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CHAPTER. 22. Options and Corporate Finance: Basic Concepts. 22.1 Options 22.2 Call Options 22.3 Put Options 22.4 Selling Options 22.5 Reading The Wall Street Journal 22.6 Combinations of Options 22.7 Valuing Options 22.8 An Option‑Pricing Formula 22.9 Stocks and Bonds as Options

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CHAPTER

22

Options and Corporate Finance: Basic Concepts


Chapter outline

22.1 Options

22.2 Call Options

22.3 Put Options

22.4 Selling Options

22.5 Reading The Wall Street Journal

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

Chapter Outline


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.

22.1 Options


22 1 options contracts preliminaries
22.1 Options Contracts: Preliminaries that are traded on organized exchanges.

  • 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 preliminaries1
22.1 Options Contracts: Preliminaries that are traded on organized exchanges.

  • 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 that are traded on organized exchanges.

  • 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.


Options contracts preliminaries1

Option Premium that are traded on organized exchanges.

Intrinsic Value

Speculative Value

+

=

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.


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.

22.2 Call Options


Basic call option pricing relationships at expiry
Basic Call Option Pricing Relationships obligation, to 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


Call option payoffs
Call Option Payoffs obligation, to

Buy a call

60

40

Option payoffs ($)

20

80

120

20

40

60

100

50

Stock price ($)

–20

Exercise price = $50

–40


Call option payoffs1
Call Option Payoffs obligation, to

60

40

Option payoffs ($)

20

80

120

20

40

60

100

50

Stock price ($)

–20

Exercise price = $50

Sell a call

–40


Call option profits

60 obligation, to

40

Option payoffs ($)

20

80

120

20

40

60

100

Stock price ($)

–20

–40

Call Option Profits

Buy a call

10

50

–10

Exercise price = $50; option premium = $10

Sell a call


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.

22.3 Put Options


Basic put option pricing relationships at expiry
Basic Put Option Pricing Relationships obligation, to 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]


Put option payoffs
Put Option Payoffs obligation, to

60

50

40

Option payoffs ($)

20

Buy a put

0

80

0

20

40

60

100

50

Stock price ($)

–20

Exercise price = $50

–40


Put option payoffs1
Put Option Payoffs obligation, to

40

Option payoffs ($)

20

Sell a put

0

80

0

20

40

60

100

50

Stock price ($)

–20

Exercise price = $50

–40

–50


Put option profits
Put Option Profits obligation, to

60

40

Option payoffs ($)

20

Sell a put

10

Stock price ($)

80

50

20

40

60

100

–10

Buy a put

–20

Exercise price = $50; option premium = $10

–40


22 4 selling options
22.4 Selling Options obligation, to

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

The purchaser of an option has an option.

Buy a call

40

Option payoffs ($)

Buy a put

Sell a call

Sell a put

10

Stock price ($)

50

40

60

100

Buy a call

–10

Buy a put

Sell a put

Exercise price = $50; option premium = $10

Sell a call

–40


22 5 reading the wall street journal
22.5 Reading obligation, to The WallStreet Journal


22 5 reading the wall street journal1
22.5 Reading obligation, to The Wall Street Journal

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 journal2
22.5 Reading obligation, to The Wall Street Journal

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 5 reading the wall street journal3
22.5 Reading obligation, to The Wall Street Journal

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


22 5 reading the wall street journal4
22.5 Reading obligation, to The Wall Street Journal

The CALL option with a strike priceof $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 journal5
22.5 Reading obligation, to The Wall Street Journal

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


22 5 reading the wall street journal6
22.5 Reading obligation, to The Wall Street Journal

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.

22.6 Combinations of Options


Protective put strategy buy a put and buy the underlying stock payoffs at expiry
Protective Put Strategy: Buy a Put and Buy the Underlying Stock: Payoffs at Expiry

Protective Put payoffs

Value at expiry

$50

Buy the stock

Buy a put with an exercise price of $50

$0

Value of stock at expiry

$50


Protective put strategy profits
Protective Put Strategy Profits Stock: Payoffs at Expiry

Value at expiry

Buy the stock at $40

$40

Protective Put strategy has downside protection and upside potential

$0

-$10

$40

$50

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

Value of stock at expiry

-$40


Covered call strategy

$10 Stock: Payoffs at Expiry

-$30

Covered Call Strategy

Value at expiry

Buy the stock at $40

Covered Call strategy

$0

Value of stock at expiry

$40

$50

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

-$40


Long straddle buy a call and a put

Stock: Payoffs at Expiry20

Long Straddle: Buy a Call and a Put

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

40

Option payoffs ($)

30

Stock price ($)

40

60

30

70

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

$50

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


Long straddle buy a call and a put1

20 Stock: Payoffs at Expiry

Long Straddle: Buy a Call and a Put

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

Option payoffs ($)

Sell a put with exercise price of

$50 for $10

Stock price ($)

30

70

40

60

$50

–30

Sell a call with an

exercise price of $50 for $10

–40


Put call parity p 0 s 0 c 0 e 1 r t

E Stock: Payoffs at Expiry

Portfolio value today = c0 +

(1+ r)T

bond

Put-Call Parity: p0 + S0 = c0 + E/(1+ r)T

Portfolio payoff

Call

Option payoffs ($)

25

Stock price ($)

25

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.


Put call parity p 0 s 0 c 0 e 1 r t1
Put-Call Parity: Stock: Payoffs at Expiryp0 + S0 = c0 + E/(1+ r)T

Portfolio payoff

Portfolio value today = p0 + S0

Option payoffs ($)

25

Stock price ($)

25

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


Put call parity p 0 s 0 c 0 e 1 r t2

Portfolio value today Stock: Payoffs at Expiry

Portfolio value today = p0 + S0

E

= c0 +

Option payoffs ($)

Option payoffs ($)

(1+ r)T

25

25

Stock price ($)

Stock price ($)

25

25

Put-Call Parity: p0 + S0 = c0 + E/(1+ r)T

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

Put-Call Parity: c0 + E/(1+r)T = p0 + S0


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.

22.7 Valuing Options


Option value determinants
Option Value Determinants option at expiry.

Call Put

  • Stock price + –

  • Exercise price – +

  • Interest rate + –

  • Volatility in the stock price + +

  • 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.


Market value time value and intrinsic value for an american call

Market Value option at expiry.

Market Value, Time Value and Intrinsic Valuefor an American Call

Profit

ST

Call

Option payoffs ($)

25

Time value

Intrinsic value

ST

E

Out-of-the-money

In-the-money

loss

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


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.

22.8 An Option‑Pricing Formula


Binomial option pricing model

S build our intuition. 1

$28.75 = $25×(1.15)

$21.25 = $25×(1 –.15)

Binomial Option Pricing Model

Suppose 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?

S0

$25


Binomial option pricing model1
Binomial Option Pricing Model build our intuition.

  • A call option on this stock with exercise price of $25 will have the following payoffs.

  • We can replicate the payoffs of the call option. With a levered position in the stock.

S0

S1

C1

$28.75

$3.75

$25

$21.25

$0


Binomial option pricing model2
Binomial Option Pricing Model build our intuition.

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.

S0

S1

debt

portfolio

C1

( – ) =

– $21.25

$7.50

$28.75

=

$3.75

$25

– $21.25

$21.25

$0

=

$0


Binomial option pricing model3
Binomial Option Pricing Model build our intuition.

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

S0

S1

debt

portfolio

C1

( – ) =

– $21.25

$7.50

$28.75

=

$3.75

$25

– $21.25

$21.25

$0

=

$0


Binomial option pricing model4
Binomial Option Pricing Model build our intuition.

We can value the call option todayas half of the value of thelevered equity portfolio:

S0

S1

debt

portfolio

C1

( – ) =

– $21.25

$7.50

$28.75

=

$3.75

$25

– $21.25

$21.25

$0

=

$0


The binomial option pricing model

C build our intuition. 0

$2.38

The Binomial Option Pricing Model

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

S0

S1

debt

portfolio

C1

( – ) =

– $21.25

$7.50

$28.75

=

$3.75

$25

– $21.25

$21.25

$0

=

$0


Binomial option pricing model5
Binomial Option Pricing Model build our intuition.

The most important lesson (so far) from the binomial option 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.


Delta and the hedge ratio

Swing of call build our intuition.

D =

Swing of stock

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:

The delta of a put option is negative.


Delta
Delta build our intuition.

  • Determining the Amount of Borrowing:

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

    $2.38 = $25 × ½ – Amount borrowed

    Amount borrowed = $10.12


The risk neutral approach to valuation

´ build our intuition.

+

-

´

q

V

(

U

)

(

1

q

)

V

(

D

)

=

V

(

0

)

+

(

1

r

)

f

The Risk-Neutral Approach to Valuation

S(U), V(U)

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

q

S(0), V(0)

1- q

S(D), V(D)


The risk neutral approach to valuation1
The Risk-Neutral Approach to Valuation build our intuition.

S(U), V(U)

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

q

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

S(0), V(0)

1- q

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.

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


The risk neutral approach to valuation2

S build our intuition. (U), V(U)

q

´

+

-

´

q

V

(

U

)

(

1

q

)

V

(

D

)

=

V

(

0

)

S(0), V(0)

+

(

1

r

)

f

1- q

S(D), V(D)

´

+

-

´

q

S

(

U

)

(

1

q

)

S

(

D

)

=

S

(

0

)

+

(

1

r

)

f

The Risk-Neutral Approach to Valuation

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

A minor bit of algebra yields:


Example of the risk neutral valuation of a call

= build our intuition.

´

$

28

.

75

$

25

(

1

.

15

)

$28.75,C(D)

q

$25,C(0)

=

´

-

$

21

.

25

$

25

(

1

.

15

)

1- q

$21.25,C(D)

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:


Example of the risk neutral valuation of a call1

+ build our intuition.

´

-

(

1

r

)

S

(

0

)

S

(

D

)

f

=

q

-

S

(

U

)

S

(

D

)

´

-

(

1

.

05

)

$

25

$

21

.

25

$

5

=

=

=

q

2

3

-

$

28

.

75

$

21

.

25

$

7

.

50

Example of the Risk-Neutral Valuation of a Call:

The next step would be to compute the risk neutral probabilities

$28.75,C(D)

2/3

$25,C(0)

1/3

$21.25,C(D)


Example of the risk neutral valuation of a call2

= build our intuition.

-

C

(

U

)

$

28

.

75

$

25

=

-

C

(

D

)

max[$

25

$

28

.

75

,

0

]

Example of the Risk-Neutral Valuation of a Call:

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

$28.75, $3.75

2/3

$25,C(0)

1/3

$21.25, $0


Example of the risk neutral valuation of a call3

´ build our intuition.

+

-

´

q

C

(

U

)

(

1

q

)

C

(

D

)

=

C

(

0

)

+

(

1

r

)

f

´

+

´

2

3

$

3

.

75

(

1

3

)

$

0

=

C

(

0

)

(

1

.

05

)

$28.75,$3.75

$

2

.

50

2/3

=

=

C

(

0

)

$

2

.

38

(

1

.

05

)

$25,C(0)

1/3

$21.25, $0

Example of the Risk-Neutral Valuation of a Call:

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

$25,$2.38


Risk neutral valuation and the replicating portfolio
Risk-Neutral Valuation build our intuition. and the Replicating Portfolio

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


The black scholes model

- build our intuition.

=

´

-

´

rT

C

S

N(

d

)

Ee

N(

d

)

0

1

2

2

σ

+

+

ln(

S

/

E

)

(

r

)

T

2

=

d

1

s

T

=

-

s

d

d

T

2

1

The Black-Scholes Model

The Black-Scholes Model is

Where

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

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.


The black scholes model1
The Black-Scholes Model build our intuition.

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 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 intrinsicvalue of the option is $10—our answer must be at least that amount.


The black scholes model2

+ build our intuition.

+

2

ln(

S

/

E

)

(

r

.

5

σ

)

T

=

d

1

s

T

+

+

2

ln(

160

/

150

)

(.

05

.

5

(

0

.

30

)

).

5

=

=

d

0

.

5282

1

0

.

30

.

5

=

-

s

=

-

=

d

d

T

0

.

52815

0

.

30

.

5

0

.

31602

2

1

The Black-Scholes Model

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

First calculate d1 and d2

Then,


The black scholes model3

- build our intuition.

=

´

-

´

rT

C

S

N(

d

)

Ee

N(

d

)

0

1

2

=

d

0

.

5282

1

=

d

0

.

31602

2

-

´

=

´

-

´

.

05

.

5

C

$

160

0

.

7013

150

e

0

.

62401

0

=

C

$

20

.

92

0

The Black-Scholes Model

N(d1) = N(0.52815) = 0.7013

N(d2) = N(0.31602) = 0.62401


22 9 stocks and bonds as options

Levered Equity is a Call Option. build our intuition.

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 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.

22.9 Stocks and Bonds as Options


22 9 stocks and bonds as options1

Levered Equity is a Put Option. build our intuition.

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.

22.9 Stocks and Bonds as Options


22 9 stocks and bonds as options2

It all comes down to put-call parity. build our intuition.

E

c0 = S0 + p0 –

(1+ r)T

Value of a risk-free bond

Value of a call on the firm

Value of a put on the firm

Value of the firm

=

+

22.9 Stocks and Bonds as Options

Stockholder’s position in terms of put options

Stockholder’s position in terms of call options


22 10 capital structure policy and options

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.

22.10 Capital-Structure Policyand Options


Balance sheet for a company in distress
Balance Sheet for a Company seen in terms of options.in Distress

Assets BV MV Liabilities BV MV

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?

$200

$0

The bondholders get $200; the shareholders get nothing.


Selfish strategy 1 take large risks

$100 seen in terms of options.

NPV = –$200 +

(1.10)

Selfish Strategy 1: Take Large Risks

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%

Expected CF from the Gamble = $1000 × 0.10 + $0 = $100

NPV = –$133


Selfish stockholders accept negative npv project with large risks

PV of Bonds With the Gamble: seen in terms of options.

PV of Stocks With the Gamble:

$30

$70

$20 =

$47 =

(1.50)

(1.50)

Selfish Stockholders Accept Negative NPV Project with Large Risks

  • Expected CF 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

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 of the firm is increased.


22 11 mergers and options
22.11 Mergers and Options seen in terms of options.

  • This is an area rich with optionality, both in the structuring of the deals and in their execution.

  • In the first half of 2000, General Mills was attempting to acquire the Pillsbury division of Diageo PLC.

  • The structure of the deal was Diageo’s stockholders received 141 million shares of General Mills stock (then valued at $42.55) plus contingent value rights of $4.55 per share.


22 11 mergers and options1

$38 seen in terms of options.

$42.55

22.11 Mergers and Options

Thecontingent value rights paidthe difference between $42.55 andGeneral Mills’ stock price in oneyear up to a maximum of $4.55.

Cash payment to newly issued shares

$4.55

$0

Value of General Mills in 1 year


22 11 mergers and options2
22.11 Mergers and Options seen in terms of options.

  • The contingent value plan can be viewed in terms of puts:

    • Each newly issued share of General Mills given to Diageo’s shareholders came with a put option with an exercise price of $42.55.

    • But the shareholders of Diageo sold a put with an exercise price of $38


22 11 mergers and options3

Own a put seen in terms of options.

Strike $42.55

$42.55

$42.55

– $38.00

$4.55

$42.55

Sell a put

Strike $38

–$38

22.11 Mergers and Options

Cash payment to newly issued shares

$0

Value of General Mills in 1 year

$38


22 11 mergers and options4
22.11 Mergers and Options seen in terms of options.

Value of General Mills in 1 year

Value of a share

Value of a share plus cash payment

$42.55

$4.55

$0

Value of General Mills in 1 year

$38

$42.55


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.

22.12 Investment in Real Projects & Options


22 13 summary and conclusions

E that real-world firms typically have.

c0–

= S0 + p0

(1+ r)T

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


22 13 summary and conclusions1
22.13 Summary and Conclusions that real-world firms typically have.

  • 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.

    • Common stock in a levered firm can be viewed as a call option on the assets of the firm.

    • Real projects often have hidden option that enhance value.


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