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Chapter 19
Normal,
Log-Normal Distribution,
and
Option Pricing Model
By
Cheng Few Lee
Joseph Finnerty
John Lee
Alice C Lee
Donald Wort
Outline
2
•
19.1 The Normal Distribution
•
19.2 The Log-Normal Distribution
•
19.3 The Log-Normal Distribution and It’s
Relationship to the Normal Distribution
•
19.4 Multivariate Normal and Log-Normal
Distributions
•
19.5 The Normal Distribution as an Application to
the Binomial and Poisson Distributions
•
19.6 Applications of the Log-Normal Distribution in
Option Pricing
Outline
3
•
19.7 THE BIVARIATE NORMAL DENSITY
FUNCTION
•
19.8 AMERICAN CALL OPTIONS
•
19.8.1 Price American Call Options by the Bivariate
Normal Distribution
•
19.8.2 Pricing an American Call Option: An
Example
•
19.9 PRICING BOUNDS FOR OPTIONS
•
19.9.1 Options Written on Nondividend-Paying
Stocks
•
19.9.2 Option Written on Dividend-Paying Stocks
19.1 The Normal Distribution
• A random
variable X is said to be normally
distributed with mean  and variance  2if it has
the probability density function (PDF)
1 x
 (
1
f ( x) 
e 2
2 

)2
  0.
(19.1)
*Useful in approximation for binomial distribution
and studying option pricing.
4
• Standard
PDF of
Z
g ( z) 
X 

1
2
is
e
z2

2
(19.2)
• This
is the PDF of the standard normal and is
independent of the parameters
2

•
and  .
5
• Cumulative
•
•
distribution function (CDF) of Z
*In many cases, value
P( Z  z )  N ( z )
N(z) is provided by
(19.3)
software.
•
• CDF
P ( X  x)  P (
X 


of X
x

)  N(
x

).
(19.4)
6
• When
X is normally distributed then the Moment
generating function (MGF) of X is
M x (t )  e
• *Useful
7
 t  t 2 2 2
(19.5)
in deriving the moment of X and moments
of log-normal distribution.
19.2 The Log-Normal Distribution
• Normally
of
and
distributed log-normality with parameters

2
• *X
Y  log X
(19.6)
8
has to be a
• positive random
• variable.
• *Useful in studying
• the behavior of
• stock prices.
• PDF for
log-normal distribution
g ( x) 
1
2 x

e
1
2
2
(log
x


)
2
, x  0.
(19.7)
•
•
9
*It is sometimes called the antilog-normal distribution,
because it is the distribution of the random variable X.
*When applied to economic data, it is often called “CobbDouglas distribution”.
• The rth
moment of X is
 r  E ( X r )  E (e rY )  e
r 2 2
 r
2
.
(19.8)
• From
equation 19.8 we have:
E( X )  e

2
2
,
2
(19.9)
2
Var( X )  e e [e
10
2
 1].
(19.10)
The CDF of X
P( X  x)  P(log X  log x)  N (
log x  

),
(19.11)
The distribution of X is unimodal with the mode
at
mod e( X )  e
11
(   2 )
.
(19.12)
Log-normal distribution is NOT symmetric.
x be the percentile for the log-normal
distribution and z  be the corresponding percentile
for the standard normal, then
• Let
P( X  x )  P(
log x  
log X  


log x  

)  N(
 z
implying
x

e
.



• Also that median( X )  e , as z 0.5  0.
• so z 
log x  

).
(19.13)
,
(19.15)
(19.14)
• Meaning
that
median( X )  mod e( X ).
12
19.3 The Log-Normal Distribution and Its
Relationship to the Normal Distribution
• Compare
PDF of normal distribution and
PDF of log-normal distribution to see that
f ( y)
f ( x) 
x
• Also
from (19.6), we can see that
dx  xdy
13
(19.16)
(19.17)
• CDF for
the log-normal distribution
F (a)  Pr( X  a)  Pr(log X  log a)
log X   log a  
 Pr(

)


(19.18)
 N (d )
• Where
d
log a  

(19.19)
*N(d) is the CDF of standard normal distribution
which can be found from Normal Table; it can also be
obtained from S-plus/other software.
•
14
• N(d)
can alternatively be approximated by the
following formula:
N (d )  a0 e
d2

2
(a1t  a2 t  a3t )
2
3
(19.20)
1
1  0.33267d
a0  0.3989423, a1  0.4361936, a2  0.1201676, a3  0.9372980
•
Where
•
In case we need Pr(X>a), then we have
t
Pr( X  a)  1  Pr( X  a)  1  N (d )  N (d )
(19.21)
15
•
Since for any h, E ( X h )  E (e hY ), the hth moment of X, the
following moment generating function of Y, which is
normally distributed.
1
t  t 
M Y (t )  e
2
2
2
(19.22)
For example,
 X  E ( X )  E (eY )  M Y (1)  e
E ( X )  E (e )  M Y ( h)  e
h
•
hY
  E ( X )  ( EX )  e
16
1
2
h  t 2 2
(19.23)
Hence
2
X
•
1
2
  t 2 2
2
2
2   2 2
e
2   2
e
2   2
2
(e  1)
(19.24)
Fractional and negative movement of a log-normal
distribution can be obtained from Equation (19.23)
•
Mean of a log-normal random variable can be
defined as


0
2
(19.25)
If the lower bound a > 0; then the partial mean of x
can be shown as
•

xf ( x)dx  e

2

0
xf ( x)dx  

log(a )
f ( y )e dy  e
y

2
2
N (d )
Where
(19.26)
•
17
This implies that
• partial mean of a log-normal
• = (mean of x )( N(d))
d
  log(a)




19.4 Multivariate Normal and LogNormal Distributions
The normal distribution with the PDF given in
Equation (19.1) can be extended to the p-dimensional
case. Let X  X 1 , , X p  be a p × 1 random vector. Then
we say that X ~ N  ,   , if it has the PDF
•
p
p
f ( x )  2 
2

1
2
 1


exp   x     1  x   
 2

• is the mean vector and 
(19.27)
is the covariance matrix
which is symmetric and positive definite.
18
• Moment
generating function of X is
 
M x (t )  E e t x  e

• Where t  t1 , , t p 
1
t   t t
2
(19.28)
is a p x 1 vector of real values.
• From Equation (19.28), it can be shown that
E ( X )   and Cov( X )  
If C is a q  p matrix of rank q  p.
Then CX ~ N q C , CC. Thus, linear
transformation of a normal random vector is also
a multivariate normal random vector.
19
  (1) 
 X (1) 
X   ( 2 )     ( 2) 
 
X 
Let
,
and      , where X (i ) and


(i )
 are pi  1 , p1  p2  p , and  ij = pi  p j
The marginal distribution is also a multivariate
normal with mean vector and covariance matrix
that’s X ~ N  ,   . The conditional
distribution of X (1) with givens where
(i )
12
21
22
(i )
pi
(19.29)
11 2  11  12  22  21
(19.30)
(1)
 12  22
and
That is,
1
x
ii

1  2  
20
11
( 2)
  ( 2)
1
X (1) X ( 2)  x ( 2) ~ N p1 1 2 , 112 
• Bivariate
•
version of correlated log-normal
distribution.
  1    11
 Y1   log  X 1  






~
N
 Y   log  X 
, 

  
2 
 2 
  2    21
 12  

 22  
Let
• Joint PDF of X 1and X 2 can be obtained from the
joint PDF of Y1 andY2by observing that
dx1dx2  x1 x2 dy1dy2
(19.31)
• (19.31)
is an extension of (19.17) to the bivariate
case.
• Hence, joint PDF of X𝟏 and X𝟐 is
g x1 , x2  
21
1
 1


exp  log x1 , log x2      1 log x1 , log x2    
2  x1 x2
 2

(19.32)
• From
• Hence,
the property of the multivariate normal
distribution, we have
Yi ~ N i ,  ii 
𝑿𝐢 is log-normal with
E( X i )  e
Var ( X i )  e
2 i
i 
 ii
 ii
2
,
e [e
(19.33)
 ii
 1].
(19.34)
22
• By
the property of the movement generating
for the bivariate normal distribution, we
have
1
       2 
Y Y
E  X 1 X 2   E e
1
 e
  exp
2
 E  X 1 E  X 2
1
2
2
11
 11 22
22

12
(19.35)
the covariance between X1 and X2 is
Cov X 1 , X 2   E X 1 X 2   E X 1 E X 2 
• Thus,
 
 
 E  X 1 E  X 2   exp   11 22  1
 
 
1


 exp  1   2   11   22   exp   11 22  1
2


(19.36)
23
• From
the property of conditional normality of
𝑌1 given 𝑌2 =𝑦2 , we also see that the
conditional distribution of 𝑌1 given 𝑌2 =𝑦2 is
log normal.



Y

Y
,

,
Y
• When
where Yi  log X i .
1
p
If Y ~ N p  ,  
where μ     and    ij . The joint PDF of
X 1 ,  , X p can be obtained from Theorem 1.
1
24
p
Theorem 1
• Let
•
the PDF of Y ,, Y be f ( y ,, y ), consider the
• p-valued functions
1
p
1
p
xi  xi ( y1 ,, y p ), i  1,, p.
(19.37)
• Assume
transformation from the y-space to
• x-space is one to one with
• inverse transformation
yi  yi ( x1 ,, x p ), i  1,, p.
25
(19.38)
•
If we let random variables X 1 ,  , X pbe defined by
X i  xi (Y1 ,, Yp ), i  1,, p.
(19.39)
Then the PDF of X 1 ,  , X p is
g ( x1 , , x p )  f y1 ( x1 , , x p ), , y p ( x1 , , x p )J ( x1 , , x p )
•
Where J(𝑥1 ,.., 𝑥𝑝 ) is Jacobian of transformations
y1
x1
J ( x1 ,, x p )  mod 
y p
x1
•
26
(19.40)



y1
x p

y p
x p
“Mod” means modulus or absolute value
(19.41)
When applying theorem 1 with
f ( y1 ,, y p ) being a p-variate normal and
J ( x1 ,, x p )  mod
1
x1
0

0
0
1
x2

0


1
xp

0
g ( x1 ,, x p ) 
(2 )
p
2


p
2
p
(
i 1


We have joint PDF of

p

i 1
1
xi
(19.42)
X 1 ,..., X p
 
*when p=2, Equation (19.43) reduces to the
bivariate case given in Equation (19.32)
27

1
 1


) exp log x1 ,  , log x p     1 log x1 ,  , log x p    
xi
 2

(19.43)
Yjij
The first two moments are
E( X i )  e
i 
 ii
2
,
(19.44)
Var ( X i )  e 2 i e ii [e ii  1].



Cor X , X  exp 
i
j
(19.45)
 
*Where  ij is the correlation between Yi and Y j
28
 
1



 ii   jj   exp  ij  ii jj  1
i  j 
2


(19.46)
19.5 The Normal Distribution as an
Application to the Binomial and
Poisson Distribution
• Cumulative
normal density function tells
us the probability that a random
variable Z will be less than x.
29
Figure 19-1
• *P(Z<x)
30
is the area under the normal curve
from   up to point x.
• Applications
of the cumulative normal
distribution function is in valuing stock
options.
• A call
option gives the option holder the right
to purchase, at a specified price known as the
exercise price, a specified number of shares of
stock during a given time period.
• A call
31
option is a function of S, X, T,  2 ,and r
• The
binomial option pricing model in Equation
(19.22) can be written as
T
T!
k
T k
C  S[ 
p ' (1  p ' ) ]
k  m k!(T  k )!
T
X
T!
k
T k

[
p
(
1

p
)
]
T 
(1  r ) k  m k!(T  k )!
X
 SB(T , p ' , m) 
B(T , p, m),
T
(1  r )
*C= 0 if m>T
32
(19.47)
S = Current price of the firm’s common stock
T = Term to maturity in years
m = minimum number of upward movements in
stock price that is necessary for the option
to terminate “in the money”
Rd
uR
p
and 1  p 
ud
ud
X = Exercise price (or strike price) of the option
R= 1+r = 1+ risk-free rate of return
u = 1 + percentage of price increase
d = 1 + percentage of price decrease
u
p'    p
R
33
n
B(n, p, m)   n C k p k (1  p) n  k
k m
•
By a form of the central limit theorem, in Section
19.7 you will see T   , the option price C
converges to C below
C  SN (d1 )  XR T N (d 2 )
•
(19.48)
C = Price of the call option
d1 
S
)
t
Xr  1 t
2
 t
log(
d 2  d1   t
N(d) is the value of the cumulative standard
normal distribution
• t is the fixed length of calendar time to expiration
and h is the elapsed time between successive stock
price changes and T=ht.
•
34
• If
future stock price is constant over time,
2

then  0
It can be shown that both N (d1 ) and N (d 2 ) are
equal to 1 and that that Equation (19.48)
becomes
C  S  Xe
 rT
(19.49)
*Equation (19.48 and 19.49) can also be
understood in terms of the following steps
35
Step 1: Future price of the stock is
constant over time
• Value
of the call option:
X
CS
.
T
(1  r )
(19.50)
X= exercise price
• C= value of the option (current price of stock –
present value of purchase price)
•
*Equation 19.50 assumes discrete compounding of
interest, whereas Equation 19.49 assumes continuous
compounding of interest.
36
*We can adjust Equation 19.50 for continuous
compounding by changing
1
(1  r) T
to
e  rT
And get
C  S  Xe
 rT
(19.51)
37
Step 2: Assume the price of the stock
fluctuates over time (S t )
• Adjust
Equation 19.49 for uncertainty
associated with fluctuation by using the
cumulative normal distribution function.
•
• Assume
S t from Equation 19.48 follows a log-
normal distribution (discussed in section 19.3).
38
Adjustment factors N (d1 ) and N (d 2 )in BlackScholes option valuation model are adjustments
made to EQ 19.49 to account for uncertainty of the
fluctuation of stock price.
•
Continuous option pricing model (EQ 19.48)
vs
• binomial option price model (EQ19.47)
N (d1 ) and N (d 2 ) are cumulative normal density
functions
while B (T , p, m) and B (T , p ' , m) are
complementary binomial distribution functions.
•
39
Application Eq. (19.48) Example
• Theoretical
value: As of November 29, 1991, of
one of IBM’s options with maturity on April
1992.
In
this
case
we
have
X
=
$90,
S
=
$92.50,

= 0.2194, r = 0.0435, and T= =0.42 (in years).
Armed with this information we can calculate
the estimated 𝑑1 and 𝑑2 .
x
{ln(
92.5
1
)  [(.0435)  (.2194) 2 ](.42)}
90
2
 0.392,
(.2194)(.42)
1
2
1
2
x   t  x  (0.2194)(0.42)  0.25 .
40
Probability of Variable Z between 0 and x
Figure 19-2
*In Equation 19.45, N (d1 ) and N (d1 )are the probabilities that a
random variable with a standard normal distribution takes on a value
less than d1 and a value less than d 2, respectively. The values for
the probabilities can be found by using the tables in the back of the
book for the standard normal distribution.
41
• To
find the cumulative normal density
function, we add the probability that Z is less
than zero to the value given in the standard
normal distribution table. Because the standard
normal distribution is symmetric around zero,
the probability that Z is less than zero is 0.5, so
P( Z  x)  P( Z  0)  P(0  Z  x)
•=
42
0.5 + value from table
•
From
N (d1 )  P( Z  d1 )  P( Z  0)  P(0  Z  d1 )
 P( Z  .392)  .5  .1517  0.6517
N (d 2 )  P( Z  d 2 )  P( Z  0)  P(0  Z  d 2 )
 P( Z  .25)  .5  .0987  0.5987
•
The theoretical value of the option is
C  (92.5)(.6517)  [(90)(.5987)] / e(.0435)(.42)
 60.282  53.883 / 1.0184  $7.373.
•
43
The actual price of the option on November
29,1991, was $7.75.
19.6 Applications of the Log-Normal
Distribution in Option Pricing
Assumptions of Black-Scholes formula :
No transaction costs
No margin requirements
No taxes
All shares are infinitely divisible
Continuous trading is possible
Economy risk is neutral
Stock price follows log-normal distribution
44
Sj
S j 1
 exp[ K j ]
*Is a random variable with a log-normal
distribution
S = current stock price
𝑆𝑗 = end period stock price
𝐾𝑗 = rate of return in period and random variable
with normal distribution
45
2


• Let Kt have the expected value k and variance k
for each j. Then K1  K 2  ...  K t is a normal random
2
variable with expected value t k and variance t k .
Thus, we can define the expected value (mean) of
St
 exp[ K1  K 2  ...  K t ] as S
t 2
S
E(
t
S
)  exp[ t k 
k
2
].
(19.52)
Under the assumption of a risk-neutral investor,
S
the expected return E( S ) becomes exp( rt ) ( where r is
the riskless rate of interest). In other words,
t
k  r 
46
 k2
2
(19.53)
In risk-neutral assumptions, call option price C
can be determined by discounting the expected
value of terminal option price by the riskless rate
of interest:
C  exp[ rt ]E[Max(ST  X ,0)]
(19.54)
T = time of expiration and X = striking price
Max ( S T  X ,0)  ( S (
0
ST X
 )), for
S
S
for
ST
X

S
S
ST
X

S
S
(19.55)
47
Eq. (19.54) and (19.55) say that the value of the call
option today will be either St  X or 0, whichever is
greater.
• If the price of stock at time t is greater than the
exercise price, the call option will expire in the money.
• In other words, the investor will exercise the call
option. The option will be exercised regardless of
whether the option holder would like to take physical
possession of the stock.
•
48
Two Choices For Investor
1.Own
Stock
St  X
49
• Since
the price the investor paid (X) is lower
that the price he or she can sell the stock for
(𝑆𝑡 ), the investor realizes an immediate the
profit of St  X .
• If the price of the stock (𝑆𝑡 ) the exercise price
(X), the option expires out of the money.
• This occurs because in purchasing shares of the
stock the investor will find it cheaper to
purchase the stock in the market than to
exercise the option.
50
• Let
ST
S
be log-normally
distributed with
2
t k
2
2


t

parameters   tr  2 and
k . Then
X 
C  exp[ rt ]E[ Max ( S t  X )]

 exp[ rt ]X
S
X
S [ x  ]g ( x)dx
S

 exp[ rt ]S X
S
• Where
of
51
X
xg( x)dx  exp[ rt ]S
S


X
S
g ( x)dx
(19.56)
g(x) is the probability density function
St
Xt 
S
2
2
2


tr

t

/
2
,


t

• By substituting
k
k and
Into eq. (19.18) and (19.26), we get


X
S
xg( x)dx  e N (d1 )
rt
(19.57)


X
S
X
a
S
g ( x)dx  N (d 2 )
(19.58)
where
t
X
S
1
tr   k2  log( )
log( )  (r   k2 )t
2
S  t 
X
2
d1 
k
t k
t k
(19.59)
S
1 2
log( )  (r   k )t
X
2
d2 
 d1  t  k
t k
52
(19.60)
• Substituting
eq. (19.58) into eq. (19.56), we get
C  SN (d1 )  X exp[ rt ]N (d 2 )
(19.61)
• This
53
is also Eq.(19.48) defined in Section 19.6
• Put
option is a contract conveying the right to sell
a designated security at a stipulated price.
• The relationship between a call option (C) and a
out option (P) can be shown as
C  Xe  rt  P  S
(19.62)
• Substituting
Eq. (19.33) into Eq. (19.34), the put
option formula becomes
P  Xe  rt N (d 2 )  SN (d1 )
(19.63)
*where S, C, r, t, 𝑑1 , 𝑑2 , are identical to those defined in the
call option model.
54
19.7 The Bivariate Normal Density
Function
• A joint
distribution of two variables is when in
correlation analysis, we assume a population where
both X and Y vary jointly.
• If both X and Y are normally distributed, then we
call this known distribution a bivariate normal
distribution.
55
• The
PDF of the normally distributed random
variables X and Y can be
f (X ) 
f (Y ) 
1
X
1
Y
 ( X   X ) 
exp 
 ,   X  
2
2
 2 X

 (Y  Y ) 
exp 
 ,   Y  
2
2
 2 Y 
(19.64)
(19.65)
 X and Y are population means for X and Y,
respectively; X and  Y are population standard
deviations of X and Y, respectively;  3.1416 ;and
exp represents the exponential function.
• Where
56

• If
 represents the population correlation between
X and Y, then the PDF of the bivariate normal
distribution can be defined as
f ( X ,Y ) 
1
2 X  Y 1   2
exp(q / 2),   X  ,   Y  
(19.66)
• Where  X  0,  Y  0
1   X   X
 
q
2 
1     X

2
and  1    1,

 X  X
  2  

 X
 Y   Y

  Y
  X  Y
  
  Y



2




(19.67)
57
•
It can be shown that the conditional mean of Y, given X, is
linear in X and given by
 Y
E (Y | X )  Y   
 X

( X   X )

(19.67)
This equation can be regarded as describing the population
linear regression line.
• Accordingly, a linear regression in terms of the bivariate
normal distribution variable is treated as though there were
a two-way relationship instead of an existing causal
relationship.
• It should be noted that regression implies a causal
relationship only under a “prediction” case.
•
58
• It
is also clear that given X, we can define the
conditional variance of Y as
 (Y | X )   (1   )
2
Y
2
(19.68)
• Eq.
(19.66) represents a joint PDF for X and Y.
• If   0 , then Equation (19.66) becomes
f ( X , Y )  f ( X ) f (Y )
• This
(19.69)
implies that the joint PDF of X and Y is equal
to the PDF of X times the PDf of Y. We also know
that both X and Y are normally distributed.
Therefore, X is independent of Y.
59
Example 19.1
Using a Mathematics Aptitude Test to
Predict Grade in Statistics
• Let
X and Y represent scores in a mathematics
aptitude test and numerical grade in elementary
statistics, respectively.
• In addition, we assume that the parameters in
Equation (19.66) are
 X  550  X  40 Y  80  Y  4
60
  .7
• Substituting
this information into Equations (19.67)
and (19.68), respectively, we obtain
E (Y | X )  80  .7(4 / 40)( X  550)  41.5  .07 X
(19.70)
 (Y | X )  (16)(1  .49)  8.16
2
(19.71)
61
•
If we know nothing about the aptitude test score of
a particular student (say, john), we have to use the
distribution of Y to predict his elementary statistics
grade.
95% interval  80  (1.96)(4)  80  7.84
• That
is, we predict with 95% probability that John’s
grade will fall between 87.84 and 72.16.
62
•
Alternatively, suppose we know that John’s
mathematics aptitude score is 650. In this case, we
can use Equations (19.70) and (19.71) to predict
John’s grade in elementary statistics.
E (Y | X  650)  41.5  (.07)(650)  87
And
 2 (Y | X )  (16)(1  .49)  8.16
63
• We
can now base our interval on a normal
probability distribution with a mean of 87 and a
standard deviation of 2.86.
95% interval  87  (1.96)(2.86)  87  5.61
• That
is, we predict with 95% probability that John’s
grade will fall between 92.61 and 81.39.
64
• Two
things have happened to this interval.
1. First, the center has shifted upward to take into
account the fact that John’s mathematics aptitude
score is above average.
2. Second, the width of the interval has been
narrowed from 87.84−72.16 = 15.68 grade points
to 92.61－81.39 = 11.22 grade points.
• In this sense, the information about John’s
mathematics aptitude score has made us less
uncertain about his grade in statistics.
65
19.8 American Call Options
• 19.8.1
Price American Call Options by the
Bivariate Normal Distribution
• An
option contract which can be exercised only on
the expiration date is called European call.
• If the contract of a call option can be exercised at
any time of the option's contract period, then this
kind of call option is called American call.
66
• When
a stock pays a dividend, the American call is
more complex.
• The valuation equation for American call option
with one known dividend payment can be defined
as C ( S , T , X )  S [ N (b )  N (a ,b ; t T )]
x
1
1
2
1
1
 Xe  rt [ N 1(b 2)e r (T t )  N 2( a 2,b 2; t T )]  De  rt N 1(b 2)
• where
S
ln 
X

a1 
x
 Sx
ln  *
St

b1 
67
 
1 2

r

 T
 
2 
 
, a 2  a1   T
 T
 
1 2

r

 t
 
2 
 
, b 2  b1   t
 t
(19.72a)
(19.72b)
(19.72c)
S x  S  De  rt
(19.73)
x
• S represents the correct stock net price of the
present value of the promised dividend per share
(D);
• t represents the time dividend to be paid.
*
• S t is the exdividend stock price for which
C(St* , T  t )  St*  D  X
(19.74)
X, r,  2, T have been defined previously in this
chapter.
• S,
68
•
Following Equation (19.66), the probability that is
less than a and that is less than b for the standardized
cumulative bivariate normal distribution
'2
' '
'2

 ' '
2
x

2

x
y

y
'
'
P ( X  a, Y  b) 
exp 
dx dy
2
2    
2(1   )
2 1  


1
x 
'
•
69
x  x
a
,y 
'
b
y  y
Where
and p is the correlation
x
y
between the random variables x’ and y’.
• The
first step in the approximation of the bivariate
normal probability N 2 (a, b;  ) is as follows:
5
5
 (a, b;  )  .31830989 1   2  wi w j f ( xi' , x 'j )
i 1 j 1
(19.75)
where
f ( xi' , x 'j )  exp[ a1 (2 xi'  a1 )  b1 (2 x 'j  b1 )  2 ( xi'  a1 )( x 'j  b1 )]
70
• The
pairs of weights, (w) and corresponding
abscissa values (x ' ) are
i, j
1
2
3
4
5
71
w
0.24840615
0.39233107
0.21141819
0.033246660
0.00082485334
x'
0.10024215
0.48281397
1.0609498
1.7797294
2.6697604
• and
the coefficients 𝑎1 and 𝑏1 are computed using
a1 
• The
a
2(1   )
2
b1 
b
2(1   2 )
second step in the approximation involves
computing the product ab𝜌; if ab𝜌 ≤ 0, compute
the bivariate normal probability, N 2 (a, b;  ) , using
certain rules.
72
• Rules:
• (1)
•
then N 2 (a, b;  )   (a, b;  ) ;
• (2)
•
If a ≥0, b ≤0, and 𝜌 >0,
then N 2 (a, b;  )  N1 (b)   (a, b;   ) ;
• (4)
•
If a ≤0, b ≥0, and 𝜌 >0,
then N 2 (a, b;  )  N1 (a)   (a, b;   ) ;
• (3)
•
If a ≤0, b ≤0, and 𝜌 ≤0,
If a ≥0, b ≥0, and 𝜌 ≤0,
Then N 2 (a, b;  )  N1 (a)  N1 (b)  1   (a, b;  ) .
(19.76)
73
• If
ab𝜌 > 0, compute the bivariate normal
probability,N 2 (a, b;  ) ,as
N 2 (a, b;  )  N 2 (a,0;  ab )  N 2 (b,0;  ab )  
(19.77)
the values of N 2 () on the right-hand side are
computed from the rules, for ab𝜌 ≤ 0
• where
 ab 
( a  b) Sgn(a )
a  2 ab  b
2
2
 ba 
1  Sgn(a)  Sgn(b)

4
•
a 2  2 ab  b 2
1
Sgn( x)  
 1
N1 (d ) is the cumulative univariate normal
probability.
74
( b  a ) Sgn(b)
x0
x0
19.8.2 Pricing an American Call Option
• An American
call option whose exercise price is
$48 has an expiration time of 90 days. Assume the
risk-free rate of interest is 8% annually, the
underlying price is $50, the standard deviation of
the rate of return of the stock is 20%, and the stock
pays a dividend of $2 exactly for 50 days.
(a) What is the European call value?
(b) Can the early exercise price predicted?
(c) What is the value of the American call?
75
(a)
The current stock net price of the present value of
the promised dividend is
S  50  2e
x
0.08( 50
)
365
 48.0218
The European call value can be calculated as
C  (48.0218) N (d1 )  48e
0.08( 90
)
365
N (d 2 )
where
[ln( 48.208 / 48)  (0.08  0.5(0.20) 2 )(90 / 365)]
d1 
 0.25285
.20 90 / 365
d 2  0.292  0.0993  0.15354.
76
• From
standard normal table, we obtain
N (0.25285)  0.5  .3438  0.599809
N (0.15354)  0.5  .3186  0.561014.
• So
77
the European call value is
C = (48.516)(0.599809) − 48(0.980)(0.561014)
= 2.40123.
(b) The present value of the interest income that
would be earned by deferring exercise until
expiration is
X (1  e r (T t ) )  48(1  e0.08(9050) / 365 )  48(1  0.991)  0.432.
Since d = 2> 0.432, therefore, the early exercise is
not precluded.
78
S t*= 46.9641. An Excel program used to calculate this value is presented in Table 19-1.
(c) The value of the American call is now calculated
as
C  48.208[ N1 (b1 )  N 2 (a1 ,b1 ; 50 90 )]
 48e 0.08(90 / 365) [ N1 (b2 )e 0.08( 40 / 365)  N 2 (a2 ,b2 ; 50 / 90 )]
 2e 0.08( 50 / 365) N1 (b2 )
(19.78)
since both and depend on the critical exdividend
stock price , which can be determined by
C ( S t* ,40 / 365;48)  S t*  2  48
using trial and error, we find thatS t* = 46.9641.
An Excel program used to calculate this value is
presented in Table 19-1.
• By
79
S t*
Table 19-1 Calculation of St*
• St*
80
(Critical ex-dividend stock price)
S*(critical exdividend
stock price)
46
46.962
46.963
46.9641
46.9
47
X(exercise price of option)
48
48
48
48
48
48
r(risk-free interest rate)
0.08
0.08
0.08
0.08
0.08
0.08
volalitity of stock
0.2
0.2
0.2
0.2
0.2
0.2
T-t(expiration date-exercise date)
0.10959
0.10959
0.10959
0.10959
0.10959
0.10959
d1
−0.4773
−0.1647
−0.1644
−0.164
−0.1846
−0.1525
d2
−0.5435
−0.2309
−0.2306
−0.2302
−0.2508
−0.2187
D(divent)
2
2
2
2
2
2
c(value of European call option to
buy one share)
0.60263
0.96319
0.96362
0.9641
0.93649
0.9798
p(value of European put option to
sell one share)
2.18365
1.58221
1.58164
1.58102
1.61751
1.56081
C(St*,T−t;X) −St*−D+X
0.60263
0.00119
0.00062
2.3E−06
0.03649
−0.0202
Caculation of St*(critical ex-dividend stock price)
Column C*
1*
2
3
4
5
6
7
8
S*(critical ex-dividend
46
stock price)
X(exercise price of
48
option)
r(risk-free interest
0.08
rate)
volatility of stock
0.2
T-t(expiration date=(90-50)/365
exercise date)
d1
=(LN(C3/C4)+(C5+C6^2/2)*(C7))/(C6*SQRT(C7))
9
d2
=(LN(C3/C4)+(C5-C6^2/2)*(C7))/(C6*SQRT(C7))
10
D(divent)
2
11
c(value of European
12 call option to buy one
share)
p(value of European
13 put option to sell one
share)
14
15 C(St*,T-t;X)-St*-D+X
81
=C3*NORMSDIST(C8)-C4*EXP(-C5*C7)*
NORMSDIST(C9)
=C4*EXP(-C5*C7)*NORMSDIST(-C9)C3*NORMSDIST(-C8)
=C12-C3-C10+C4
Sx = 48.208，X =＄48 and St* into
Equations (19.72b) and (19.72c), we can calculate
a1, a2, b1, and b2:
a1 = d1 =0.25285.
a2 = d2 =0.15354.
• Substituting
48.208
0.2 2 50
ln(
)  (0.08 
)(
)
46.9641
2 365  0.4859
b1 
(.20) 50 365
b2 = 0.485931–0.074023 = 0.4119.
82
• In
addition, we also know   
• From
50 90  0.7454.
the above information, we now calculate
related normal probability as follows:
N1（b1）= N1（0.4859）=0.6865
N1（b2）= N1（0.7454）=0.6598
83
• Following
Equation (19.77), we now calculate the
value of N2（0.25285,−0.4859; −0.7454）and N2
（0.15354, −0.4119; −0.7454）as follows:
abρ > 0 for both cumulative bivariate normal
density function, therefore, we can use Equation N2
（a, b;ρ）= N2（a, 0;ρab）+ N2（b, 0;ρba）-δ
• Since
calculate the value of both N2（a, b;ρ）as
follows:
• to
84
 ab 
 ba 
[( 0.7454)(0.25285)  0.4859](1)
(0.25285)  2(0.7454)(0.25285)( 0.4859)  (0.4859)
2
2
[( 0.7454)( 0.4859)  0.25285](1)
(0.25285)  2(0.7454)(0.25285)( 0.4859)  (0.4859)
2
2
 0.87002
 0.31979
δ =（1−（1）（−1））/4 = ½
N2（0.292,−0.4859; −0.7454）=N2（0.292,0.0844）+N2
（−0.5377,0.0656）− 0.5 = N1（0）+ N1（−0.5377）−Φ
（−0.292, 0; − 0.0844）−Φ（−0.5377,0; −0.0656）−0.5 =
0.07525
85
• Using
a Microsoft Excel programs presented in
Appendix 19A, we obtain
• N2（0.1927,
• Then
−0.4119; −0.7454）= 0.06862.
substituting the related information into the
Equation (19.78), we obtain C=＄3.08238 and all
related results are presented in Appendix 19B.
86
19.9 Price Bounds for Options
19.9.1 Options Written on Nondividend- Paying
Stocks
• To
derive the lower price bounds and the put–call
parity relations for options on nondividend-paying
stocks, simply set
cost-of-carry rate (b) = risk-less rate of interest (r)
• Note that, the only cost of carrying the stock is
interest.
87
• The
lower price bounds for the European call and
put options are
c(S , T ; X )  max[ 0, S  Xe  rT ]
p(S , T ; X )  max[ 0, Xe  rT  S ]
(19.79a)
(19.79b)
respectively, and the lower price bounds for the
American call and put options are
C (S , T ; X )  max[ 0, S  Xe
P(S , T ; X )  max[ 0, Xe
respectively.
88
 rT
 rT
]
(19.80a)
 S]
(19.80b)
• The
put–call parity relation for nondividend-paying
European stock options is
c(S , T ; X )  p(S , T ; X )  S  Xe  rT
(19.81a)
and the put–call parity relation for American options
on nondividend-paying stocks is
S  X  C (S , T ; X )  P(S , T ; X )  S  Xe  rT
• For
(19.81b)
nondividend-paying stock options, the
American call option will not rationally be
exercised early, while the American put option may
be done so.
89
19.9.2 Options Written on DividendPaying Stocks
• If
dividends are paid during the option's life, the
above relations must reflect the stock's drop in
value when the dividends are paid.
• To manage this modification, we assume that the
underlying stock pays a single dividend during the
option’s life at a time that is known with certainty.
• he dividend amount is D and the time to exdividend
is t.
90
• If
the amount and the timing of the dividend
payment are known, the lower price bound for the
European call option on a stock is
c(S , T ; X )  max[ 0, S  De  rt  Xe  rT ]
(19.82a)
• In
this relation, the current stock price is reduced by
the present value of the promised dividend.
• Because a European-style option cannot be
exercised before maturity, the call option holder has
no opportunity to exercise the option while the
stock is selling cum dividend.
91
• In
other words, to the call option holder, the current
value of the underlying stock is its observed market
price less the amount that the promised dividend
contributes to the current stock value, that is, S  De  .rt
• To
prove this pricing relation, we use the same
arbitrage transactions, except we use the reduced
stock price S  De  rt in place of S. The lower price
bound for the European put option on a stock is
p(S ,T ; X )  max[ 0, Xe rT  S  De  rt ]
(19.82b)
92
• In
the case of the American call option, for
example, it may be optimal to exercise just prior to
the dividend payment because the stock price falls
by an amount D when the dividend is paid.
• The lower price bound of an American call option
expiring at the exdividend instant would be 0 or ,
whichever is greater.
• On the other hand, it may be optimal to wait until
the call option’s expiration to exercise.
93
• The
lower price bound for a call option expiring
normally is (19.82a). Combining the two results,
we get
 rt
C (S , T ; X )  max[ 0, S  Xe , S  De
 rt
 Xe
 rT
]
(19.83a)
• The
last two terms on the right-hand side of
(19.83a) provide important guidance in deciding
whether to exercise the American call option early,
just prior to the exdividend instant.
• The second term in the squared brackets is the
present value of the early exercise proceeds of the
call.
94
• If
the amount is less than the lower price bound of
the call that expires normally, that is, if
S  Xe  rt  S  De  rT  Xe  rt
(19.84)
the American call option will not be exercised just
prior to the exdividend instant.
• To see why, simply rewrite (19.84) so it reads
D  X [1  e  r (T t ) ]
• In
(19.85)
other words, the American call will not be
exercised early if the dividend captured by
exercising prior to the exdividend date is less than
the interest implicitly earned by deferring exercise
until expiration.
95
•
Figure 19-3 depicts a case in which early exercise could
occur at the exdividend instant, t. Just prior to exdividend,
the call option may be exercised yielding proceeds S t  D  X,
where 𝑆𝑡 , is the exdividend stock price.
•
An instant later, the option is left unexercised with value
c(𝑆𝑡 ,T –t; X), where c is the European call option formula.
Thus, if the exdividend stock price, 𝑆𝑡 is above the critical
exdividend stock price where the two functions intersect, 𝑆𝑡∗ ,
the option holder will choose to exercise his or her option
early just prior to the exdividend instant.
∗
• On the other hand, if 𝑆𝑡 ≤ 𝑆𝑡 , the option holder will choose
to leave her position open until the option’s expiration.
•
96
Figure 19-3 *Early exercise may be optimal.
Figure 19-4 *Early exercise will not be optimal.
97
• Figure
19-4 depicts a case in which early exercise
will not occur at the exdividend instant, t.
• Early exercise will not occur if the functions, 𝑆𝑡 +
𝐷 − 𝑋 and c(𝑆𝑡 ,T-t,X) do not intersect, as is
depicted in Figure 19-4. In this case, the lower
 r (T  t )
S

Xe
boundary condition of the European call, t
,
lies above the early exercise proceeds, 𝑆𝑡 + 𝐷 − 𝑋 ,
and hence the call option will not be exercised
early. Stated explicitly, early exercise is not rational
if
S t  D  X  S t  Xe
98
 r (T t )
• This
condition for no early exercise is the same as
(19.84), where 𝑆𝑡 is the exdividend stock price and
where the investor is standing at the exdividend
instant, t.
• In words, if exdividend stock price decline, the
dividend is less than present value of the interest
income that would be earned by deferring exercise
until expiration, early exercise will not occur.
• When condition of Eq. (19.85) is met, the value of
American call is the value of corresponding
European call.
99
• In
the absence of a dividend, an American put may
be exercised early.
• In the presence of a dividend payment, however,
there is a period just prior to the exdividend date
when early exercise is suboptimal.
• In that period, the interest earned on the exercise
proceeds of the option is less than the drop in the
stock price from the payment of the dividend.
• If t 𝑛 represents a time prior to the dividend payment
at time t, early exercise is suboptimal, where
( X  S )e
100
 r ( t tn )
 ( X  S  D)
• Rearranging,
𝑡𝑛 and t if
early exercise will not occur between
D
ln( 1 
)
X S
tn  t 
r
(19.86)
• Early
exercise will become a possibility again
immediately after the dividend is paid. Overall, the
lower price bound of the American put option is
P(S , T ; X )  max[ o, X  (S  De  rt )]
(19.83b)
101
• Put–call
parity for European options on dividendpaying stocks also reflects the fact that the current
stock price is deflated by the present value of the
promised dividend, that is
c(S , T ; X )  p(S , T ; X )  S  De  rt  Xe  rT
(19.87)
• That
the presence of the dividend reduces the value
of the call and increases the value of the put is
again reflected here by the fact that the term on the
right-hand side of (19.87) is smaller than it would
be if the stock paid no dividend.
102
•
Put–call parity for American options on dividendpaying stocks is represented by a pair of inequalities,
that is
 rt
 rt
S  De  X  C (S , T ; X )  P(S , T ; X )  S  De  Xe
 rT
(19.88)
•
To prove the put–call parity relation (19.88), we
consider each inequality in turn. The left-hand side
condition of (19.88) can be derived by considering the
values of a portfolio that consists of buying a call,
selling a put, selling the stock, and lending X + 𝐷𝑒 −𝑟𝑡
risklessly. Table 19-2 contains these portfolio values
103
•
•
•
•
•
In Table 19-2, if all of the security positions stay open until
expiration, the terminal value of the portfolio will be positive,
independent of whether the terminal stock price is above or below
the exercise price of the options.
If the terminal stock price is above the exercise price, the call option
is exercised, and the stock acquired at exercise price X is used to
deliver, in part, against the short stock position.
If the terminal stock price is below the exercise price, the put is
exercised. The stock received in the exercise of the put is used to
cover the short stock position established at the outset.
In the event the put is exercised early at time T, the investment in the
riskless bonds is more than sufficient to cover the payment of the
exercise price to the put option holder, and the stock received from
the exercise of the put is used to cover the stock sold when the
portfolio was formed.
In addition, an open call option position that may still have value
104
Table 19-2 Arbitrage Transactions for Establishing
Put–Call Parity for American Stock Options
S  De  rt  X  C (S , T ; X )  P(S , T ; X )
ExDividend
Day(t)
Position
Buy American Call
Sell American Put
Sell Stock
Lend D e  rt
Lend X
Net Portfolio Value
105
Initial Value
Put Exercised
Early(γ)
Intermediate
Value
Put Exercised
normally(T)
Terminal Value
~
ST  X
~
−C
0
C
~
P
S
−D
 De  rt
−X
−C+P+S
 De  rt −X
D
~
 (X  S  )
 (X  ST )
~
~
 S
~
ST  X
~
ST  X
0
~
 ST
Xe rT
 ST
Xe rT
X (e rT  1)
X (e rT  1)
Xe r
0
~
C  X (e r  1)
• In
other words, by forming the portfolio of
securities in the proportions noted above, we have
formed a portfolio that will never have a negative
future value.
• If the future value is certain to be non-negative, the
initial value must be nonpositive, or the left-hand
inequality of (19.88) holds.
106
Summary
•
In this chapter, we first introduced univariate and multivariate normal distribution
and log-normal distribution.
•
Then we showed how normal distribution can be used to approximate binomial
distribution.
•
Finally, we used the concepts normal and log-normal distributions to derive Black–
Scholes formula under the assumption that investors are risk neutral.
•
•
In this chapter, we first reviewed the basic concept of the Bivariate normal density
function and present the Bivariate normal CDF.
•
The theory of American call stock option pricing model for one dividend payment
is also presented.
•
The evaluations of stock option models without dividend payment and with
dividend payment are discussed, respectively.
•
Finally, we provided an excel program for evaluating American option pricing
model with one dividend payment.
107