Corporate finance 7e ross ch22

An option gives the holder the right, but not the obligation , to buy or sell a given quantity of an asset on or perhaps before a given date, at prices agreed upon today.. Calls versus

22.8 An Option Pricing Formula ‑

22.8 An Option Pricing Formula ‑

22.9 Stocks and Bonds as Options

22.10 Capital-Structure Policy and Options

22.11 Mergers and Options

22.12 Investment in Real Projects and Options

22.13 Summary and Conclusions

22.8 An Option Pricing Formula ‑

22.8 An Option Pricing Formula ‑

22.9 Stocks and Bonds as Options

22.10 Capital-Structure Policy and Options

22.11 Mergers and Options

22.12 Investment in Real Projects and Options

22.13 Summary and Conclusions

22.1 Options

Many corporate securities are similar to the

stock options that are traded on organized

exchanges

Almost every issue of corporate stocks and

bonds has option features.

In addition, capital structure and capital

budgeting decisions can be viewed in terms of options.

Many corporate securities are similar to the

stock options that are traded on organized

exchanges

Almost every issue of corporate stocks and

bonds has option features.

In addition, capital structure and capital

budgeting decisions can be viewed in terms of options.

An option gives the holder the right, but not the obligation , to buy or sell a given quantity of an asset on (or perhaps before)

a given date, at prices agreed upon today.

Calls versus Puts

Call options gives the holder the right, but not the obligation, to buy

a given quantity of some asset at some time in the future, at prices agreed upon today When exercising a call option, you “call in” the asset.

Put options gives the holder the right, but not the obligation, to sell a given quantity of an asset at some time in the future, at prices agreed upon today When exercising a put, you “put” the asset to someone.

An option gives the holder the right, but not the obligation , to buy or sell a given quantity of an asset on (or perhaps before)

a given date, at prices agreed upon today.

Calls versus Puts

Call options gives the holder the right, but not the obligation, to buy

a given quantity of some asset at some time in the future, at prices agreed upon today When exercising a call option, you “call in” the asset.

Put options gives the holder the right, but not the obligation, to sell a given quantity of an asset at some time in the future, at prices agreed upon today When exercising a put, you “put” the asset to someone.

22.1 Options Contracts: Preliminaries

Exercising the Option

The act of buying or selling the underlying asset through the option contract.

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

The maturity date of the option is referred to as the expiration date, or the expiry

European versus American options

European options can be exercised only at expiry.

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

Exercising the Option

The act of buying or selling the underlying asset through the option contract.

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

The maturity date of the option is referred to as the expiration date, or the expiry

European versus American options

European options can be exercised only at expiry.

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

Options Contracts: Preliminaries

Options Contracts: Preliminaries

Intrinsic Value

Speculative

Value +

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.

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.

Basic Call Option Pricing Relationships

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 S T – E.

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

C = Max[ S T – E , 0]

Where

S T 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

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 S T – E.

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

C = Max[ S T – E , 0]

Where

S T 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

120

20 40 60

Call Option Payoffs

10

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 – S T .

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

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

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

worthless.

P = Max [ E – S T , 0]

Put Option Payoffs

Put Option Profits

–20

–40

20 40 60

The seller (or writer) of an option has an obligation.

The purchaser of an option has an option.

22.5 Reading The Wall

Street Journal

Option/Strike Exp Vol Last Vol Last

IBM 130 Oct 364 15¼ 107 5¼

138¼ 130 Jan 112 19½ 420 9¼ 138¼ 135 Jul 2365 4¾ 2431 13/16

138¼ 135 Aug 1231 9¼ 94 5½ 138¼ 140 Jul 1826 1¾ 427 2¾ 138¼ 140 Aug 2193 6½ 58 7½

Call

Put Option/Strike Exp Vol Last Vol Last IBM 130 Oct 364 15¼ 107 5¼ 138¼ 130 Jan 112 19½ 420 9¼

138¼ 135 Jul 2365 4¾ 2431 13/16

138¼ 135 Aug 1231 9¼ 94 5½ 138¼ 140 Jul 1826 1¾ 427 2¾ 138¼ 140 Aug 2193 6½ 58 7½

Call

Put This option has a strike price of $135;

a recent price for the stock is $138.25

July is the expiration month

22.5 Reading The Wall Street Journal

Option/Strike Exp Vol Last Vol Last

IBM 130 Oct 364 15¼ 107 5¼

138¼ 130 Jan 112 19½ 420 9¼

138¼ 135 Jul 2365 4¾ 2431 13/16

138¼ 135 Aug 1231 9¼ 94 5½ 138¼ 140 Jul 1826 1¾ 427 2¾ 138¼ 140 Aug 2193 6½ 58 7½

Call

Put This makes a call option with this exercise price money by $3.25 = $138¼ – $135

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

22.5 Reading The Wall Street Journal

Option/Strike Exp Vol Last Vol Last IBM 130 Oct 364 15¼ 107 5¼ 138¼ 130 Jan 112 19½ 420 9¼

138¼ 135 Jul 2365 4¾ 2431 13/16

138¼ 135 Aug 1231 9¼ 94 5½ 138¼ 140 Jul 1826 1¾ 427 2¾ 138¼ 140 Aug 2193 6½ 58 7½

Call

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

22.5 Reading The Wall Street Journal

Option/Strike Exp Vol Last Vol Last IBM 130 Oct 364 15¼ 107 5¼ 138¼ 130 Jan 112 19½ 420 9¼

138¼ 135 Jul 2365 4¾ 2431 13/16

138¼ 135 Aug 1231 9¼ 94 5½ 138¼ 140 Jul 1826 1¾ 427 2¾ 138¼ 140 Aug 2193 6½ 58 7½

Call

Put 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.

22.5 Reading The Wall Street Journal

Option/Strike Exp Vol Last Vol Last IBM 130 Oct 364 15¼ 107 5¼ 138¼ 130 Jan 112 19½ 420 9¼

138¼ 135 Jul 2365 4¾ 2431 13/16

138¼ 135 Aug 1231 9¼ 94 5½ 138¼ 140 Jul 1826 1¾ 427 2¾ 138¼ 140 Aug 2193 6½ 58 7½

Call

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

22.5 Reading The Wall Street Journal

Option/Strike Exp Vol Last Vol Last IBM 130 Oct 364 15¼ 107 5¼ 138¼ 130 Jan 112 19½ 420 9¼

138¼ 135 Jul 2365 4¾ 2431 13/16

138¼ 135 Aug 1231 9¼ 94 5½ 138¼ 140 Jul 1826 1¾ 427 2¾ 138¼ 140 Aug 2193 6½ 58 7½

Call

Put 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.

22.6 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.

Puts and calls can serve as the building

blocks for more complex option

Protective Put Strategy: Buy a Put and Buy the Underlying Stock: Payoffs at Expiry

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

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

Covered Call Strategy

Sell a call with exercise price

Long Straddle: Buy a Call and a Put

Call

Portfolio payoff

Portfolio value today = c0 +

(1+ r) T E

Put-Call Parity: p 0 + S 0 = c 0 + E/(1+ r)

Since these portfolios have identical payoffs, they must have the same

value today: hence

Put-Call Parity: c + E/(1+r)T = p + S

22.7 Valuing Options

The last section

concerned itself with

the value of an

option at expiry.

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.

This section considers the value

of an option prior to the expiration date.

A much more interesting question.

Option Value Determinants

for an American Call

The value of a call option C0 must fall within max (S0 – E, 0) < C0 < S0.

Market Value

In-the-money Out-of-the-money

22.8 An Option Pricing Formula ‑

We will start with

approximation to the binomial for some real-world option valuation.

Then we will graduate to the normal

approximation to the binomial for some real-world option valuation.

Binomial Option Pricing Model

Suppose a stock is worth $25 today and in one period will either

be worth 15% more or 15% less S 0 = $25 today and in one

year S 1 is either $28.75 or $21.25 The risk-free rate is 5%

What is the value of an at-the-money call option?

Suppose a stock is worth $25 today and in one period will either

be worth 15% more or 15% less S 0 = $25 today and in one

year S 1 is either $28.75 or $21.25 The risk-free rate is 5%

What is the value of an at-the-money call option?

1 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.

1 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.

Binomial Option Pricing Model

Borrow 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.

Borrow the present value of $21.25 today and buy 1 share

The net payoff for this levered equity portfolio in one period is

The value today of the levered equity portfolio is today’s value of one share less the present value of a $21.25 debt:

The 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

Binomial Option Pricing Model

We can value the call option today

as half of the value of the

levered equity portfolio:

We can value the call option today

as half of the value of the

(

25 21

$ 25

$ 2

1

0

f

r C

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

( $ 25 20 24 ) $ 2 38 2

1 )

05 1 (

25 21

$ 25

$ 2

the replicating portfolio intuition.

the replicating portfolio intuition.

Binomial Option Pricing Model

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:

Delta and the Hedge Ratio

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:

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

$

0 75

3

Determining the Amount of Borrowing:

Value of a call = Stock price × Delta – Amount borrowed

$2.38 = $25 × ½ – Amount borrowed

Amount borrowed = $10.12

Determining the Amount of Borrowing:

Value of a call = Stock price × Delta – Amount borrowed

$2.38 = $25 × ½ – Amount borrowed

Amount borrowed = $10.12

( $ 25 $ 20 24 ) $ 2 38 2

1 )

05 1 (

25 21

$ 25

$ 2

We could value V (0) as the value of the replicating

valuation

(

) ( )

1 ( ) ( )

0

(

fr

D V q

U V q V

+

×

− +

×

=

The Risk-Neutral Approach to Valuation

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 asset in the next period

following an up move and a down move, respectively.

q is the risk-neutral

probability of an

“up” move.

The key to finding q is to note that it is already impounded

into an observable security price: the value of S (0):

(

) ( )

0 ( )

1

(

D S U

S

D S S

(

) ( )

1 ( ) ( )

0

(

fr

D S q

U S q S

+

×

− +

×

=

) 1

(

) ( )

1 ( ) ( )

0

(

fr

D V q

U V q V

+

×

− +

×

=

Example of the Risk-Neutral Valuation of a Call:

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:

$ 75 28

) 15 1 ( 25

$ 25 21

) ( )

0 ( )

1

(

D S U

S

D S S

$

5

$ 25

21

$ 75 28

$

25 21

$ 25

$ ) 05 1

Example of the Risk-Neutral Valuation of a Call:

$ 25 max[$

$ )

C

Finally, find the value of the call at time 0:

(

) ( )

1 ( ) ( )

0

(

fr

D C q

U C q C

+

×

− +

×

=

) 05 1 (

0

$ ) 3 1 ( 75 3

$ 3 2 ) 0

C

38 2

$ ) 05 1 (

50 2

$ ) 0

C

This risk-neutral result is consistent with

valuing the call using a replicating

portfolio.

This risk-neutral result is consistent with

valuing the call using a replicating

portfolio.

Risk-Neutral Valuation and the Replicating Portfolio

( $ 25 20 24 ) $ 2 38 2

1 )

05 1 (

25 21

$ 25

$ 2

$ 05

1

50 2

$ )

05 1 (

0

$ )

3 1 ( 75

3

$ 3

2

C

The Black-Scholes Model is

Where

r = the risk-free interest rate.

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.

) N(

E

S d

σ

) 2

( ) / ln(

The Black-Scholes Model

Find the value of a six-month call option on the

Microsoft with an exercise price of $150

The current value of a share of Microsoft is $160

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

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

is $10—our answer must be at least that amount.

Find the value of a six-month call option on the

Microsoft with an exercise price of $150

The current value of a share of Microsoft is $160

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

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

is $10—our answer must be at least that amount.