Financial Analysis, Planning and
Forecasting
Theory and Application
Chapter 10
Option Pricing Theory and firm Valuation
By
Alice C. Lee
San Francisco State University
John C. Lee
J.P. Morgan Chase
Cheng F. Lee
Rutgers University
1
Outline
10.1 Introduction
 10.2 Basic concepts of options
 10.3 Factors affecting option value
 10.4 Determining the value of options
 10.5 Option pricing theory and capital structure
 10.6 Warrants
 10.7 Summary
 Appendix 10A. Application of the binomial
distribution to evaluate call options

2
10.2 Basic concepts of Options
 Option
price information
3
10.2 Basic concepts of Options
Exhibit 10-1 Listed Options Quotations
Close
Price
Calls
Puts
Strike
Price
Sep
Oct
Jan
Sep
Oct
Jan
65.12
45.00
20.40
N/A
N/A
N/A
N/A
N/A
65.12
50.00
N/A
N/A
N/A
N/A
N/A
N/A
65.12
55.00
10.30
10.40
11.10
N/A
0.02
0.30
65.12
60.00
5.30
N/A
6.40
N/A
N/A
0.70
65.12
65.00
0.10
1.15
2.60
0.05
0.85
1.95
65.12
70.00
N/A
0.05
0.55
4.80
4.50
5.00
JNJ
4
10.2 Basic concepts of Options
Exhibit 10-1 Listed Options Quotations (Cont’d)
Close
Price
Calls
Puts
Strike
Price
Sep
Oct
Jan
Sep
Oct
Jan
51.82
47.50
4.34
4.90
6.04
N/A
N/A
1.30
51.82
50.00
1.85
2.70
4.40
0.03
0.65
2.10
51.82
52.50
0.03
1.10
2.85
0.55
1.50
3.10
51.82
55.00
0.01
0.35
1.70
3.20
3.30
4.60
51.82
57.50
N/A
N/A
0.95
5.70
N/A
N/A
51.82
60.00
N/A
N/A
0.50
8.20
N/A
8.40
MRK
5
10.2 Basic concepts of Options
Exhibit 10-1 Listed Options Quotations (Cont’d)
Close
Price
Puts
Calls
Strike
Price
Sep
Oct
Jan
Sep
Oct
Jan
69.39
40.00
29.70
29.70
30.10
N/A
N/A
N/A
69.39
45.00
24.60
24.70
N/A
N/A
N/A
N/A
69.39
55.00
14.50
N/A
15.20
N/A
N/A
0.15
69.39
60.00
9.70
9.80
10.70
N/A
0.08
0.42
69.39
65.00
4.50
4.90
6.19
N/A
0.21
1.00
69.39
70.00
0.05
0.90
2.75
0.60
1.50
2.65
69.39
75.00
N/A
0.05
0.70
5.50
5.70
5.70
69.39
80.00
N/A
N/A
0.15
10.40
10.50
N/A
PG
6
10.2 Basic concepts of Options
 The
value of a call option on the maturity
date is
Vc  MAX (0, P  E )
 The
value of a put option on the maturity
date is
V p  MAX (0, E  P)
7
10.2 Basic concepts of Options
Figure 10-1
Value of $50 Exercise Price Call Option (a) to Holder, (b) to Seller
8
10.2 Basic concepts of Options
Figure 10-1
Value of $50 Exercise Price Call Option (a) to Holder, (b) to Seller
9
10.3 Factors affecting option value
There are five factors that influence the
value of a call option:
(1)
(2)
(3)
(4)
(5)
Market price of the stock
The exercise price
The risk-free interest rule
The volatility of the stock price
Time to expiration
10
10.3 Factors affecting option value
Figure 10-2 Value of Call Option
11
10.3 Factors affecting option value
Figure 10-3 Call Option Value as a Function of Stock Price
12
10.3 Factors affecting option value
TABLE 10-1
Probabilities for Future Prices of Two Stocks
Less Volatile Stock
More Volatile Stock
Future Price($)
Probability
Future Price ($)
Probability
42
47
52
57
62
.10
.20
.40
.20
.10
32
42
52
62
72
.15
.20
.30
.20
.15
13
10.3 Factors affecting option value
Figure 10-4 Call-Option Value as Function of Stock Price for High-,
Moderate-, and Low-Volatility Stocks
14
10.3 Factors affecting option value
TABLE 10-2 Data for a Hedging Example
Current price per share:
Future price per share:
$100
$125 with probability .6
$85 with probability .4
Exercise price of call option: $100
TABLE 10-3 Possible Expiration-Date Outcomes for Hedging Example
Expiration-Date
Stock Price
Value per Share
of Stock Holdings
Value per Share
of Options Written
$125
$85
$125
$85
－$25
$0
15
10.3 Factors affecting option value



Suppose that the investor wants to form a hedged portfolio by both
purchasing stock and writing call options, so as to guarantee as large a
total value of holdings per dollar invested as possible, whatever the
stock price on the expiration date. The hedge is constructed to be
riskless, since any profit (or loss) from the stock is exactly offset with a
loss (or profit) from the call option.
A riskless hedge can be accomplished by purchasing H shares stock
for each option written.
125H  25  85H
25 5
H

40 8
This ratio is known as the hedge ratio of stocks to options. More
generally, the hedge ratio is given by:
PU  E
H
PU  PL
16
10.3 Factors affecting option value
Let Vc denote the price per share of a call option. Then, the purchase of
five shares costs $500, but $8Vc is received from writing call options on
eight shares, so that the net outlay will be $500-$8Vc. For this outlay, a
value one year hence of $425 is assured.
(5)(125)－(8)(25) = $425
(5) (85)＋(8) (0) = $425
An investment could be made in government securities at the risk-free
interest rate. Suppose that this rate is 8% per annual. On the expiration
date, an initial investment of $500-$8Vc will be worth 1.08($500-$8Vc). If
this is to be equal to the value of the hedge portfolio, then
1.08(500  8 Vc )  425
1.08  500   425

Vc 
 $13.31
1.088
17
10.3 Factors affecting option value

Assumptions of Black-Scholes Model
1)
Only European options are considered
2)
Options and stocks can be traded in any quantities in a
perfectly competitive market. (no transaction costs, and all
information is freely available to market participants.)
3)
Short-selling of stocks and options in a perfectly competitive
market is possible.
4)
The risk-free interest rate is known and is constant up to the
expiration date.
5)
Market participants are able to borrow or lend at risk-free rate.
6)
No dividends are paid on the stock.
7)
The stock price follows a random path in continuous time such
that the variance of the rate of return is constant over time and
known to market participants. The logarithm of future stock
prices follows a normal distribution.
18
10.4 Determining the value of options
FIGURE 10-6 Probability Distribution of Stock Prices
This figure is used to explain the area of N(d1). On page 386 of the book, the last
19
sentence of the second paragraph need to be rewritten as “N(d1) can be
graphically described by Figure 10-6.”
10.4 Determining the value of options
Black-Scholes Option pricing Model
 rt
Vc  P[ N (d1 )]  e E[ N (d 2 )]
(10-1)
N() is the cumulative distribution function of standard normal distribution
t
is the time to expiration
P
E
is the spot price of the underlying asset
is the strike price
r
is the risk free rate (annual rate, expressed in terms of continuous compounding)
is the volatility in the log-returns of the underlying

20
10.4 Determining the value of options
Black-Scholes Option pricing Model
 rt
Vc  P[ N (d1 )]  e E[ N (d 2 )]
P
2 t
ln    rt  
E
2

d1 
 t
P
2 t
ln    rt  
E
2

d2 
 d1   t
 t
(10-1)
(10-2A)
(10-2B)
21
10.4 Determining the value of options
Example 10-1
Suppose that the current market price of a share of stock is
$90 with a standard deviation 0.6. An option is written to
purchase the stock for $100 in 6 months. The current risk-free
rate of interest is 8%.
P
ln    ln .9   .1054
E
t
P
ln    rt   2
E
2
d1   
 t
1
.1054  .08 .50   .36 .50 
2

 .06
.6 .50
t
P
ln    rt   2
E
2
d2   
 t

.1054  .08.50  
1
.36 .50 
2
.6 .50
 .37
22
10.4 Determining the value of options

Example 10-1
N  d1   N .06  .5239
N  d2   N  .37   .3557
e
 rt
e
.08.5
 .9608
Vc  PN  d1   e rt EN  d 2 
  90 .5239   .9608 100 .3557   $12.98
23
10.5 Option pricing theory and capital structure
V  MAX  0, V f  B 
（10-3）
V = value of stockholder
V f = value of the firm
FIGURE 10-7 Option Approach to Capital Structure
24
10.5 Option pricing theory and capital structure
Proportion
of debt in capital structure
Example 10.2
An unlevered corporation is valued at $14 million. The corporation issues
debt, payable in 6 years, with a face value of $10 million. The standard
deviation of the continuously compounded rate of return on the total value
of this corporation is 0.2. Assume that the risk-free rate of interest is 8%
per annum.
P
ln( )  ln(1.4)  .3365
E
t
P
ln    rt   2
E
2
d2   
 t
t
P
ln    rt   2
E
2
d1   
 t

.3365  .08  6  
.2 6
1
.2  6 
2
 1.91

.3365  .08  6  
.2 6
1
.2  6 
2
 1.42
25
10.5 Option pricing theory and capital structure
 Example
10-2
N  d1   N 1.91  .9719
N  d2   N 1.42  .9222
e
 rt
e
.08 6
 .6188
V  PN (d1 )  e rt EN (d 2 )
 (14)(.9719)  (.6188)(10)(.9222)  $7.90 million
26
10.5 Option pricing theory and capital structure

Following Example 10.2, only $5miilion face
value of debt is to be issued.
 rt
V  PN (d1 )  e EN (d 2 )
 (14)(.9996)  (.6188)(5)(.9977)
 $10.91 million
value of debt = 14－10.91 = $3.09 million
27
10.5 Option pricing theory and capital structure
Table 10-4
Effect of Different Levels of Debt on Debt Value
Face Value of Debt
($ millions)
Actual Value of Debt
($ millions)
Actual Value per Dollar Debt
Face Value of Debt
5
3.09
$.618
10
6.10
$.610
28
10.5 Option pricing theory and capital structure
Riskiness
of Business Operations
Following Example 10.2, which is about to issue debt with
face value of $10 million. Leaving the other variables
unchanged, suppose that the standard deviation of the
continuously compounded rate of return on the corporation’s
total value is 0.4 rather than 0.2.
V  PN (d1 )  e  rt EN (d 2 )
 (14)(.9066)  (.6188)(10)(.6331)
 $8.77 million
value of debt = 14－8.77 = $5.23 million
29
10.5 Option pricing theory and capital structure
Table 10-5
Effect of Different Levels of Business Risk on the Value of $10
Million Face Value of Debt
Variance of Rate of
Return
Value of Equity
($ millions)
Value of Debt
($ millions)
.2
7.90
6.10
.4
8.77
5.23
30
10.6 Warrants
A warrant is an option issued by a corporation to individual investors to
purchase, at a stated price, a specified number of the shares of common
stock of that corporation.
Vw  MAX  0, NP  E 
Since using warrants, a corporation can extract more favorable terms from
bondholders, it follows that the company must have transferred something
of value it bondholders. This transfer can be visualized as giving the
bondholders a stake in the corporation’s equity. Thus, we should regard
equity comprising both stockholdings and warrant value. We will refer to
this total equity, prior to the exercise of the options, as old equity so that
old equity = stockholders’ equity + warrants
If the warrants are exercised, the corporation then receives additional
money from the purchase of new shares, so that total equity is
new equity = old equity + exercise money
31
10.6 Warrants
We denote by N the number of shares outstanding and by Nw the
number of shares that warrantholders can purchase. If the options are
exercised, there will be a total of N+Nw shares, a fraction Nw/(N+Nw) of
which is owned by former warrantholders. These holders then own this
fration of the new equity,
H(new equity) = H (old equity) + H (exercise money)
Nw
H
Nw  N
32
10.6 Warrants

Thus, a fraction, Nw/(N+Nw), of the exercise money is effectively
returned to the former warrantholders. Therefore, in valuing the
warrants, the Black-Scholes formula must be modified. We need to
make appropriate substitutions in Eq. (10.1) for the current stock price,
P, and the exercise price of the option, E.
 Nw 
P
   Value of old equity 
 Nw  N 
 N 
E 
   exercise money 
 Nw  N 

It also follows that the appropriate measure of volatility, , is the
variance of rate of return on the total old equity (including the value of
warrants), not simply on stockholders’ equity.
33
10.7 Summary
In Chapter 10, we have discussed the basic concepts of call
and put options and have examined the factors that
determine the value of an option. One procedure used in
option valuation is the Black-Scholes model, which allows us
to estimate option value as a function of stock price, optionexercise price, time-to-expiration date, and risk-free interest
rate. The option pricing approach to investigating capital
structure is also discussed, as is the value of warrants.
34
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
 What
is an option?
 The simple binomial option pricing
model
 The Generalized Binomial Option
Pricing Model
35
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
Cu  Max(0, uS  X )
(10A.1)
Cd  Max(0, dS  X )
(10A.2)
h(uS )  Cu  h(dS )  Cd
Cu  Cd
h
(u  d ) S
(10A.3)
(10A.4)
36
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
(1  r )(hS  C )  h(uS )  Cu  h(dS )  Cd
 R  d 
u  R 
C  
Cu  
C d  R
ud  
 u  d 
Rd
p
ud
u  R 
so 1  p  

u  d 
C   pCu  (1  p)Cd  R
(10A.5)
(10A.6)
(10A.7)
(10A.8)
37
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
Table 10A.1 Possible Option Value at Maturity
Today
Stock (S)
Option (C)
Next Period (Maturity)
uS = $110
$100
Cu =
Max (0,uS – X)
=
Max (0,110 – 100)
=
Max (0,10)
=
$ 10
Cd =
Max (0,dS – X)
=
Max (0,90 – 100)
=
Max (0, –10)
=
$0
C
dS = $ 90
38
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
CT = Max [0, ST – X]
(10A.9)
Cu = [pCuu + (1 – p)Cud] / R
(10A.10)
Cd= [pCdu + (1 – p)Cdd] / R
(10A.11)


C  p 2 Cuu  2 p(1  p)Cud  (1  p) 2 C dd R 2
(10A.12)
39
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
1
C n
R
n
n!
k
nk
k nk
p
(
1

p
)
Max
[
0
,
u
d S  X ] (10A.13)

k  0 k!( n  k )!
C1 = Max [0, (1.1)3(.90)0(100) – 100] = 33.10
C2 = Max [0, (1.1)2(.90) (100) – 100] = 8.90
C3 = Max [0, (1.1) (.90)2(100) – 100] = 0
C4 = Max [0, (1.1)0(.90)3(100) – 100] = 0
40
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
1  3!
3!
0
3
1
2
C
(.85)
(.15)
X
0

(.85)
(.15)
X0
3 
(1.07)  0!3!
1!2!
3!
3!

2
1
3
0

(.85) (.15) X 8.90 
(.85) (.15) X 33.10 
2!1!
3!0!

1 
3X 2 X1
3X 2 X1

0

0

(.7225)(.15)(8.90)

X
(.61413)(1)(33.10)

1.225 
2 X 1X 1
3 X 2 X 1X 1
1

(.32513 X 8.90)  (.61413 X 33.10)]
1.225
 $18.96

41
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
k nk
 n
 X  n
n!
n!
K
nk u d
k
nk 
C  S 
p (1  p)
p (1  p) 
  n 
n
R  R k m k!(n  k )!

k m k!(n  K )!
(10A.14)
P (1  p)
k
nk
k
nk
u d
Rn
 p rk (1  p ) nk
X
C  SB1 (n, p , m)  n B2 (n, p, m) (10A.15)
R
42
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
n
B1 (n, p , m)   C p  (1  p )
k m
B2 (n, p, m) 
n
k
n
C
k m
n
k
k
p (1  p )
k
nk
nk
43
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
190.61
162.22
Figure 10A.1
Price Path of
Underlying
Stock
Source:
R.J.Rendelman,
Jr., and
B.J.Bartter (1979),
“Two-State
Option Pricing,”
Journal of
Finance 34
(December), 1906.
138.06
117.50
117.35
137.89
99.75
117.35
137.89
99.75
84.90
99.75
72.16
117.35
137.89
99.75
84.90
99.75
72.16
84.90
99.75
72.16
61.41
72.16
52.20
99.88
$100.00
99.88
85.00
72.25
0
1
2
137.89
3
4
44
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
16
 Pi
190.61  137.89  . . .  52.20
P

16
16
i 1
 $105.09
12
 (190.61  105.09)  . . .  (52.20  105.09) 
P  

16


2
2
 $34.39
45