Chapter Outline

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Chapter Outline

• 22.1 Options
• 22.2 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

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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.

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22.1 Options Contracts Preliminaries

• 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.

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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.

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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.

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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


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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.

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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 ST - E.
• If the call is out-of-the-money, it is worthless.
• CaT CeT MaxST - 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

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Call Option Payoffs
60
40
Buy a call
20
0
Option payoffs ()
100
90
80
70
60
0
10
20
30
40
50
Stock price ()
-20
-40
-60
Exercise price 50
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Call Option Payoffs
Write a call
Exercise price 50
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Call Option Profits
Buy a call
Write a call
Exercise price 50 option premium 10
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22.3 Put Options

• Put options give 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.

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Basic Put Option Pricing Relationships 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.
• PaT PeT MaxE - ST, 0

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Put Option Payoffs
60
40
Buy a put
20
0
Option payoffs ()
100
90
80
70
60
0
10
20
30
40
50
Stock price ()
-20
-40
-60
Exercise price 50
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Put Option Payoffs
60
40
20
0
Option payoffs ()
100
90
80
70
60
0
10
20
30
40
50
Stock price ()
-20
-40
write a put
-60
Exercise price 50
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Put Option Profits
60
Option profits ()
40
20
Write a put
10
0
100
90
80
70
60
0
10
20
30
40
50
-10
Buy a put
Stock price ()
-20
-40
-60
Exercise price 50 option premium 10
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22.4 Selling Options

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

• The purchaser of an option has an option.

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22.5 Stock Option Quotations
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22.5 Stock Option Quotations
A recent price for the stock is 9.35
This option has a strike price of 8
June is the expiration month
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22.5 Stock Option Quotations
This makes a call option with this exercise price
in-the-money by 1.35 9.35 8.
Puts with this exercise price are
out-of-the-money.
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22.5 Stock Option Quotations
On this day, 15 call options with this exercise
price were traded.
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22.5 Stock Option Quotations
The holder of this CALL option can sell it for
1.95.
Since the option is on 100 shares of stock,
selling this option would yield 195.
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22.5 Stock Option Quotations
Buying this CALL option costs 2.10.
Since the option is on 100 shares of stock,
buying this option would cost 210.
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22.5 Stock Option Quotations
On this day, there were 660 call options with
this exercise outstanding in the market.
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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 clients needs.

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Protective Put Strategy Buy a Put and Buy the
Underlying Stock Payoffs at Expiry
Value at expiry
Protective Put strategy has downside protection
and upside potential
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Buy the stock
Buy a put with an exercise price of 50
0
Value of stock at expiry
50
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Protective Put Strategy Profits
Value at expiry
Buy the stock at 40
40
Protective Put strategy has downside protection
and upside potential
0
Buy a put with exercise price of 50 for 10
40
50
-40
Value of stock at expiry
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Covered Call Strategy
Value at expiry
Buy the stock at 40
40
Covered call
10
0
Value of stock at expiry
40
30
50
Sell a call with exercise price of 50 for 10
-30
-40
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Long Straddle Buy a Call and a Put
Value at expiry
Buy a call with an exercise price of 50 for 10
40
30
0
-10
Buy a put with an exercise price of 50 for 10
-20
40
50
60
30
70
Value of stock at expiry
A Long Straddle only makes money if the stock
price moves 20 away from 50.
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Short Straddle Sell a Call and a Put
Value at expiry
A Short Straddle only loses money if the stock
price moves 20 away from 50.
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Sell a put with exercise price of 50 for 10
10
0
Value of stock at expiry
40
50
60
30
70
-30
Sell a call with an exercise price of 50 for 10
-40
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Long Call Spread
Value at expiry
Buy a call with an exercise price of 50 for 10
5
long call spread
0
-5
-10
Value of stock at expiry
50
60
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Sell a call with exercise price of 55 for 5
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Put-Call Parity
In market equilibrium, it mast be the case that
option prices are set such that
Otherwise, riskless portfolios with positive
payoffs exist.
Buy the stock at 40
Value at expiry
Buy the stock at 40 financed with some debt FV
X
Buy a call option with an exercise price of 40
Sell a put with an exercise price of 40
0
Value of stock at expiry
40
-40-P0
40-P0
-40
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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.

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Option Value Determinants

• 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) lt C0 lt S0.
• The precise position will depend on these factors.

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Market Value, Time Value, and Intrinsic Value for
an American Call
The value of a call option C0 must fall within
max (S0 E, 0) lt C0 lt S0.
Profit
ST
ST - E

• CaT gt MaxST - E, 0

Market Value
Time value
Intrinsic value
ST
E
loss
Out-of-the-money
In-the-money
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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.

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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 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
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Binomial Option Pricing Model

• 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
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Binomial Option Pricing Model

• Borrow the present value of 21.25 today and buy
  one 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
  options payoff so the portfolio is worth twice
  the call option value.

S1
S0
debt
portfolio
C1
( - )
- 21.25
7.50
28.75

3.75
25
21.25
- 21.25
0

0
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Binomial Option Pricing Model

• The levered equity portfolio value today is
  todays value of one share less the present value
  of a 21.25 debt

S1
S0
debt
portfolio
C1
( - )
- 21.25
7.50
28.75

3.75
25
21.25
- 21.25
0

0
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Binomial Option Pricing Model

• We can value the option today as half of the
  value of the levered equity portfolio

S1
S0
debt
portfolio
C1
( - )
- 21.25
7.50
28.75

3.75
25
21.25
- 21.25
0

0
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The Binomial Option Pricing Model

• If the interest rate is 5, the call is worth

S1
S0
debt
portfolio
C1
( - )
- 21.25
7.50
28.75

3.75
25
21.25
- 21.25
0

0
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The Binomial Option Pricing Model

• If the interest rate is 5, the call is worth

S1
S0
debt
portfolio
C1
( - )
- 21.25
7.50
28.75

3.75
25
21.25
- 21.25
0

0
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Binomial Option Pricing Model
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.
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The Risk-Neutral Approach to Valuation
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
  risk-neutral valuation

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The Risk-Neutral Approach to Valuation
S(U), V(U)
q
q is the risk-neutral probability of an up move.
S(0), V(0)
1- q

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

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.
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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
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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

28.75,C(D)
q
25,C(0)
1- q
21.25,C(D)
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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)
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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
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Example of the Risk-Neutral Valuation of a Call

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

25,2.38
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Risk-Neutral Valuation and the Replicating
Portfolio

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

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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 two-state world.
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The Black-Scholes Model

• Find the value of a six-month call option on
  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 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
  the option is 10our answer must be at least
  that amount.

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The Black-Scholes Model

• Lets try our hand at using the model. If you
  have a calculator handy, follow along.

First calculate d1 and d2
Then,
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The Black-Scholes Model
N(d1) N(0.52815) 0.7013 N(d2) N(0.31602)
0.62401
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Another Black-Scholes Example
Assume S 50, X 45, T 6 months, r 10,
and ? 28, calculate the value of a call and a
put.
From a standard normal probability table, look
up N(d1) 0.812 and N(d2) 0.754 (or use
Excels normsdist function)
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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.

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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.

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22.9 Stocks and Bonds as Options

• It all comes down to put-call parity.

Stockholders position in terms of call options
Stockholders position in terms of put options
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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.

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Balance Sheet for a Company 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?

The bondholders get 200 the shareholders get
nothing.
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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 firms cash)
• Required return is 50
• Expected CF from the Gamble 1000 0.10 0
  100

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Selfish Stockholders Accept Negative NPV Project
with Large Risks

• Expected cash flow 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.
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22.11 Mergers and Options

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

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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.

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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

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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.
• Common stock in a levered firm can be viewed as a
  call option on the assets of the firm.
• Real projects often have hidden options that
  enhance value.