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# Options

## 1.

22.1 Options22.2 Call Options

22.3 Put Options

22.4 Selling Options

22.5 Stock Option Quotations

22.6 Combinations of Options

22.7 Valuing Options

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

## 2. 22.1 Options

• Many corporate securities are similar to the stockoptions 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.

## 3. 22.1 Options Contracts: Preliminaries

• An option gives the holder the right, but not theobligation, 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.

## 4. 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.

## 5. Options Contracts: Preliminaries

• In-the-Money– The exercise price is less than the spot price of

the underlying asset.

• At-the-Money

– The exercise price is equal to the spot price of

the underlying asset.

• Out-of-the-Money

– The exercise price is more than the spot price of

the underlying asset.

## 6. Options Contracts: Preliminaries

• Intrinsic Value– The difference between the exercise price of the

option and the spot price of the underlying

asset.

• Speculative Value

– The difference between the option premium and

the intrinsic value of the option.

Option

Premium

=

Intrinsic

Value

Speculative

+

Value

## 7. 22.2 Call Options

• Call options gives the holder the right, but not theobligation, 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.

## 8. Basic Call Option Pricing Relationships at Expiry

• At expiry, an American call option is worth thesame 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.

CaT = CeT = Max[ST - E, 0]

• Where

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

E is the exercise price.

CaT is the value of an American call at expiry

CeT is the value of a European call at expiry

## 9. Call Option Payoffs

60Option payoffs ($)

40

Buy a call

20

0

0

10

20

30

40

50

60

70

80

90

100

Stock price ($)

-20

-40

-60

Exercise price = $50

## 10. Call Option Payoffs

60Option payoffs ($)

40

20

0

-20

0

10

20

30

40

50

60

70

80

Stock price ($)

Write a call

-40

-60

Exercise price = $50

90

100

## 11. Call Option Profits

60Option profits ($)

40

Buy a call

20

0

-20

0

10

20

30

40

50

60

70

80

90

Stock price ($)

Write a call

-40

-60

Exercise price = $50; option premium = $10

100

## 12. 22.3 Put Options

• Put options give the holder the right, but not theobligation, 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.

## 13. Basic Put Option Pricing Relationships at Expiry

• At expiry, an American put option is worth thesame 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.

PaT = PeT = Max[E - ST, 0]

## 14. Put Option Payoffs

60Option payoffs ($)

40

Buy a put

20

0

-20

0

10

20

30

40

50

60

70 80

90

100

Stock price ($)

-40

-60

Exercise price = $50

## 15. Put Option Payoffs

60Option payoffs ($)

40

20

0

0

10

20

30

-20

-40

40

50

60

70 80

90

100

Stock price ($)

write a put

-60

Exercise price = $50

## 16. Put Option Profits

Option profits ($)Put Option Profits

60

40

20

10

0

-10

-20

Write a put

0

10

20

30

Stock price ($)

40

50

60

70 80

Buy a put

90

-40

-60

Exercise price = $50; option premium = $10

100

## 17. 22.4 Selling Options

The seller (or writer) of an

option has an obligation.

Option profitsOption

($)

profits ($)

The purchaser of an option has

an option.

60

40

20

10

0

-10

-20

-40

-60

Buy a call

Write a put

0

10

20

30

Stock price ($)

40

50

60

70 80

Buy a put

Write a call

90

100

## 18. 22.5 Stock Option Quotations

StkExp

P/C

Vol

Nortel Networks (NT)

9

Mar

C

446

9

Mar

P

155

8

June

C

15

8

June

P

35

11 Sept

C

11

11 Sept

P

5

Bid

0.50

0.20

1.95

0.55

1.10

2.65

Ask

0.55

0.30

2.10

0.65

1.25

2.80

Opint

9.35

2461

841

660

1310

459

279

## 19. 22.5 Stock Option Quotations

A recent price for the stock is $9.35Stk

Exp

P/C

Vol

Nortel Networks (NT)

9

Mar

C

446

9

Mar

P

155

8

June

C

15

8

June

P

35

11 Sept

C

11

11 Sept

P

5

Bid

0.50

0.20

1.95

0.55

1.10

2.65

Ask

0.55

0.30

2.10

0.65

1.25

2.80

Opint

9.35

2461

841

660

1310

459

279

This option has a strike price of $8;

June is the expiration month

## 20. 22.5 Stock Option Quotations

This makes a call option with this exercise price in-themoney by $1.35 = $9.35 – $8.Stk

Exp

P/C

Vol

Nortel Networks (NT)

9

Mar

C

446

9

Mar

P

155

8

June

C

15

8

June

P

35

11 Sept

C

11

11 Sept

P

5

Bid

0.50

0.20

1.95

0.55

1.10

2.65

Ask

0.55

0.30

2.10

0.65

1.25

2.80

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

Opint

9.35

2461

841

660

1310

459

279

## 21. 22.5 Stock Option Quotations

StkExp

P/C

Vol

Nortel Networks (NT)

9

Mar

C

446

9

Mar

P

155

8

June

C

15

8

June

P

35

11 Sept

C

11

11 Sept

P

5

Bid

0.50

0.20

1.95

0.55

1.10

2.65

Ask

0.55

0.30

2.10

0.65

1.25

2.80

Opint

9.35

2461

841

660

1310

459

279

On this day, 15 call options with this exercise price were

traded.

## 22. 22.5 Stock Option Quotations

The holder of this CALL option can sell it for $1.95.Stk

Exp

P/C

Vol

Nortel Networks (NT)

9

Mar

C

446

9

Mar

P

155

8

June

C

15

8

June

P

35

11 Sept

C

11

11 Sept

P

5

Bid

0.50

0.20

1.95

0.55

1.10

2.65

Ask

0.55

0.30

2.10

0.65

1.25

2.80

Opint

9.35

2461

841

660

1310

459

279

Since the option is on 100 shares of stock, selling this option

would yield $195.

## 23.

22.5 Stock Option QuotationsBuying this CALL option costs $2.10.

Stk

Exp

P/C

Vol

Nortel Networks (NT)

9

Mar

C

446

9

Mar

P

155

8

June

C

15

8

June

P

35

11 Sept

C

11

11 Sept

P

5

Bid

0.50

0.20

1.95

0.55

1.10

2.65

Ask

0.55

0.30

2.10

0.65

1.25

2.80

Opint

9.35

2461

841

660

1310

459

279

Since the option is on 100 shares of stock, buying this option

would cost $210.

## 24. 22.5 Stock Option Quotations

StkExp

P/C

Vol

Nortel Networks (NT)

9

Mar

C

446

9

Mar

P

155

8

June

C

15

8

June

P

35

11 Sept

C

11

11 Sept

P

5

Bid

0.50

0.20

1.95

0.55

1.10

2.65

Ask

0.55

0.30

2.10

0.65

1.25

2.80

Opint

9.35

2461

841

660

1310

459

279

On this day, there were 660 call options with this exercise

outstanding in the market.

## 25. 22.6 Combinations of Options

• Puts and calls can serve as the building blocksfor 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.

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

Value atexpiry

Protective Put strategy has

downside protection and

upside potential

$50

Buy the

stock

Buy a put with an exercise

price of $50

$0

$50

Value of

stock at

expiry

## 27. Protective Put Strategy Profits

Value atexpiry

$40

Buy the stock at $40

Protective Put

strategy has

downside protection

and upside potential

$0

$40 $50

-$40

Buy a put with

exercise price of

$50 for $10

Value of

stock at

expiry

## 28. Covered Call Strategy

Value atexpiry

$40

Buy the stock at $40

Covered call

$10

$0

Value of stock at expiry

$30 $40 $50

-$30

-$40

Sell a call with

exercise price of

$50 for $10

## 29. Long Straddle: Buy a Call and a Put

Value atexpiry

Buy a call with an

exercise price of

$50 for $10

$40

$30

$0

-$10

-$20

$30 $40 $50 $60

Buy a put with an

$70 exercise price of

$50 for $10

A Long Straddle only makes money if the

stock price moves $20 away from $50.

Value of

stock at

expiry

## 30. Short Straddle: Sell a Call and a Put

Value atexpiry

$20

$10

$0

A Short Straddle only loses money if the stock

price moves $20 away from $50.

Sell a put with exercise price of

$50 for $10

Value of stock at

expiry

-$30

-$40

$30 $40 $50 $60 $70

Sell a call with an

exercise price of $50 for $10

## 31. Long Call Spread

Value atexpiry

Buy a call with an

exercise price of

$50 for $10

$5

$0

-$5

-$10

long call spread

$50 $60

Value of

stock at

expiry

$55

Sell a call with exercise

price of $55 for $5

## 32. Put-Call Parity

In market equilibrium, it mast be the case that option pricesrT

are set such that: C0 Xe

P0 S0

Otherwise, riskless portfolios with positive payoffs exist.

Value at

expiry

Buy a call option with

an exercise price of $40

Buy the

Buy the stock at $40 stock at $40

financed with some

debt: FV = $X

P0

Sell a put with an

exercise price of $40

$0

C0

-[$40-P0]

rT

($40 Xe )

-$40

$40-P0

$40

$40 Xe rT

$40 C0

Value of

stock at

expiry

## 33. 22.7 Valuing Options

• The last sectionconcerned 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.

## 34. Option Value Determinants

Call1.

2.

3.

4.

5.

Stock price

Exercise price

Interest rate

Volatility in the stock price

Expiration date

Put

+

–

+

+

+

–

+

–

+

+

The value of a call option C0 must fall within

max (S0 – E, 0) < C0 < S0.

The precise position will depend on these factors.

## 35. Market Value, Time Value, and Intrinsic Value for an American Call

ProfitThe value of a call option C0 must fall

within max (S0 – E, 0) < C0 < S0.

ST

CaT > Max[ST - E, 0]

Market Value

Time value

loss

Out-of-the-money

ST

-E

Intrinsic value

E

In-the-money

ST

## 36. 22.8 An Option‑Pricing Formula

• We will start with abinomial option pricing

formula to build our

intuition.

• Then we will graduate

to the normal

approximation to the

binomial for some realworld option valuation.

## 37. Binomial Option Pricing Model

Suppose a stock is worth $25 today and in one period willeither be worth 15% more or 15% less. S0= $25 today and in

one year S1 is either $28.75 or $21.25. The risk-free rate is 5%.

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

S0

S1

$28.75

$25

$21.25

## 38. Binomial Option Pricing Model

1. A call option on this stock with exercise priceof $25 will have the following payoffs.

2. We can replicate the payoffs of the call

option. With a levered position in the stock.

S0

S1

C1

$28.75

$3.75

$21.25

$0

$25

## 39. Binomial Option Pricing Model

Borrow the present value of $21.25 today andbuy one share.

The net payoff for this levered equity portfolio

in one period is either $7.50 or $0.

TheS levered equity

portfolio

has portfolio

twice the C

S

debt

(

)

=

0

1

1

option’s payoff$28.75

so the

portfolio

is worth$3.75

twice

- $21.25

= $7.50

the call option value.

$25

$21.25- $21.25 =

$0

$0

## 40. Binomial Option Pricing Model

The levered equity portfolio value today istoday’s value of one share less the present value

of a $21.25 debt:

$21.25

$25

S0

( S1 - debt

)=

(1 rf )

portfolio C1

$28.75- $21.25 = $7.50

$3.75

$25

$21.25- $21.25 =

$0

$0

## 41. Binomial Option Pricing Model

We can value the option todayas half of the value of the

1

$21.25

levered equity portfolio: C0 2 $25 (1 r )

f

S0

( S1 - debt

)=

portfolio C1

$28.75- $21.25 = $7.50

$3.75

$25

$21.25- $21.25 =

$0

$0

## 42. The Binomial Option Pricing Model

If the interest rate is 5%, the call is worth:1

$21.25 1

$25 20.24 $2.38

C0 $25

2

(1.05) 2

S0

( S1 - debt

)=

portfolio C1

$28.75- $21.25 = $7.50

$3.75

$25

$21.25- $21.25 =

$0

$0

## 43. The Binomial Option Pricing Model

If the interest rate is 5%, the call is worth:1

$21.25 1

$25 20.24 $2.38

C0 $25

2

(1.05) 2

S0

C0

( S1 - debt

)=

portfolio C1

$28.75- $21.25 = $7.50

$3.75

$25 $2.38

$21.25- $21.25 =

$0

$0

## 44. Binomial Option Pricing Model

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

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.

## 45. The Risk-Neutral Approach to Valuation

The Risk-NeutralValuation

Approach

to

S(U), V(U)

q

S(0), V(0)

1- q

S(D), V(D)

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

replicating portfolio. An equivalent method is

qvaluation

V (U ) (1 q) V ( D)

risk-neutral

V (0)

(1 rf )

## 46. The Risk-Neutral Approach to Valuation

The Risk-NeutralValuation

Approach

to

S(U), V(U)

q

q is the risk-neutral

probability of an

“up” move.

S(0), V(0)

1- q

S(D), V(D)

S(0) is the value

of

the

underlying

S(U) and S(D) are the values of the asset in

asset

the

next today.

period following an up move and a

down move, respectively.

V(U) and V(D) are the values of the asset in the next period

following an up move and a down move, respectively.

## 47. The Risk-Neutral Approach to Valuation

S(U), V(U)q

V (0)

S(0), V(0)

q V (U ) (1 q) V ( D)

(1 rf )

1- q

S(D), V(D)

• The key to finding q is to note that it is

already impounded into an observable

q S (U ) (1 q) S ( D)

S (0) the value of S(0):

security price:

(1 r )

f

A minor bit of algebra yields: q

(1 rf ) S (0) S ( D)

S (U ) S ( D)

## 48. Example of the Risk-Neutral Valuation of a Call:

Suppose a stock is worth $25 today and inone 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?

$28.75 $25 (1.15)

The binomial tree would look like this:

q

$25,C(0)

$28.75,C(D)

$21.25 $25 (1 .15)

1- q

$21.25,C(D)

## 49. Example of the Risk-Neutral Valuation of a Call:

The next step would be to compute the risk(1 r ) S (0) S ( D)

neutral probabilities

q

f

S (U ) S ( D)

q

(1.05) $25 $21.25

$5

2 3

$28.75 $21.25

$7.50

2/3

$28.75,C(D)

$25,C(0)

1/3

$21.25,C(D)

## 50. Example of the Risk-Neutral Valuation of a Call:

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

C (U ) $28.75 $25

2/3

$25,C(0)

$28.75, $3.75

C ( D) max[$25 $28.75,0]

1/3

$21.25, $0

## 51. Example of the Risk-Neutral Valuation of a Call:

Finally, find the value of the call at time 0:q C (U ) (1 q) C ( D)

C (0)

(1 rf )

C ( 0)

2 3 $3.75 (1 3) $0

(1.05)

$

2

.

50

C

(

0

)

$

2

.

38

(

1

.

05

)

2/3

$28.75,$3.75

$25,$2.38

$25,C(0)

1/3

$21.25, $0

## 52. Risk-Neutral Valuation and the Replicating Portfolio

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

2 3 $3.75 (1 3) $0 $2.50

C0

$2.38

(1.05)

1.05

1

$21.25 1

$25 20.24 $2.38

C0 $25

2

(1.05) 2

## 53. The Black-Scholes Model

The Black-Scholes Model isC0 S N(d1 ) Ee rT N(d 2 )

Where

C0 = the value of a European option at time t = 0

r = the risk-free interest rate.

σ2

ln(S / E ) (r )T

2

d1

T

d 2 d1 T

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 two-state world.

## 54. The Black-Scholes Model

Find the value of a six-month call option onMicrosoft with an exercise price of $150.

The current value of a share of Microsoft is

$160.

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

The option maturity is six months (half of a

year).

The volatility of the underlying asset is 30% per

annum.

Before we start, note that the intrinsic value of

## 55. The Black-Scholes Model

Let’s try our hand at using the model. If youhave a calculator handy, follow along.

First calculate d1 and d2

ln(S / E ) (r .5σ 2 )T

d1

T

ln(160 / 150) (.05 .5(0.30) 2 ).5

d1

0.5282

0.30 .5

Then,

d 2 d1 T 0.52815 0.30 .5 0.31602

## 56. The Black-Scholes Model

C0 S N(d1 ) Ee rT N(d 2 )d1 0.5282

N(d1) = N(0.52815) = 0.7013

d 2 0.31602

N(d2) = N(0.31602) = 0.62401

C0 $160 0.7013 150e .05 .5 0.62401

C0 $20.92

## 57. Another Black-Scholes Example

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.

## 58. 22.9 Stocks and Bonds as Options

• Levered Equity is a Put Option.– The underlying asset comprise 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.

## 59. 22.9 Stocks and Bonds as Options

• It all comes down to put-call parity.C0 S P0 X e

Value of a

call on the

firm

Value of

= the firm

Stockholder’s

position in terms

of call options

rT

Value of a

+ put on the –

firm

Stockholder’s

position in terms

of put options

Value of a

risk-free

bond

## 60. 22.9 Stocks and Bonds as Options

22.10 Capital-Structure Policy andOptions

• Recall some of the agency costs of debt: they can

all be seen in terms of options.

• For example, recall the incentive shareholders in

a levered firm have to take large risks.

## 61. 22.10 Capital-Structure Policy and Options

Balance Sheet for a Company inDistress

Assets BVMV Liabilities BVMV

Cash $200 $200 LT bonds $300 ?

Fixed Asset $400 $0 Equity $300 ?

Total $600 $200 Total $600 $200

What happens if the firm is liquidated today?

The bondholders get $200; the shareholders get nothing.

## 62. Balance Sheet for a Company in Distress

Selfish Strategy 1: Take Large Risks(Think of a Call Option)

The Gamble

Probability Payoff

Win Big

10%

$1,000

Lose Big

90%

$0

Cost of investment is $200 (all the firm’s cash)

Required return is 50%

$100

NPV $200

1.50 × 0.10 +

Expected CF from the Gamble = $1000

NPV $133

$0 = $100

## 63. Selfish Strategy 1: Take Large Risks (Think of a Call Option)

Selfish Stockholders AcceptNegative NPV Project with Large

• Expected cash flowRisks

from the Gamble

– To Bondholders = $300 × 0.10 + $0 = $30

– To Stockholders = ($1000 - $300) × 0.10 + $0 =

$70

PV of Bonds Without the Gamble = $200

PV of Stocks Without the Gamble = $0

PV of Bonds With the Gamble = $30 / 1.5 = $20

PV of Stocks With the Gamble = $70 / 1.5 = $47

The stocks are worth more with the high risk project because

the call option that the shareholders of the levered firm hold

is worth more when the volatility is increased.

## 64. Selfish Stockholders Accept Negative NPV Project with Large Risks

22.11 Mergers and Options• This is an area rich with optionality, both in the

structuring of the deals and in their execution.

## 65. 22.11 Mergers and Options

22.12 Investment in Real Projects & Options• Classic NPV calculations typically ignore the

flexibility that real-world firms typically have.

• The next chapter will take up this point.

## 66. 22.12 Investment in Real Projects & Options

22.13 Summary and Conclusions• The most familiar options are puts and calls.

– Put options give the holder the right to sell stock

at a set price for a given amount of time.

– Call options give the holder the right to buy stock

at a set price for a given amount of time.

• Put-Call parity

C0 X e

rT

S P0

## 67. 22.13 Summary and Conclusions

• The value of a stock option depends on six factors:1. Current price of underlying stock.

2. Dividend yield of the underlying stock.

3. Strike price specified in the option contract.

4. Risk-free interest rate over the life of the contract.

5. Time remaining until the option contract expires.

6. Price volatility of the underlying stock.

• Much of corporate financial theory can be presented

in terms of options.

1. Common stock in a levered firm can be viewed as

a call option on the assets of the firm.

2. Real projects often have hidden options that