Chap 022
-
Upload
florensia-restian -
Category
Documents
-
view
222 -
download
1
description
Transcript of Chap 022
![Page 1: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/1.jpg)
Options and Corporate Finance
Chapter 22
Copyright © 2010 by the McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin
![Page 2: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/2.jpg)
22-2
Key Concepts and Skills Understand option terminology Be able to determine option payoffs and profits Understand the major determinants of option
prices Understand and apply put-call parity Be able to determine option prices using the
binomial and Black-Scholes models
![Page 3: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/3.jpg)
22-3
Chapter Outline22.1 Options22.2 Call Options22.3 Put Options22.4 Selling Options22.5 Option Quotes22.6 Combinations of Options22.7 Valuing Options22.8 An Option Pricing Formula22.9 Stocks and Bonds as Options22.10 Options and Corporate Decisions: Some Applications22.11 Investment in Real Projects and Options
![Page 4: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/4.jpg)
22-4
22.1 Options An option gives the holder the right, but not the
obligation, to buy or sell a given quantity of an asset on (or before) a given date, at prices agreed upon today.
Exercising the Option The act of buying or selling the underlying asset
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 (Expiration Date)
The maturity date of the option
![Page 5: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/5.jpg)
22-5
Options European versus American options
European options can be exercised only at expiry. American options can be exercised at any time up to expiry.
In-the-Money Exercising the option would result in a positive payoff.
At-the-Money Exercising the option would result in a zero payoff (i.e.,
exercise price equal to spot price). Out-of-the-Money
Exercising the option would result in a negative payoff.
![Page 6: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/6.jpg)
22-6
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.
![Page 7: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/7.jpg)
22-7
Call Option Pricing 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
![Page 8: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/8.jpg)
22-8
Call Option Payoffs
–20
12020 40 60 80 100
–40
20
40
60
Stock price ($)
Op
tion
pay
offs
($) Buy
a ca
llExercise price = $50
50
![Page 9: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/9.jpg)
22-9
Call Option Profits
Exercise price = $50; option premium = $10
Buy a call
–20
12020 40 60 80 100
–40
20
40
60
Stock price ($)
Op
tion
pro
fits
($)
50–10
10
![Page 10: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/10.jpg)
22-10
22.3 Put Options Put options gives the holder the right,
but not the obligation, to sell a given quantity of an asset on or before some time in the future, at prices agreed upon today.
When exercising a put, you “put” the asset to someone.
![Page 11: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/11.jpg)
22-11
Put Option Pricing at Expiry At expiry, an American put option is
worth the same as a European option with the same characteristics.
If the put is in-the-money, it is worth E – ST.
If the put is out-of-the-money, it is worthless.
P = Max[E – ST, 0]
![Page 12: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/12.jpg)
22-12
Put Option Payoffs
–20
0 20 40 60 80 100
–40
20
0
40
60
Stock price ($)
Op
tion
pay
offs
($)
Buy a put
Exercise price = $50
50
50
![Page 13: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/13.jpg)
22-13
Put Option Profits
–20
20 40 60 80 100
–40
20
40
60
Stock price ($)
Op
tion
pro
fits
($)
Buy a put
Exercise price = $50; option premium = $10
–10
10
50
![Page 14: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/14.jpg)
22-14
Option Value Intrinsic Value
Call: Max[ST – E, 0]
Put: Max[E – ST , 0]
Speculative Value The difference between the option premium and the intrinsic
value of the option.
Option Premium
=Intrinsic Value
Speculative Value
+
![Page 15: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/15.jpg)
22-15
22.4 Selling Options The seller (or writer) of an option has an
obligation. The seller receives the option premium in
exchange.
![Page 16: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/16.jpg)
22-16
Call Option Payoffs
–20
12020 40 60 80 100
–40
20
40
60
Stock price ($)
Op
tion
pay
offs
($)
Sell a callExercise price = $50
50
![Page 17: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/17.jpg)
22-17
Put Option Payoffs
–20
0 20 40 60 80 100
–40
20
0
40
–50
Stock price ($)
Op
tion
pay
offs
($)
Sell a put
Exercise price = $50
50
![Page 18: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/18.jpg)
22-18
Option Diagrams Revisited
Exercise price = $50; option premium = $10
Sell a call
Buy a call
50 6040 100
–40
40
Stock price ($)
Op
tion
pro
fits
($)
Buy a put
Sell a put
–10
10
Buy a call
Sell a
put
Buy a put
Sell a call
![Page 19: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/19.jpg)
22-19
22.5 Option Quotes
Option/Strike Exp. Vol. Last Vol. LastIBM 130 Oct 364 15¼ 107 5¼138¼ 130 Jan 112 19½ 420 9¼138¼ 135 Jul 2365 4¾ 2431 13/16
138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½
--Put----Call--
![Page 20: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/20.jpg)
22-20
Option Quotes
Option/Strike Exp. Vol. Last Vol. LastIBM 130 Oct 364 15¼ 107 5¼138¼ 130 Jan 112 19½ 420 9¼138¼ 135 Jul 2365 4¾ 2431 13/16
138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½
--Put----Call--
This option has a strike price of $135;
a recent price for the stock is $138.25;
July is the expiration month.
![Page 21: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/21.jpg)
22-21
Option Quotes
Option/Strike Exp. Vol. Last Vol. LastIBM 130 Oct 364 15¼ 107 5¼138¼ 130 Jan 112 19½ 420 9¼138¼ 135 Jul 2365 4¾ 2431 13/16
138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½
--Put----Call--
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.
![Page 22: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/22.jpg)
22-22
Option Quotes
Option/Strike Exp. Vol. Last Vol. LastIBM 130 Oct 364 15¼ 107 5¼138¼ 130 Jan 112 19½ 420 9¼138¼ 135 Jul 2365 4¾ 2431 13/16
138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½
--Put----Call--
On this day, 2,365 call options with this exercise price were traded.
![Page 23: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/23.jpg)
22-23
Option Quotes
Option/Strike Exp. Vol. Last Vol. LastIBM 130 Oct 364 15¼ 107 5¼138¼ 130 Jan 112 19½ 420 9¼138¼ 135 Jul 2365 4¾ 2431 13/16
138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½
--Put----Call--
The CALL option with a strike price of $135 is trading for $4.75.
Since the option is on 100 shares of stock, buying this option would cost $475 plus commissions.
![Page 24: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/24.jpg)
22-24
Option Quotes
Option/Strike Exp. Vol. Last Vol. LastIBM 130 Oct 364 15¼ 107 5¼138¼ 130 Jan 112 19½ 420 9¼138¼ 135 Jul 2365 4¾ 2431 13/16
138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½
--Put----Call--
On this day, 2,431 put options with this exercise price were traded.
![Page 25: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/25.jpg)
22-25
Option Quotes
Option/Strike Exp. Vol. Last Vol. LastIBM 130 Oct 364 15¼ 107 5¼138¼ 130 Jan 112 19½ 420 9¼138¼ 135 Jul 2365 4¾ 2431 13/16
138¼ 135 Aug 1231 9¼ 94 5½138¼ 140 Jul 1826 1¾ 427 2¾138¼ 140 Aug 2193 6½ 58 7½
--Put----Call--
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.
![Page 26: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/26.jpg)
22-26
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.
![Page 27: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/27.jpg)
22-27
Protective Put Strategy (Payoffs)
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
![Page 28: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/28.jpg)
22-28
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 protection and upside potential
$40
$0
-$40
$50
Value at expiry
Value of stock at expiry
-$10
![Page 29: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/29.jpg)
22-29
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
$10
![Page 30: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/30.jpg)
22-30
Long Straddle
30 40 60 70
30
40
Stock price ($)
Op
tion
pay
offs
($)
Buy a put with exercise price 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.
$50
–20
![Page 31: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/31.jpg)
22-31
Short Straddle
–30
30 40 60 70
–40
Stock price ($)
Op
tion
pay
offs
($)
$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 $10
20
![Page 32: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/32.jpg)
22-32
bond
Put-Call Parity: P0 + S0 = C0 + E/(1+ r)T
25
25
Stock price ($)
Op
tion
pay
offs
($)
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 $25.
Call
Portfolio payoffPortfolio value today = C0 +
(1+ r)T
E
![Page 33: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/33.jpg)
22-33
Put-Call Parity
25
25
Stock price ($)
Op
tion
pay
offs
($)
Consider the payoffs from holding a portfolio consisting of a share of stock and a put with a $25 strike.
Portfolio value today = P0 + S0
Portfolio payoff
![Page 34: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/34.jpg)
22-34
Put-Call Parity
Since these portfolios have identical payoffs, they must have the same value today: hence
Put-Call Parity: C0 + E/(1+r)T = P0 + S0
25
25
Stock price ($)
Opt
ion
payo
ffs
($)
25
25
Stock price ($)O
ptio
n pa
yoff
s ($
) Portfolio value today = P0 + S0
Portfolio value today
(1+ r)T
E= C0 +
![Page 35: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/35.jpg)
22-35
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.
![Page 36: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/36.jpg)
22-36
American Call
C0 must fall within max (S0 – E, 0) < C0 < S0.
25
Op
tion
pay
offs
($) Call
ST
loss
E
Profit
ST
Time value
Intrinsic value
Market Value
In-the-moneyOut-of-the-money
![Page 37: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/37.jpg)
22-37
Option Value DeterminantsCall Put
1. 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.
![Page 38: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/38.jpg)
22-38
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.
![Page 39: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/39.jpg)
22-39
Binomial Option Pricing ModelSuppose 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?
$25
$21.25 = $25×(1 –.15)
$28.75 = $25×(1.15)S1S0
![Page 40: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/40.jpg)
22-40
Binomial Option Pricing Model1. 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
$28.75S1S0 C1
$3.75
$0
![Page 41: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/41.jpg)
22-41
Binomial Option Pricing ModelBorrow 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.
$25
$21.25
$28.75S1S0 debt
– $21.25portfolio$7.50
$0
( – ) ==
=
C1
$3.75
$0– $21.25
![Page 42: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/42.jpg)
22-42
Binomial Option Pricing ModelThe value today of the levered equity portfolio is today’s value of one share less the present value of a $21.25 debt: )1(
25.21$25$
fR
$25
$21.25
$28.75S1S0 debt
– $21.25portfolio$7.50
$0
( – ) ==
=
C1
$3.75
$0– $21.25
![Page 43: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/43.jpg)
22-43
Binomial Option Pricing ModelWe can value the call option today as half of the value of the levered equity portfolio:
)1(
25.21$25$
2
10
fRC
$25
$21.25
$28.75S1S0 debt
– $21.25portfolio$7.50
$0
( – ) ==
=
C1
$3.75
$0– $21.25
![Page 44: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/44.jpg)
22-44
If the interest rate is 5%, the call is worth:
Binomial Option Pricing Model
38.2$24.2025$2
1
)05.1(
25.21$25$
2
10
C
$25
$21.25
$28.75S1S0 debt
– $21.25portfolio$7.50
$0
( – ) ==
=
C1
$3.75
$0– $21.25
$2.38
C0
![Page 45: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/45.jpg)
22-45
Binomial Option Pricing Model
the replicating portfolio intuition.the replicating portfolio intuition.
Many derivative securities can be valued by valuing portfolios of primitive securities when those portfolios have the same payoffs as the derivative securities.
The most important lesson (so far) from the binomial option pricing model is:
![Page 46: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/46.jpg)
22-46
Delta This practice of the construction of a riskless
hedge is called delta hedging. The delta of a call option is positive.
Recall from the example:
• The delta of a put option is negative.
2
1
5.7$
75.3$
25.21$75.28$
075.3$
Swing of callSwing of stock
![Page 47: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/47.jpg)
22-47
Delta Determining the Amount of Borrowing:
Value of a call = Stock price × Delta
– Amount borrowed
$2.38 = $25 × ½ – Amount borrowed
Amount borrowed = $10.12
38.2$24.20$25$2
1
)05.1(
25.21$25$
2
10
C
![Page 48: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/48.jpg)
22-48
The Risk-Neutral Approach
We could value the option, V(0), as the value of the replicating portfolio. An equivalent method is risk-neutral valuation:
S(0), V(0)
S(U), V(U)
S(D), V(D)
q
1- q
)1(
)()1()()0(
fR
DVqUVqV
![Page 49: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/49.jpg)
22-49
The Risk-Neutral Approach
S(0) is the value of the underlying asset today.
S(0), V(0)
S(U), V(U)
S(D), V(D)
S(U) and S(D) are the values of the asset in the next period following an up move and a down move, respectively.
q
1- q
V(U) and V(D) are the values of the option in the next period following an up move and a down move, respectively.
q is the risk-neutral probability of an “up” move.
![Page 50: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/50.jpg)
22-50
The Risk-Neutral Approach
The key to finding q is to note that it is already impounded into an observable security price: the value of S(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
![Page 51: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/51.jpg)
22-51
Example of Risk-Neutral Valuation
$21.25,C(D)
q
1- q
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:
$25,C(0)
$28.75,C(U)
)15.1(25$75.28$
)15.1(25$25.21$
![Page 52: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/52.jpg)
22-52
Example of Risk-Neutral Valuation
$21.25,C(D)
2/3
1/3
The next step would be to compute the risk neutral probabilities
$25,C(0)
$28.75,C(U)
)()(
)()0()1(
DSUS
DSSRq f
3250.7$
5$
25.21$75.28$
25.21$25$)05.1(
q
![Page 53: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/53.jpg)
22-53
Example of Risk-Neutral Valuation
$21.25, $0
2/3
1/3
After that, find the value of the call in the up state and down state.
$25,C(0)
$28.75, $3.75
25$75.28$)( UC
]0,75.28$25max[$)( DC
![Page 54: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/54.jpg)
22-54
Example of Risk-Neutral ValuationFinally, 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
![Page 55: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/55.jpg)
22-55
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$05.1
50.2$
)05.1(
0$)31(75.3$320
C
![Page 56: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/56.jpg)
22-56
The Black-Scholes Model
)N()N( 210 dEedSC Rt
Where
C0 = the value of a European option at time t = 0R = the risk-free interest rate.
t
tσ
RESd
)2
()/ln(2
1
tdd 12
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.
![Page 57: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/57.jpg)
22-57
The Black-Scholes ModelFind the value of a six-month call option on Hardcraft, Inc. with an exercise price of $150.
The current value of a share of Hardcraft 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 intrinsic value of the option is $10—our answer must be at least that amount.
![Page 58: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/58.jpg)
22-58
The Black-Scholes ModelLet’s try our hand at using the model. If you have a calculator handy, follow along.
Then,
t
tσRESd
)5.()/ln( 2
1
First calculate d1 and d2
31602.05.30.052815.012 tdd
52815.05.30.0
5).)30.0(5.05(.)150/160ln( 2
1
d
![Page 59: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/59.jpg)
22-59
The Black-Scholes Model
N(d1) = N(0.52815) = 0.7013
N(d2) = N(0.31602) = 0.62401
52815.01 d
31602.02 d
)N()N( 210 dEedSC Rt
92.20$
62401.01507013.0160$
0
5.05.0
C
eC
![Page 60: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/60.jpg)
22-60
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.
![Page 61: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/61.jpg)
22-61
Stocks and Bonds as Options Levered equity is a put 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 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.
![Page 62: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/62.jpg)
22-62
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= + –
Stockholder’s position in terms of call options
Stockholder’s position in terms of put options
C0 = S0 + P0 – (1+ R)t
E
![Page 63: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/63.jpg)
22-63
Mergers and Diversification Diversification is a frequently mentioned reason for
mergers. Diversification reduces risk and, therefore, volatility. Decreasing volatility decreases the value of an option. Assume diversification is the only benefit to a merger:
Since equity can be viewed as a call option, should the merger increase or decrease the value of the equity?
Since risky debt can be viewed as risk-free debt minus a put option, what happens to the value of the risky debt?
Overall, what has happened with the merger and is it a good decision in view of the goal of stockholder wealth maximization?
![Page 64: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/64.jpg)
22-64
Example Consider the following two merger candidates. The merger is for diversification purposes only with no synergies
involved. Risk-free rate is 4%.
Company A Company B
Market value of assets $40 million $15 million
Face value of zero coupon debt
$18 million $7 million
Debt maturity 4 years 4 years
Asset return standard deviation
40% 50%
![Page 65: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/65.jpg)
22-65
Example Use the Black and Scholes OPM (or an options
calculator) to compute the value of the equity. Value of the debt = value of assets – value of equity
Company A Company B
Market Value of Equity 25.72 9.88
Market Value of Debt 14.28 5.12
![Page 66: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/66.jpg)
22-66
Example The asset return standard deviation for the combined firm is 30% Market value assets (combined) = 40 + 15 = 55 Face value debt (combined) = 18 + 7 = 25
Combined Firm
Market value of equity 34.18
Market value of debt 20.82
Total MV of equity of separate firms = 25.72 + 9.88 = 35.60
Wealth transfer from stockholders to bondholders = 35.60 – 34.18 = 1.42 (exact increase in MV of debt)
![Page 67: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/67.jpg)
22-67
M&A Conclusions Mergers for diversification only transfer wealth
from the stockholders to the bondholders. The standard deviation of returns on the assets
is reduced, thereby reducing the option value of the equity.
If management’s goal is to maximize stockholder wealth, then mergers for reasons of diversification should not occur.
![Page 68: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/68.jpg)
22-68
Options and Capital Budgeting Stockholders may prefer low NPV projects to high
NPV projects if the firm is highly leveraged and the low NPV project increases volatility.
Consider a company with the following characteristics: MV assets = 40 million Face Value debt = 25 million Debt maturity = 5 years Asset return standard deviation = 40% Risk-free rate = 4%
![Page 69: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/69.jpg)
22-69
Example: Low NPV Current market value of equity = $22.706 million Current market value of debt = $17.294 million
Project I Project II
NPV $3 $1
MV of assets $43 $41
Asset return standard deviation
30% 50%
MV of equity $23.831 $25.381
MV of debt $19.169 $15.169
![Page 70: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/70.jpg)
22-70
Example: Low NPV Which project should management take? Even though project B has a lower NPV, it is
better for stockholders. The firm has a relatively high amount of
leverage: With project A, the bondholders share in the NPV
because it reduces the risk of bankruptcy. With project B, the stockholders actually appropriate
additional wealth from the bondholders for a larger gain in value.
![Page 71: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/71.jpg)
22-71
Example: Negative NPV We have seen that stockholders might prefer a low
NPV to a high one, but would they ever prefer a negative NPV?
Under certain circumstances, they might. If the firm is highly leveraged, stockholders have
nothing to lose if a project fails, and everything to gain if it succeeds.
Consequently, they may prefer a very risky project with a negative NPV but high potential rewards.
![Page 72: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/72.jpg)
22-72
Example: Negative NPV Consider the previous firm. They have one additional project they are
considering with the following characteristics Project NPV = -$2 million MV of assets = $38 million Asset return standard deviation = 65%
Estimate the value of the debt and equity MV equity = $25.453 million MV debt = $12.547 million
![Page 73: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/73.jpg)
22-73
Example: Negative NPV In this case, stockholders would actually prefer
the negative NPV project to either of the positive NPV projects.
The stockholders benefit from the increased volatility associated with the project even if the expected NPV is negative.
This happens because of the large levels of leverage.
![Page 74: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/74.jpg)
22-74
Options and Capital Budgeting As a general rule, managers should not accept
low or negative NPV projects and pass up high NPV projects.
Under certain circumstances, however, this may benefit stockholders: The firm is highly leveraged The low or negative NPV project causes a
substantial increase in the standard deviation of asset returns
![Page 75: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/75.jpg)
22-75
22.11 Investment in Real Projects and Options Classic NPV calculations generally ignore
the flexibility that real-world firms typically have. Option to expand Option to abandon
![Page 76: Chap 022](https://reader035.fdocuments.in/reader035/viewer/2022062308/563db8b0550346aa9a9603e3/html5/thumbnails/76.jpg)
22-76
Quick Quiz What is the difference between call and put options? What are the major determinants of option prices? What is put-call parity? What would happen if it does
not hold? What is the Black-Scholes option pricing model? How can equity be viewed as a call option? Should a firm do a merger for diversification
purposes only? Why or why not? Should management ever accept a negative NPV
project? If yes, under what circumstances?