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Sixth Edition
23
Chapter Twenty Three
Options and Corporate
Finance: Basic Concepts
Prepared by
Gady Jacoby
University of Manitoba
McGraw-Hill Ryerson 2005 McGrawHill Ryerson Limited
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Chapter Outline
23.1 Options23.2 Call Options
23.3 Put Options
23.4 Selling Options
23.5 Stock Option Quotations23.6 Combinations of Options
23.7 Valuing Options
23.8 An Option-Pricing Formula
23.9 Stocks and Bonds as Options
23.10 Capital-Structure Policy and Options
23.11 Mergers and Options
23.12 Investment in Real Projects and Options
23.13 Summary and Conclusions
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23.1 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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23.1 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, tobuy 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 theobligation, 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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23.1 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 theunderlying 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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23.2 Call Options
Call options gives the holder the right, but not the
obligation, tobuy 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=Max[ST- E, 0]
Where
STis the value of the stock at expiry (time T)
Eis the exercise price.
CaTis the value of an American call at expiry
CeTis the value of a European call at expiry
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Call Option Payoffs
20
12020 40 60 80 100
40
20
40
60
Stock price ($)
Option
payoffs($)
Exercise price = $50
50
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Call Option Payoffs
20
12020 40 60 80 100
40
20
40
60
Stock price ($)
Option
payoffs($)
Exercise price = $50
50
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Call Option Profits
Exercise price = $50;
option premium = $10Sell a call
Buy a call
20
12020 40 60 80 100
40
20
40
60
Stock price ($)
Option
payoffs($)
50
10
10
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23.3 Put Options
Put options give the holder the right, but not the
obligation, to sell a given quantity of an asset on
or before some time in the future, at prices
agreed upon today. When exercising a put, you put the asset to
someone.
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Basic Put Option Pricing Relationships at Expiry
At expiry, an American put option is worth the
same as a European option with the same
characteristics.
If the put is in-the-money, it is worthE - ST. If the put is out-of-the-money, it is worthless.
PaT=PeT=Max[E - ST, 0]
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Put Option Payoffs
20
0 20 40 60 80 100
40
20
0
40
60
Stock price ($)
Option
payoffs($)
Buy a put
Exercise price = $50
50
50
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Put Option Payoffs
20
0 20 40 60 80 100
40
20
0
40
50
Stock price ($)
Optionpayoffs($)
Sell a put
Exercise price = $50
50
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Put Option Profits
20
20 40 60 80 100
40
20
40
60
Stock price ($)
Option
payoffs($)
Buy a put
Exercise price = $50; option premium = $10
10
10Sell a put
50
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23.4 Selling Options
Exercise price = $50;
option premium = $10Sell a call
Buy a call
50 6040 100
40
40
Stock price ($)
Option
profits($)
Buy a put
Sell a put
The seller (or writer) of an option has an obligation.
The purchaser of an option has an option (right).
10
10
Buy a call
Sell a call
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23.5 Stock Option Quotations
Stk Exp P/C Vol Bid Ask Opint
Nortel Networks (NT) 9.35
9 Mar C 446 0.50 0.55 2461
9 Mar P 155 0.20 0.30 841
8 June C 15 1.95 2.10 660
8 June P 35 0.55 0.65 1310
11 Sept C 11 1.10 1.25 45911 Sept P 5 2.65 2.80 279
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Stk Exp P/C Vol Bid Ask Opint
Nortel Networks (NT) 9.35
9 Mar C 446 0.50 0.55 2461
9 Mar P 155 0.20 0.30 841
8 June C 15 1.95 2.10 660
8 June P 35 0.55 0.65 1310
11 Sept C 11 1.10 1.25 45911 Sept P 5 2.65 2.80 279
23.5 Stock Option Quotations
This option has a strike price of $8;
A recent price for the stock is $9.35
June is the expiration month
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23.5 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.
Stk Exp P/C Vol Bid Ask Opint
Nortel Networks (NT) 9.35
9 Mar C 446 0.50 0.55 2461
9 Mar P 155 0.20 0.30 841
8 June C 15 1.95 2.10 660
8 June P 35 0.55 0.65 1310
11 Sept C 11 1.10 1.25 45911 Sept P 5 2.65 2.80 279
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Stk Exp P/C Vol Bid Ask Opint
Nortel Networks (NT) 9.35
9 Mar C 446 0.50 0.55 2461
9 Mar P 155 0.20 0.30 841
8 June C 15 1.95 2.10 660
8 June P 35 0.55 0.65 1310
11 Sept C 11 1.10 1.25 45911 Sept P 5 2.65 2.80 279
23.5 Stock Option Quotations
On this day, 15 call options with this exercise price were traded.
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Stk Exp P/C Vol Bid Ask Opint
Nortel Networks (NT) 9.35
9 Mar C 446 0.50 0.55 2461
9 Mar P 155 0.20 0.30 841
8 June C 15 1.95 2.10 660
8 June P 35 0.55 0.65 1310
11 Sept C 11 1.10 1.25 45911 Sept P 5 2.65 2.80 279
23.5 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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Stk Exp P/C Vol Bid Ask Opint
Nortel Networks (NT) 9.35
9 Mar C 446 0.50 0.55 2461
9 Mar P 155 0.20 0.30 841
8 June C 15 1.95 2.10 660
8 June P 35 0.55 0.65 1310
11 Sept C 11 1.10 1.25 45911 Sept P 5 2.65 2.80 279
23.5 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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Stk Exp P/C Vol Bid Ask Opint
Nortel Networks (NT) 9.35
9 Mar C 446 0.50 0.55 2461
9 Mar P 155 0.20 0.30 841
8 June C 15 1.95 2.10 660
8 June P 35 0.55 0.65 1310
11 Sept C 11 1.10 1.25 45911 Sept P 5 2.65 2.80 279
23.5 Stock Option Quotations
On this day, there were 660 call options with this exercise
outstanding in the market.
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23.6 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-returnprofile to meet your clients needs.
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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
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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 protectionand upside potential
$40
$0
-$40
$50
Value at
expiry
Value of
stock at
expiry
-$10
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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
Value of stock at expiry
-$30
$10
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Long Straddle: Buy a Call and a Put
30 40 60 70
30
40
Stock price ($)
Buy a put with exerciseprice 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.
$50
20
Value atexpiry
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Long Straddle: Buy a Call and a Put
30
30 40 60 70
40
Stock price ($)
$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
Value atexpiry
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Long Call Spread
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Sell a call with exercise
price of $55 for $5
$55
long call spread$5
$0
$50
Buy a call with an
exercise price of
$50 for $10
-$10-$5
$60
Value of
stock at
expiry
Value at
expiry
Long Call Spread
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Call options and Slope
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Option Combo
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Bond
Put-Call Parity:p0 + S0 = c0 +E/(1+ r)T
25
25
Stock price ($)
Optionpayoffs($)
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 $25.
Call
Portfolio payoffPortfolio value today = c0 +
(1+ r)TE
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Put-Call Parity:p0 + S0 = c0 +E/(1+ r)T
25
25
Stock price ($)
Optionpay
offs($)
Consider the payoffs from holding a portfolio
consisting of a share of stock and a put with a $25
strike.
Portfolio value today =p0 + S0
Portfolio payoff
Put
Stock
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Put-Call Parity:p0 + S0 = c0 +E/(1+ r)T
Since these portfolios have identical payoffs, they must have
the same value today: hence
Put-Call Parity: c0 +E/(1+r)T=p0 + S0
25
25
Stock price ($)
Optionpayoffs($)
25
25
Stock price ($)
Optionpayoffs($) Portfolio value today
=p0 + S0
Portfolio value today
(1+ r)T
E= c0 +
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23.7 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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American Option Value Determinants
Call Put1. 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.
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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) < C0 < S0.
25
Call
ST
loss
E
$
ST
Time value
Intrinsic value
Market Value
In-the-moneyOut-of-the-money
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23.8 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 yearS1 is 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
$28.75
S1S0
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Binomial Option Pricing Model
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.
$25
$21.25
$28.75
S1S0 C1
$3.75
$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.
$25
$21.25
$28.75
S1S0 debt- $21.25
portfolio$7.50
$0
( - ) ==
=
C1$3.75
$0- $21.25
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Binomial Option Pricing Model
The levered equity portfolio value today istodays value of one share less the present valueof a $21.25 debt:
)1(
25.21$25$
f
r+-
$25
$21.25
$28.75
S1S0 debt- $21.25
portfolio$7.50
$0
( - ) ==
=
C1$3.75
$0- $21.25
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Binomial Option Pricing Model
We can value the option today as
half of the value of the levered
equity portfolio:
+-=
)1(
25.21$25$
2
10
frC
$25
$21.25
$28.75
S1S0 debt- $21.25
portfolio$7.50
$0
( - ) ==
=
C1$3.75
$0- $21.25
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If the interest rate is 5%, the call is worth:
The Binomial Option Pricing Model
( ) 38.2$24.2025$2
1
)05.1(
25.21$25$
2
10 =-=
-=C
$25
$21.25
$28.75
S1S0 debt- $21.25
portfolio$7.50
$0
( - ) ==
=
C1$3.75
$0- $21.25
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If the interest rate is 5%, the call is worth:
The Binomial Option Pricing Model
( ) 38.2$24.2025$2
1
)05.1(
25.21$25$
2
10 =-=
-=C
$25
$21.25
$28.75
S1S0 debt- $21.25
portfolio$7.50
$0
( - ) ==
=
C1$3.75
$0- $21.25
$2.38
C0
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Binomial Option Pricing Model
the replicating portfolio intuition.the replicating portfolio intuition.
Many derivative securities can be valued by
valuing portfolios of primitive securitieswhen those portfolios have the same
payoffs as the derivative securities.
The most important lesson (so far) from the binomialoption pricing model is:
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Delta and the Hedge Ratio
This practice of the construction of a riskless hedge
is called delta hedging.
The delta of a call option is positive.
Recall from the example:
The delta of a put option is negative.
2
1
5.7$
75.3$
25.21$75.28$
075.3$==
-
-=D =
Swing of call
Swing of stock
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Delta
Determining the Amount of Borrowing:
( ) 38.2$24.20$25$2
1
)05.1(
25.21$25$
2
10 =-=
-=C
Value of a call = Stock price DeltaAmount borrowed
$2.38 = $25 Amount borrowed
Amount borrowed = $10.12
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The Risk-Neutral Approach to Valuation
We could value V(0) as the value of the replicating portfolio.
An equivalent method is risk-neutral valuation
S(0), V(0)
S(U), V(U)
q
S(D), V(D)
1- q
)1(
)()1()()0(
fr
DVqUVqV
+
-+=
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The Risk-Neutral Approach to Valuation
S(0) is the value of theunderlying 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 inthe next period following an up move and adown move, respectively.
q
1- q
V(U) and V(D) are the values of the asset in the next periodfollowing an up move and a down move, respectively.
q is the risk-neutral
probability of an
up move.
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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 ofS(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
-
-+=
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Example of the Risk-Neutral Valuation of a Call:
$21.25,C(D)
q
1- q
Suppose a stock is worth $25 today and in one period willeither be worth 15% more or 15% less. The risk-free rate is5%. What is the value of an at-the-money call option?
The binomial tree would look like this:
$25,C(0)
$28.75,C(D)
)15.1(25$75.28$ =
)15.1(25$25.21$ -=
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Example of the Risk-Neutral Valuation of a Call:
$21.25,C(D)
2/3
1/3
The next step would be to compute the risk neutralprobabilities
$25,C(0)
$28.75,C(D)
)()(
)()0()1(
DSUS
DSSrq
f
-
-+=
3250.7$5$
25.21$75.28$25.21$25$)05.1( ==-
-=q
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Example of the Risk-Neutral Valuation of a Call:
$21.25, $0
2/3
1/3
After that, find the value of the call in the up state and downstate.
$25,C(0)
$28.75, $3.75
25$75.28$)( -=UC
]0,75.28$25max[$)( -=DC
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Example of the Risk-Neutral Valuation of a Call:
Finally, 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
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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$05.150.2$
)05.1(0$)31(75.3$32
0 ==+=C
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The Black-Scholes Model
The Black-Scholes Model is)N()N( 210 dEedSC
rT -= -
Where
C0 = the value of a European option at time t= 0
r= the risk-free interest rate.
T
T
rESd
s
)2
()/ln(2
1
++=
Tdd s-= 12
N(d) = Probability that a
standardized, normally
distributed, random
variable will be less thanor 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 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 the
option is $10our answer must be at least that
amount.
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The Black-Scholes Model
Lets try our hand at using the model. If you have acalculator handy, follow along.
Then,
T
TrESd s
)5.()/ln( 2
1
++=
First calculate d1 and d2
31602.05.30.052815.012 =-=-= Tdd s
5282.05.30.0
5).)30.0(5.05(.)150/160ln( 2
1 =++
=d
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The Black-Scholes Model
N(d1) = N(0.52815) = 0.7013
N(d2) = N(0.31602) = 0.62401
5282.01 =d
31602.02 =d
)N()N( 210 dEedSCrT -= -
92.20$
62401.01507013.0160$
0
5.05.
0
=
-= -
C
eC
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Assume S= $50,X= $45, T= 6 months, r= 10%,and s = 28%, calculate the value of a call and a put.
125.1$45$50$32.8$ )50.0(10.0 =+-= -eP
32.8$)754.0(45)812.0(50 )50.0(10.0)5.0(0 =-= -- eeC
( )884.0
50.028.0
50.02
28.0010.0
4550ln
2
1 =
+-+
=d
686.050.028.0884.02 =-=d
From a standard normal probability table, look upN(d1) =
0.812 andN(d2) = 0.754 (or use Excels normsdist function)
Another Black-Scholes Example
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23.9 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 firmare greater in value than the debt, the shareholdershave 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 havean out-of-the-money call, they will not pay the
bondholders (i.e., the shareholders will declarebankruptcy), and let the call expire.
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23.9 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 havean 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 theoption (i.e.,NOT declare bankruptcy) and let the
put expire.
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23.9 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= +
Stockholders
position in terms
of call options
Stockholders
position in terms
of put options
c0 = S0 +p0 (1+ r)TE
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23.10 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.
$200
$0
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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
NPV =$200 +$100
(1.10)
NPV =$133
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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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23.11 Mergers and 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 Diageos stockholders
received 141 million shares of General Mills stock
(then valued at $42.55) plus contingent value rights
of $4.55 per share.
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23.11 Mergers and Options
The contingent value rights paid the difference
between $42.55 and General Mills stock price in
one year up to a maximum of $4.55.
Cash
payment tonewly
issued
shares
$0
Value of General
Mills in 1 year$42.55$38
$4.55
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23.11 Mergers and Options
The contingent value plan can be viewed in terms of
puts:
Each newly issued share of General Mills given
to Diageos 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
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23.11 Mergers and Options
$38
$0
Value of General
Mills in 1 year$42.55
$42.55
$38
Own a put
Strike $42.55
Sell a put
Strike $38
$38.00
$4.55
$42.55
Cash payment to newly issued shares
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23.11 Mergers and Options
Value of a share
$38
$4.55
$0
$42.55
Value of
GeneralMills in 1
year
Value of General
Mills in 1 year
Value of a share
plus cash
payment
$42.55
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23.12 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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23.13 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 stockat a set price for a given amount of time.
Put-Call parity
00 PSeXC Tr +=+ -
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23.13 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 enhance
value.
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Binomial Option Pricing Model Example
European Call Option Example
European Call Option that at t = 1 matures and has a strike price of $27
Future Call Value
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B.O.P.M Example Cont.
Replicating Portfolio for the call option involving the
underlying stock and risk-free debt.
Scale down by 3/10 so as to make a replicating portfolio of
the same future call value.
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B.O.P.M Example Cont.
Co = Price of this portfolio =
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B.O.P.M Example Cont
European Put Option Example
European Put Option that at t = 1 matures with a strike price of $27
Future Put Value
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B.O.P.M Example Cont
Replicating Portfolio for the put involving the underlying
stock and risk-free debt
Scale down by 7/10 so as to make a replicating portfolio
of the same future put value.
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B.O.P.M Example Cont
Po = Price of this portfolio =
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B.O.P.M Example Cont
Remember
26.59 = 26.59
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B.O.P.M Example Cont/ Risk-Neutral Pricing
Risk-Neutral Pricing
Example
American call option with strike price $18 and r = 10%
What is the price of the call?
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B.O.P.M Example Cont/ Risk-Neutral Pricing
We know that by using Binomial Option pricing
For American call, PV of exercising at t=0 is $2
PV of not exercising is $4.45
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B.O.P.M Example Cont/ Risk-Neutral Pricing
Is there another way of pricing this option?
Fact. Option pricing depends on price of stock, not (directly)
on Beta (), the discount factor, or probabilities of high-vs-
low state.
Price of call option should be the same for all (, )combinations such that S0 is constant
Idea: Assume = 0 compute compute value of call
option
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B.O.P.M Example Cont/ Risk-Neutral Pricing
Solve for S0 = (option payoff in good state) + (1- )
(option payoff in bad state)
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