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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
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
10.2 Basic concepts of Options
Option price information
10.2 Basic concepts of Options
Exhibit 10-1 Listed Options Quotations
Close Strike Calls Puts
Price Price Sep Oct Jan Sep Oct Jan
JNJ
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
10.2 Basic concepts of Options
Exhibit 10-1 Listed Options Quotations (Cont’d)
Close Strike Calls Puts
Price Price Sep Oct Jan Sep Oct Jan
MRK
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
10.2 Basic concepts of Options
Exhibit 10-1 Listed Options Quotations (Cont’d)
Close Strike Calls Puts
Price Price Sep Oct Jan Sep Oct Jan
PG
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
10.2 Basic concepts of Options
Vc MAX (0, P E )
V p MAX (0, E P)
10.2 Basic concepts of Options
Figure 10-1
Value of $50 Exercise Price Call Option (a) to Holder, (b) to Seller
10.2 Basic concepts of Options
Figure 10-1
Value of $50 Exercise Price Call Option (a) to Holder, (b) to Seller
10.3 Factors affecting option value
Determining the value of a call
option before the expiration date
10.3 Factors affecting option value
Figure 10-2 Value of Call Option
10.3 Factors affecting option value
Figure 10-3 Call Option Value as a Function of Stock Price
10.3 Factors affecting option value
TABLE 10-1
Probabilities for Future Prices of Two Stocks
Less Volatile Stock More Volatile Stock
Future Price($) Probability Future Price ($) Probability
42 .10 32 .15
47 .20 42 .20
52 .40 52 .30
57 .20 62 .20
62 .10 72 .15
10.3 Factors affecting option value
Figure 10-4 Call-Option Value as Function of Stock Price for High-,
Moderate-, and Low-Volatility Stocks
10.3 Factors affecting option value
TABLE 10-2 Data for a Hedging Example
Current price per share: $100
Future price per share: $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 Value per Share Value per Share
Stock Price of Stock Holdings of Options Written
$125 $125 －$25
$85 $85 $0
10.3 Factors affecting option value
125H 25 85H
25 5
H
40 8
P E
H U
P PL
U
10.3 Factors affecting option value
(5)(125)－(8)(25) = $425
(5) (85)＋(8) (0) = $425
1.08(500 8 Vc ) 425
Vc
1.08 500 425 $13.31
1.088
10.4 Determining the value of options
Expected value estimation
The Black-Scholes option
Pricing model
Taxation of options
American options
10.4 Determining the value of options
expected value of share = (.6)(125) + (.4)(85) = $109
expected value of call = (.6)(25) + (.4)(0) = $15
15 13.31
expected rate of return on call 100 12.7%
13.31
VP MAX E P,0
10.4 Determining the value of options
Figure 10-5 Put-Option Value
10.4 Determining the value of options
Vc P[ N (d1 )] e rt E[ N (d 2 )] (10-1)
P 2 t
ln rt
d1 E 2
(10-2A)
t
P 2 t
ln rt
d2 E 2
d1 t (10-2B)
t
10.4 Determining the value of options
FIGURE 10-6 Probability Distribution of Stock Prices
10.4 Determining the value of options
Example 10-1
P
.6 ln ln .9 .1054
E
P t
P t ln rt 2
ln rt 2
d2
E 2
d1
E 2
t
t
1
1 .1054 .08 .50 .36 .50
.1054 .08.50 .36 .50 2
2
.06 .6 .50
.6 .50
.37
10.4 Determining the value of options
Example 10-1
N d1 N .06 .5239
N d2 N .37 .3557
rt .08.5
e e .9608
Vc PN d1 e rt EN d 2
90 .5239 .9608 100 .3557 $12.98
10.5 Option pricing theory and capital structure
Proportion of debt in capital structure
Riskiness of business operations
10.5 Option pricing theory and capital structure
V MAX 0,V f B （10-3）
FIGURE 10-7 Option Approach to Capital Structure
10.5 Option pricing theory and capital structure
Example 10-2
P
ln( ) ln(1.4) .3365
E
P t P t
ln rt 2 ln rt 2
d1 d2
E 2 E 2
t t
1 1
.3365 .08 6 .2 6 .3365 .08 6 .2 6
2 1.91 2
1.42
.2 6 .2 6
10.5 Option pricing theory and capital structure
Example 10-2
N d1 N 1.91 .9719
N d2 N 1.42 .9222
.08 6
e rt e .6188
V PN (d1 ) e rt EN (d 2 )
(14)(.9719) (.6188)(10)(.9222) $7.90 million
10.5 Option pricing theory and capital structure
rt
V PN (d1 ) e EN (d 2 )
(14)(.9996) (.6188)(5)(.9977)
$10.91 million
value of debt = 14－10.91 = $3.09 million
10.5 Option pricing theory and capital structure
Table 10-4
Effect of Different Levels of Debt on Debt Value
Face Value of Debt Actual Value of Debt Actual Value per Dollar Debt
($ millions) ($ millions) Face Value of Debt
5 3.09 $.618
10 6.10 $.610
10.5 Option pricing theory and capital structure
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
10.5 Option pricing theory and capital structure
Table 10-5
Effect of Different Levels of Business Risk on the Value of $10
Million Face Value of Debt
Variance of Rate of Value of Equity Value of Debt
Return ($ millions) ($ millions)
.2 7.90 6.10
.4 8.77 5.23
10.6 Warrants
Vw MAX 0, NP E
old equity = stockholders’ equity + warrants
new equity = old equity + exercise money
H(new equity) = H (old equity) + H (exercise money)
10.6 Warrants
Nw
H
Nw N
Nw
P Value of old equity
Nw N
N
E exercise money
Nw N
10.6 Warrants
old equity = 100(lm) + 20(.5m) = $110 million
new equity = $110m + $40m = $150 million
Nw 500, 000
H or
Nw N 500, 000 1, 000, 000
1 1 1
150m 110m 40m 50 million
3 3 3
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, option-exercise
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.
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
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 (10A.3)
Cu Cd
h (10A.4)
(u d ) S
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
(1 r )(hS C ) h(uS ) Cu h(dS ) Cd (10A.5)
R d u R
C Cu Cd R (10A.6)
u d ud
Rd u R
p so 1 p (10A.7)
ud u d
C pCu (1 p)C d R (10A.8)
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 Cu = Max (0,uS – X)
= Max (0,110 – 100)
= Max (0,10)
= $ 10
$100 C
dS = $ 90 Cd = Max (0,dS – X)
= Max (0,90 – 100)
= Max (0, –10)
= $0
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 C uu 2 p(1 p)C ud (1 p) 2 C dd R 2
(10A.12)
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
n
1 n!
C n
R
k!(n k )!
k 0
p k (1 p) n k Max[0, u k d n k S X ] (10A.13)
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
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
1 3! 3!
C (.85) (.15) X 0
0 3
(.85)1 (.15) 2 X 0
(1.07)3 0!3!
1!2!
3! 3!
(.85) (.15) X 8.90
2 1 3 0
(.85) (.15) X 33.10
2!1! 3!0!
1 3X 2 X1 3X 2 X1
00 (.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
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
n n! k nk
nk u d X n n! nk
C S p (1 p)
K
n p (1 p)
k
k m k!(n K )! R R k m k!(n k )!
n
(10A.14)
k nk
u d
P (1 p)
k nk
p rk (1 p ) n k
Rn
X
C SB1 (n, p , m) n B2 (n, p, m) (10A.15)
R
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
n
B1 (n, p , m) C p (1 p )
n
k
k nk
k m
n
B2 (n, p, m) C p (1 p)
n
k
k nk
k m
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options 190.61
162.22
Figure 10A.1 138.06
137.89
Price Path of 117.35
137.89
117.50 99.75
Underlying
Stock 117.35
137.89
99.75
Source: 99.88
R.J.Rendelman, 99.75
84.90
Jr., and 72.16
$100.00
B.J.Bartter (1979),
137.89
“Two-State 117.35
99.75
Option Pricing,” 99.88
Journal of 99.75
84.90
Finance 34 72.16
85.00
(December), 1906.
99.75
84.90
72.16
72.25
72.16
61.41
52.20
0 1 2 3 4
Appendix 10A: Applications of the Binomial Distribution
to Evaluate Call Options
16
Pi190.61 137.89 . . . 52.20
P
i 1
16 16
$105.09
12
(190.61 105.09) . . . (52.20 105.09)
2 2
P
16
$34.39
