Appendix 27A:
An Alternative
Method to Derive
The Black-Scholes
Option Pricing
Model
(Related to 27.5)
By
Cheng Few Lee
Joseph Finnerty
John Lee
Alice C Lee
Donald Wort
Appendix 27A: An Alternative Method to Derive The
Black-Scholes Option Pricing Model
•
•
•
•
2
Perhaps it is unclear why it is assumed that investors have riskneutral preferences when the usual assumption in finance
courses is that investors are risk averse.
It is feasible to make this simplistic assumption because
investors are able to create riskless portfolios by combining call
options with their underlying securities.
Since the creation of a riskless hedge places no restrictions on
investor preferences other than nonsatiation, the valuation of the
option and its underlying asset will be independent of investor
risk preferences.
Therefore, a call option will trade at the same price in riskneutral economy as it will in a risk-averse or risk-preferent
economy.
Appendix 27A.1:
Assumptions and
the Present Value of
the Expected
Terminal Option
Price
By
Cheng Few Lee
Joseph Finnerty
John Lee
Alice C Lee
Donald Wort
Appendix 27A.1: Assumptions and the Present Value
of the Expected Terminal Option Price
•
•
•
In the risk-neutral assumptions of Cox and Ross (1976) and
Rubinstein (1976), today’s option price can be determined by
discounting the expected value of the terminal option price by
the riskless rate of interest.
Today’s call option price is:
(27A.1)
C  exp  rt  Max  St  X ,0
where
C  the market value of the call option;
r  riskless rate of interest;
t  time to expiration;
St  the market value of the underlying stock at time t; and
X  exercise or striking price.
4
Appendix 27A.1: Assumptions and the Present Value
of the Expected Terminal Option Price
•
Assuming that the call option expires in the money, then the present
value of the expected terminal option is equal to the present value of
the difference between the expected terminal stock price and the
exercise price as:
C  exp  rt  E  Max  St  X , 0  
(27A.2)
 exp  rt  

x
 St  X  h  St  dSt
where h  St  is the log-normal density function of St
• To evaluate the integral in (27A.2), rewrite it as the difference between
two integrals:



C  exp  rt   St h  St  dSt  X  h  St  dSt 
(27A.3)
 x

x
 Ex  St   exp  rt   X  exp  rt   1  H  X 
Ex  St   the partial expectation of St , truncated from below at x; and
H  X   the probability that St  X
5
Appendix 27A.1: Assumptions and the Present Value
of the Expected Terminal Option Price
•
The terminal stock price can be rewritten as the product of the
current price and the t-period log-normally distributed price
ratio. So, Equation (27A.3) can also be rewritten as:

  St  St  dSt 
 St  dSt   (27A.4)
C  exp   rt   S 
g  
  X x s g  S  S  
x s S
S
S
 

 


S
 S exp  rt  Ex S  t
S


X

X
exp

rt
1

G
 


S





where
S
g t
S
S
Ex S  t
S

  log normal density function of St S ;


  the partial expextation of St S , truncated from below at x S ;

X
G    the probability that St S  X S .
S 
6
Appendix 27A.2:
Present Value of the
Partial Expectation
of the Terminal
Stock Price
By
Cheng Few Lee
Joseph Finnerty
John Lee
Alice C Lee
Donald Wort
Appendix 27A.2: Present Value of the Partial
Expectation of the Terminal Stock Price
•
•
The right-hand side of Equation (27A.4) is evaluated by
considering the two integrals separately.
The first integral, S exp  rt  Ex S  St S  , can be solved by assuming
the return on the underlying stock follows a stationary random
walk:
St
 exp  Kt 
(27A.5)
S
S
ln  t
S
•
8

   Kt 

Following Garven (1986), it can be transformed into a density
function of a normally distributed variable Kt according to the
relationship St S  exp  Kt  as:
(27A.6)
 St 
 St 
g    f  Kt   
S
S
Appendix 27A.2: Present Value of the Partial
Expectation of the Terminal Stock Price
•
These transformations allow the first integral in Equation
(27A.4) to be rewritten as:

 St 
S exp  rt  Ex S    S exp  rt  
f  Kt  exp  Kt  t dK
ln
x
S


S

12
2
2
f  Kt   2 K t
exp  12  Kt  K t   K2 



•

Substitution yields:
 St
S exp  rt  Ex S 
S
1 2

2
  S exp  rt   2 K t 


2
exp  Kt  exp   12  Kt   K t   K2 t  t dK
ln  x S 


(27A.7)

•
9
Equation (27A.7)’s integrand can be simplified by adding the
terms in the two exponents, multiplying and dividing the result
by exp   12  K2 t 
Appendix 27A.2: Present Value of the Partial
Expectation of the Terminal Stock Price
First, expand the term  Kt   K t  and factor out t so that:
2
exp  Kt  exp  12  Kt  K t   K2 t 


Next, factor out t so:
exp  Kt  exp  12 t  K 2  2K K  K2   K2 
2


Now combine the two exponents:


exp  12 t ( K 2  2 K K   K2  2 K2 K )  K2 
Now, multiply and divide this result byexp   12  K2 t  to get:


exp  12 t ( K 2  2 K K   K2  2 K2 K   K4   K4 )  K2 
Next, rearrange and combine terms to get:

2 2

exp   t   K   K   K    K4  2 K  K2   K2


1
2
 exp   K  
1
2
10
2
K
 t  exp 
1
2
 Kt    K  

2
K
 t 

2
 K2 t

(27A.8)
Appendix 27A.2: Present Value of the Partial
Expectation of the Terminal Stock Price
In Equation (27A.8), exp      t   E  S S 
K
1
2
2
K
t
Therefore, Equation (27A.7) becomes:
(27A.9)
S 
S exp  rt  Ex S  t 
S
2

1 2
 St 
2
2
1 
 S E   exp  rt   2 K t   
exp ( Kt ) exp  2  Kt    K   K  t   K2 t
ln  x S 
S

•
•
Since the equilibrium rate of return in a risk-neutral economy is
the riskless rate:
S 
S E  t  exp  rt   S exp  rt  exp  rt   S
S
So Equation (27A.9) becomes
 St 
(27A.10)
S exp  rt  Ex S  
S
 S  2 t 
2
K
11
1 2

exp   Kt    K  
ln  x S 


1
2
2
K
 t 
2

 K2 t t dK

Appendix 27A.2: Present Value of the Partial
Expectation of the Terminal Stock Price
•
To complete the simplification of this part of the Black–Scholes
formula, define a standard normal random variable y:
y   Kt    K   K2  t   K2 t1 2
Kt    K   K2  t   K t1 2 y,
•
The lower limit of integration becomes:
•
Further simplify the integrand by noting that the assumption of a
risk-neural economy implies:
ln  x S    K   K2  t   K t1 2


exp   K  12  K2  t   exp  rt  ,
•
12
t dK   K t1 2 dy
Hence,

K

K
 12  K2  t   rt 
 12  K2  t   r  12  K2  t
Appendix 27A.2: Present Value of the Partial
Expectation of the Terminal Stock Price
•
The lower limit of integration is now:
 ln  S x    r  12  K2  t   K t1 2  d1
•
•
Substituting this into Equation (27A.10) and making the
transformation to y yields:

2
12
 St 
S exp  rt  Ex S    S  exp   12 y   2  dy
 d1
S
Since y is a standard normal random variable (distribution is
symmetric around zero) the limits of integration can be exchanged:
S
S exp  rt  Ex S  t
S
13
 d1

1 2

S
exp

y 

2




 S N  d1 
 2 
12
dy
(27A.11)
Appendix 27A.3:
Present Value of the
Exercise Price under
Uncertainty
By
Cheng Few Lee
Joseph Finnerty
John Lee
Alice C Lee
Donald Wort
Appendix 27A.3: Present Value of the Exercise Price
under Uncertainty
•
Start by making the logarithmic transformation:
•
The differential can be written as:
S
d t  exp  Kt  t dK
S
Therefore,

X exp  rt  1  G  X S    X exp  rt  
f  Kt  t dK 
ln  X S 
•
S
ln  t
S
X exp  rt   2 t 
2
K
•
1 2


ln X

  Kt

2
exp   12  Kt   K t   K2 t  t dK
S


The integrand is now simplified by following the same
procedure used in simplifying the previous integral. Define a
standard normal random variable Z:
 Kt   K t 
Z 
12 
  Kt

15
(27A.12)
Appendix 27A.3: Present Value of the Exercise Price
under Uncertainty
•
Making the transformation from Kt to Z means the lower
limit of integration becomes:
ln  X S   K t
 K t1 2
•
Again, note that the assumption of a risk-neutral economy
implies:
2
exp   K  12  K t   exp  rt 
•
Taking the natural logarithm of both sides yields:

•
K
 12  K2  t  rt ,
 K t   r  12  K2  t
Therefore, the lower limit of integration becomes:
 ln  S x    r  12  K2  t 
 K t1 2
16
   d1   K t1 2    d 2
Appendix 27A.3: Present Value of the Exercise Price
under Uncertainty
• Substitution yields:

x exp  rt  1  G  X S    x exp  rt   exp   12 Z 2
 d2

 x exp  rt   exp   12 Z 2


d2
 dZ  x exp  rt  N  d 2 

(27A.13)
•
Substituting, Equations (27A.11) and (27A.13) into Equation
(27A.4) completes the derivation of the Black–Scholes formula:
(27A.14)
C  S N  d1   X exp  rt  N  d2 
•
This appendix provides a simple derivation of the Black–Scholes
call-option pricing formula.
Under an assumption of risk neutrality the Black–Scholes
formula was derived using only differential and integral calculus
and a basic knowledge of normal and log-normal distributions.
•
17
 2 
12
12
2

   dZ