-free rate is 5%. Find the price of a call option on the stock that has an exercise
price of $42 and that expires in 1 year. (Hint: Use daily compounding.)
Use the Black-Scholes Model to find the price for a call option with the following
inputs: (1) current stock price is $22, (2) strike price is $20, (3) time to expiration is
6 months, (4) annualized risk-free rate is 5%, and (5) standard deviation of stock
return is 0.7.
Answers Appear in Appendix B
Easy Problems 1–2
(8-1)
Options
(8-2)
Options
A call option on the stock of Bedrock Boulders has a market price of $7. The stock sells for
$30 a share, and the option has a strike price of $25 a share. What is the exercise value of
the call option? What is the option’s time value?
The exercise price on one of Flanagan Company’s options is $15, its exercise value
is $22, and its time value is $5. What are the option’s market value and the price of
the stock?
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Intermediate
Problems 3–4
(8-3)
Assume that you have been given the following information on Purcell Industries:
Black-Scholes Model
Current stock price = $15
Strike price of option = $15
Time to maturity of option = 6 months
Risk-free rate = 6%
Variance of stock return = 0.12
d1 = 0.24495
N(d1) = 0.59675
d2 = 0.00000
N(d2) = 0.50000
According to the Black-Scholes option pricing model, what is the option’s value?
(8-4)
Put–Call Parity
The current price of a stock is $33, and the annual risk-free rate is 6%. A call option with a
strike price of $32 and with 1 year until expiration has a current value of $6.56. What is
the value of a put option written on the stock with the same exercise price and expiration
date as the call option?
Challenging
Problems 5–7
(8-5)
Black-Scholes Model
(8-6)
Binomial Model
(8-7)
Binomial Model
Use the Black-Scholes Model to find the price for a call option with the following inputs:
(1) current stock price is $30, (2) strike price is $35, (3) time to expiration is 4 months,
(4) annualized risk-free rate is 5%, and (5) variance of stock return is 0.25.
The current price of a stock is $20. In 1 year, the price will be either $26 or $16. The
annual risk-free rate is 5%. Find the price of a call option on the stock that has a strike
price of $21 and that expires in 1 year. (Hint: Use daily compounding.)
The current price of a stock is $15. In 6 months, the price will be either $18 or $13. The
annual risk-free rate is 6%. Find the price of a call option on the stock that has a strike
price of $14 and that expires in 6 months. (Hint: Use daily compounding.)
SPREADSHEET PROBLEM
(8-8)
Build a Model: BlackScholes Model
resource
Start with the partial model in the file Ch08 P08 Build a Model.xls on the textbook’s
Web site. You have been given the following information for a call option on the stock of
Puckett Industries: P = $65.00, X = $70.00, t = 0.50, rRF = 5.00% and σ = 50.00%.
a. Use the Black-Scholes option pricing model to determine the value of the call option.
b. Suppose there is a put option on Puckett’s stock with exactly the same inputs as the call
option. What is the value of the put?
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. Because
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 can use to gain a cursory understanding
of the topic.
Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
To begin, you gathered some outside materials on 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 report 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?
b. Options have a unique set of terminology. Define the following terms:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
Call option
Put option
Strike price or exercise price
Expiration date
Exercise value
Option price
Time value
Writing an option
Covered option
Naked option
In-the-money call
Out-of-the-money call
LEAPS
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 that 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.
(2) What happens to the time value as the stock price rises? Why?
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 that 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?
(2) Suppose you write one call option and buy Ns shares of stock. How many shares
must you buy to create a portfolio with a riskless payoff (i.e., a hedge portfolio)?
What is the payoff of the portfolio?
(3) What is the present value of the hedge portfolio? What is the value of the call option?
(4) What is a replicating portfolio? What is arbitrage?
Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
e. In 1973, Fischer Black and Myron Scholes developed the Black-Scholes option pricing
model (OPM).
(1) What assumptions underlie the OPM?
(2) Write out the three equations that constitute the model.
(3) According to the OPM, what is the value of a call option with the following
characteristics?
Stock price = $27.00
Strike price = $25.00
Time to expiration = 6 months = 0.5 years
Risk-free rate = 6.0%
Stock return standard deviation = 0.49
f. What impact does each of the following parameters have on the value of a call option?
(1) Current stock price
(2) Strike price
(3) Option’s term to maturity
(4) Risk-free rate
(5) Variability of the stock price
g. What is put–call parity?
Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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