chap 022

Options and Corporate Finance

Chapter 22

Copyright © 2010 by the McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

22-2

Key Concepts and Skills Understand option terminology Be able to determine option payoffs and profits Understand the major determinants of option

prices Understand and apply put-call parity Be able to determine option prices using the

binomial and Black-Scholes models

22-3

Chapter Outline22.1 Options22.2 Call Options22.3 Put Options22.4 Selling Options22.5 Option Quotes22.6 Combinations of Options22.7 Valuing Options22.8 An Option Pricing Formula22.9 Stocks and Bonds as Options22.10 Options and Corporate Decisions: Some Applications22.11 Investment in Real Projects and Options

22-4

22.1 Options An option gives the holder the right, but not the

obligation, to buy or sell a given quantity of an asset on (or before) a given date, at prices agreed upon today.

Exercising the Option The act of buying or selling the underlying asset

Strike Price or Exercise Price Refers to the fixed price in the option contract at which the

holder can buy or sell the underlying asset Expiry (Expiration Date)

The maturity date of the option

22-5

Options European versus American options

European options can be exercised only at expiry. American options can be exercised at any time up to expiry.

In-the-Money Exercising the option would result in a positive payoff.

At-the-Money Exercising the option would result in a zero payoff (i.e.,

exercise price equal to spot price). Out-of-the-Money

Exercising the option would result in a negative payoff.

22-6

22.2 Call Options Call options gives the holder the right,

but not the obligation, to buy a given quantity of some asset on or before some time in the future, at prices agreed upon today.

When exercising a call option, you “call in” the asset.

22-7

Call Option Pricing at Expiry At expiry, an American call option is worth the same

as a European option with the same characteristics. If the call is in-the-money, it is worth ST – E.

If the call is out-of-the-money, it is worthless:

C = Max[ST – E, 0]Where

ST is the value of the stock at expiry (time T)

E is the exercise price.

C is the value of the call option at expiry

22-8

Call Option Payoffs

–20

12020 40 60 80 100

–40

20

40

60

Stock price ($)

Op

tion

pay

offs

($) Buy

a ca

llExercise price = $50

50

22-9

Call Option Profits

Exercise price = $50; option premium = $10

Buy a call

–20

12020 40 60 80 100

–40

20

40

60

Stock price ($)

Op

tion

pro

fits

($)

50–10

10

22-10

22.3 Put Options Put options gives the holder the right,

but not the obligation, to sell a given quantity of an asset on or before some time in the future, at prices agreed upon today.

When exercising a put, you “put” the asset to someone.

22-11

Put Option Pricing at Expiry At expiry, an American put option is

worth the same as a European option with the same characteristics.

If the put is in-the-money, it is worth E – ST.

If the put is out-of-the-money, it is worthless.

P = Max[E – ST, 0]

22-12

Put Option Payoffs

–20

0 20 40 60 80 100

–40

20

0

40

60

Stock price ($)

Op

tion

pay

offs

($)

Buy a put

Exercise price = $50

50

50

22-13

Put Option Profits

–20

20 40 60 80 100

–40

20

40

60

Stock price ($)

Op

tion

pro

fits

($)

Buy a put


–10

10

50

22-14

Option Value Intrinsic Value

Call: Max[ST – E, 0]

Put: Max[E – ST , 0]

Speculative Value The difference between the option premium and the intrinsic

value of the option.

Option Premium

=Intrinsic Value

Speculative Value

+

22-15

22.4 Selling Options The seller (or writer) of an option has an

obligation. The seller receives the option premium in

exchange.

22-16

Call Option Payoffs

–20

12020 40 60 80 100

–40

20

40

60

Stock price ($)

Op

tion

pay

offs

($)

Sell a callExercise price = $50

50

22-17

Put Option Payoffs

–20

0 20 40 60 80 100

–40

20

0

40

–50

Stock price ($)

Op

tion

pay

offs

($)

Sell a put

Exercise price = $50

50

22-18

Option Diagrams Revisited


Sell a call

Buy a call

50 6040 100

–40

40

Stock price ($)

Op

tion

pro

fits

($)

Buy a put

Sell a put

–10

10

Buy a call

Sell a

put

Buy a put

Sell a call

22-19

22.5 Option Quotes

Option/Strike Exp. Vol. Last Vol. LastIBM 130 Oct 364 15¼ 107 5¼138¼ 130 Jan 112 19½ 420 9¼138¼ 135 Jul 2365 4¾ 2431 13/16

138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--

22-20

Option Quotes


138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--

This option has a strike price of $135;

a recent price for the stock is $138.25;

July is the expiration month.

22-21

Option Quotes


138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--

This makes a call option with this exercise price in-the-money by $3.25 = $138¼ – $135.

Puts with this exercise price are out-of-the-money.

22-22

Option Quotes


138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--

On this day, 2,365 call options with this exercise price were traded.

22-23

Option Quotes


138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--

The CALL option with a strike price of $135 is trading for $4.75.

Since the option is on 100 shares of stock, buying this option would cost $475 plus commissions.

22-24

Option Quotes


138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--

On this day, 2,431 put options with this exercise price were traded.

22-25

Option Quotes


138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½

--Put----Call--

The PUT option with a strike price of $135 is trading for $.8125.

Since the option is on 100 shares of stock, buying this option would cost $81.25 plus commissions.

22-26

22.6 Combinations of Options Puts and calls can serve as the

building blocks for more complex option contracts.

If you understand this, you can become a financial engineer, tailoring the risk-return profile to meet your client’s needs.

22-27

Protective Put Strategy (Payoffs)

Buy a put with an exercise price of $50

Buy the stock

Protective Put payoffs

$50

$0

$50

Value at expiry

Value of stock at expiry

22-28

Protective Put Strategy (Profits)

Buy a put with exercise price of $50 for $10

Buy the stock at $40

$40

Protective Put strategy has

downside protection and upside potential

$40

$0

-$40

$50

Value at expiry


-$10

22-29

Covered Call Strategy

Sell a call with exercise price of $50 for $10

Buy the stock at $40

$40

Covered Call strategy

$0

-$40

$50

Value at expiry


-$30

$10

22-30

Long Straddle

30 40 60 70

30

40

Stock price ($)

Op

tion

pay

offs

($)

Buy a put with exercise price of $50 for $10

Buy a call with exercise price of $50 for $10

A Long Straddle only makes money if the stock price moves $20 away from $50.

$50

–20

22-31

Short Straddle

–30

30 40 60 70

–40

Stock price ($)

Op

tion

pay

offs

($)

$50

This Short Straddle only loses money if the stock price moves $20 away from $50.

Sell a put with exercise price of$50 for $10

Sell a call with an exercise price of $50 for $10

20

22-32

bond

Put-Call Parity: P0 + S0 = C0 + E/(1+ r)T

25

25

Stock price ($)

Op

tion

pay

offs

($)

Consider the payoffs from holding a portfolio consisting of a call with a strike price of $25 and a bond with a future value of $25.

Call

Portfolio payoffPortfolio value today = C0 +

(1+ r)T

E

22-33

Put-Call Parity

25

25

Stock price ($)

Op

tion

pay

offs

($)

Consider the payoffs from holding a portfolio consisting of a share of stock and a put with a $25 strike.

Portfolio value today = P0 + S0

Portfolio payoff

22-34

Put-Call Parity

Since these portfolios have identical payoffs, they must have the same value today: hence

Put-Call Parity: C0 + E/(1+r)T = P0 + S0

25

25

Stock price ($)

Opt

ion

payo

ffs

($)

25

25

Stock price ($)O

ptio

n pa

yoff

s ($

) Portfolio value today = P0 + S0

Portfolio value today

(1+ r)T

E= C0 +

22-35

22.7 Valuing Options The last section

concerned itself with the value of an option at expiry.

This section considers the value of an option prior to the expiration date. A much more

interesting question.

22-36

American Call

C0 must fall within max (S0 – E, 0) < C0 < S0.

25

Op

tion

pay

offs

($) Call

ST

loss

E

Profit

ST

Time value

Intrinsic value

Market Value

In-the-moneyOut-of-the-money

22-37

Option Value DeterminantsCall Put

1. Stock price + –2. Exercise price – +3. Interest rate + –4. Volatility in the stock price + +5. Expiration date + +

The value of a call option C0 must fall within

max (S0 – E, 0) < C0 < S0.

The precise position will depend on these factors.

22-38

22.8 An Option Pricing Formula We will start with

a binomial option pricing formula to build our intuition.

Then we will graduate to the normal approximation to the binomial for some real-world option valuation.

22-39

Binomial Option Pricing ModelSuppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. S0= $25 today, and in one year S1is either $28.75 or $21.25. The risk-free rate is 5%. What is the value of an at-the-money call option?

$25

$21.25 = $25×(1 –.15)

$28.75 = $25×(1.15)S1S0

22-40

Binomial Option Pricing Model1. A call option on this stock with exercise price of $25 will

have the following payoffs.

2. We can replicate the payoffs of the call option with a levered position in the stock.

$25

$21.25

$28.75S1S0 C1

$3.75

$0

22-41

Binomial Option Pricing ModelBorrow the present value of $21.25 today and buy 1 share.

The net payoff for this levered equity portfolio in one period is either $7.50 or $0.

The levered equity portfolio has twice the option’s payoff, so the portfolio is worth twice the call option value.

$25

$21.25

$28.75S1S0 debt

– $21.25portfolio$7.50

$0

( – ) ==

=

C1

$3.75

$0– $21.25

22-42

Binomial Option Pricing ModelThe value today of the levered equity portfolio is today’s value of one share less the present value of a $21.25 debt: )1(

25.21$25$

fR

$25

$21.25

$28.75S1S0 debt


$0

( – ) ==

=

C1

$3.75

$0– $21.25

22-43

Binomial Option Pricing ModelWe can value the call option today as half of the value of the levered equity portfolio:

)1(

25.21$25$

2

10

fRC

$25

$21.25

$28.75S1S0 debt


$0

( – ) ==

=

C1

$3.75

$0– $21.25

22-44

If the interest rate is 5%, the call is worth:

Binomial Option Pricing Model

38.2$24.2025$2

1

)05.1(

25.21$25$

2

10

C

$25

$21.25

$28.75S1S0 debt


$0

( – ) ==

=

C1

$3.75

$0– $21.25

$2.38

C0

22-45

Binomial Option Pricing Model

the replicating portfolio intuition.the replicating portfolio intuition.

Many derivative securities can be valued by valuing portfolios of primitive securities when those portfolios have the same payoffs as the derivative securities.

The most important lesson (so far) from the binomial option pricing model is:

22-46

Delta This practice of the construction of a riskless

hedge is called delta hedging. The delta of a call option is positive.

Recall from the example:

• The delta of a put option is negative.

2

1

5.7$

75.3$

25.21$75.28$

075.3$

Swing of callSwing of stock

22-47

Delta Determining the Amount of Borrowing:

Value of a call = Stock price × Delta

– Amount borrowed

$2.38 = $25 × ½ – Amount borrowed

Amount borrowed = $10.12

38.2$24.20$25$2

1

)05.1(

25.21$25$

2

10

C

22-48

The Risk-Neutral Approach

We could value the option, V(0), as the value of the replicating portfolio. An equivalent method is risk-neutral valuation:

S(0), V(0)

S(U), V(U)

S(D), V(D)

q

1- q

)1(

)()1()()0(

fR

DVqUVqV

22-49


S(0) is the value of the underlying asset today.

S(0), V(0)

S(U), V(U)

S(D), V(D)

S(U) and S(D) are the values of the asset in the next period following an up move and a down move, respectively.

q

1- q

V(U) and V(D) are the values of the option in the next period following an up move and a down move, respectively.

q is the risk-neutral probability of an “up” move.

22-50


The key to finding q is to note that it is already impounded into an observable security price: the value of S(0):

S(0), V(0)

S(U), V(U)

S(D), V(D)

q

1- q

)1(

)()1()()0(

fR

DVqUVqV

)1(

)()1()()0(

fR

DSqUSqS

A minor bit of algebra yields:)()(

)()0()1(

DSUS

DSSRq f

22-51

Example of Risk-Neutral Valuation

$21.25,C(D)

q

1- q

Suppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. The risk-free rate is 5%. What is the value of an at-the-money call option?The binomial tree would look like this:

$25,C(0)

$28.75,C(U)

)15.1(25$75.28$

)15.1(25$25.21$

22-52


$21.25,C(D)

2/3

1/3

The next step would be to compute the risk neutral probabilities

$25,C(0)

$28.75,C(U)

)()(

)()0()1(

DSUS

DSSRq f

3250.7$

5$

25.21$75.28$

25.21$25$)05.1(

q

22-53


$21.25, $0

2/3

1/3

After that, find the value of the call in the up state and down state.

$25,C(0)

$28.75, $3.75

25$75.28$)( UC

]0,75.28$25max[$)( DC

22-54

Example of Risk-Neutral ValuationFinally, find the value of the call at time 0:

$21.25, $0

2/3

1/3

$25,C(0)

$28.75,$3.75

)1(

)()1()()0(

fR

DCqUCqC

)05.1(

0$)31(75.3$32)0(

C

38.2$)05.1(

50.2$)0( C

$25,$2.38

22-55

This risk-neutral result is consistent with valuing the call using a replicating portfolio.

Risk-Neutral Valuation and the Replicating Portfolio

38.2$24.2025$2

1

)05.1(

25.21$25$

2

10

C

38.2$05.1

50.2$

)05.1(

0$)31(75.3$320

C

22-56

The Black-Scholes Model

)N()N( 210 dEedSC Rt

Where

C0 = the value of a European option at time t = 0R = the risk-free interest rate.

t

tσ

RESd

)2

()/ln(2

1

tdd 12

N(d) = Probability that a standardized, normally distributed, random variable will be less than or equal to d.

The Black-Scholes Model allows us to value options in the real world just as we have done in the 2-state world.

22-57

The Black-Scholes ModelFind the value of a six-month call option on Hardcraft, Inc. with an exercise price of $150.

The current value of a share of Hardcraft is $160.

The interest rate available in the U.S. is R = 5%.

The option maturity is 6 months (half of a year).

The volatility of the underlying asset is 30% per annum.

Before we start, note that the intrinsic value of the option is $10—our answer must be at least that amount.

22-58

The Black-Scholes ModelLet’s try our hand at using the model. If you have a calculator handy, follow along.

Then,

t

tσRESd

)5.()/ln( 2

1

First calculate d1 and d2

31602.05.30.052815.012 tdd

52815.05.30.0

5).)30.0(5.05(.)150/160ln( 2

1

d

22-59

The Black-Scholes Model

N(d1) = N(0.52815) = 0.7013

N(d2) = N(0.31602) = 0.62401

52815.01 d

31602.02 d

)N()N( 210 dEedSC Rt

92.20$

62401.01507013.0160$

0

5.05.0

C

eC

22-60

22.9 Stocks and Bonds as Options Levered equity is a call option.

The underlying asset comprises the assets of the firm. The strike price is the payoff of the bond.

If at the maturity of their debt, the assets of the firm are greater in value than the debt, the shareholders have an in-the-money call. They will pay the bondholders and “call in” the assets of the firm.

If at the maturity of the debt the shareholders have an out-of-the-money call, they will not pay the bondholders (i.e. the shareholders will declare bankruptcy) and let the call expire.

22-61

Stocks and Bonds as Options Levered equity is a put option.

The underlying asset comprises the assets of the firm. The strike price is the payoff of the bond.

If at the maturity of their debt, the assets of the firm are less in value than the debt, shareholders have an in-the-money put.

They will put the firm to the bondholders. If at the maturity of the debt the shareholders have

an out-of-the-money put, they will not exercise the option (i.e. NOT declare bankruptcy) and let the put expire.

22-62

Stocks and Bonds as Options It all comes down to put-call parity.

Value of a call on the

firm

Value of a put on the

firm

Value of a risk-free

bond

Value of the firm= + –

Stockholder’s position in terms of call options

Stockholder’s position in terms of put options

C0 = S0 + P0 – (1+ R)t

E

22-63

Mergers and Diversification Diversification is a frequently mentioned reason for

mergers. Diversification reduces risk and, therefore, volatility. Decreasing volatility decreases the value of an option. Assume diversification is the only benefit to a merger:

Since equity can be viewed as a call option, should the merger increase or decrease the value of the equity?

Since risky debt can be viewed as risk-free debt minus a put option, what happens to the value of the risky debt?

Overall, what has happened with the merger and is it a good decision in view of the goal of stockholder wealth maximization?

22-64

Example Consider the following two merger candidates. The merger is for diversification purposes only with no synergies

involved. Risk-free rate is 4%.

Company A Company B

Market value of assets $40 million $15 million

Face value of zero coupon debt

$18 million $7 million

Debt maturity 4 years 4 years

Asset return standard deviation

40% 50%

22-65

Example Use the Black and Scholes OPM (or an options

calculator) to compute the value of the equity. Value of the debt = value of assets – value of equity

Company A Company B

Market Value of Equity 25.72 9.88

Market Value of Debt 14.28 5.12

22-66

Example The asset return standard deviation for the combined firm is 30% Market value assets (combined) = 40 + 15 = 55 Face value debt (combined) = 18 + 7 = 25

Combined Firm

Market value of equity 34.18

Market value of debt 20.82

Total MV of equity of separate firms = 25.72 + 9.88 = 35.60

Wealth transfer from stockholders to bondholders = 35.60 – 34.18 = 1.42 (exact increase in MV of debt)

22-67

M&A Conclusions Mergers for diversification only transfer wealth

from the stockholders to the bondholders. The standard deviation of returns on the assets

is reduced, thereby reducing the option value of the equity.

If management’s goal is to maximize stockholder wealth, then mergers for reasons of diversification should not occur.

22-68

Options and Capital Budgeting Stockholders may prefer low NPV projects to high

NPV projects if the firm is highly leveraged and the low NPV project increases volatility.

Consider a company with the following characteristics: MV assets = 40 million Face Value debt = 25 million Debt maturity = 5 years Asset return standard deviation = 40% Risk-free rate = 4%

22-69

Example: Low NPV Current market value of equity = $22.706 million Current market value of debt = $17.294 million

Project I Project II

NPV $3 $1

MV of assets $43 $41

Asset return standard deviation

30% 50%

MV of equity $23.831 $25.381

MV of debt $19.169 $15.169

22-70

Example: Low NPV Which project should management take? Even though project B has a lower NPV, it is

better for stockholders. The firm has a relatively high amount of

leverage: With project A, the bondholders share in the NPV

because it reduces the risk of bankruptcy. With project B, the stockholders actually appropriate

additional wealth from the bondholders for a larger gain in value.

22-71

Example: Negative NPV We have seen that stockholders might prefer a low

NPV to a high one, but would they ever prefer a negative NPV?

Under certain circumstances, they might. If the firm is highly leveraged, stockholders have

nothing to lose if a project fails, and everything to gain if it succeeds.

Consequently, they may prefer a very risky project with a negative NPV but high potential rewards.

22-72

Example: Negative NPV Consider the previous firm. They have one additional project they are

considering with the following characteristics Project NPV = -$2 million MV of assets = $38 million Asset return standard deviation = 65%

Estimate the value of the debt and equity MV equity = $25.453 million MV debt = $12.547 million

22-73

Example: Negative NPV In this case, stockholders would actually prefer

the negative NPV project to either of the positive NPV projects.

The stockholders benefit from the increased volatility associated with the project even if the expected NPV is negative.

This happens because of the large levels of leverage.

22-74

Options and Capital Budgeting As a general rule, managers should not accept

low or negative NPV projects and pass up high NPV projects.

Under certain circumstances, however, this may benefit stockholders: The firm is highly leveraged The low or negative NPV project causes a

substantial increase in the standard deviation of asset returns

22-75

22.11 Investment in Real Projects and Options Classic NPV calculations generally ignore

the flexibility that real-world firms typically have. Option to expand Option to abandon

22-76

Quick Quiz What is the difference between call and put options? What are the major determinants of option prices? What is put-call parity? What would happen if it does

not hold? What is the Black-Scholes option pricing model? How can equity be viewed as a call option? Should a firm do a merger for diversification

purposes only? Why or why not? Should management ever accept a negative NPV

project? If yes, under what circumstances?

chap 022

Documents

option premium

optionsan option

option quotes22

option contract

option profitsexercise

option pricing formula22

putexercise price

fixed price