UNIVERSITY OF HAIFA
DEPARTMENT OF ECONOMICS
AVNER BAR-ILAN
European Options
The value of a European option at time t, when the stock price is P, is
(1) V ( P, t )  e  r (T t ) EV P, T .
T is the expiration date, r is the risk free interest rate, and E denotes
expectations taken at time t.
In order to find the value of the option at time t, we need to specify the
value of the option at the expiration date. For a call option this is
(2) V ( P, T )  0P (T )c
forP(T )  c
otherwise
where P(T) is the stock price at time T and c is the exercise price. V(P,T) for
a put option is
(3) V ( P, T )  c0 P (T )
forP(T )  c
otherwise
Substituting (2) and (3), respectively, into equation (1) yields the following
expressions for the value of a call option, equation (4), and put option,
equation (5),

(4) V ( P, t )  e r (T t )  ( P(T )  c) f ( P(T )) dP(T )
c
c
(5) V ( P, t )  e  r (T t )  (c  P(T )) f ( P(T )) dP(T )

Where f(P(T)) is the probability density function of the price P(T). Assume
that the price follows a geometric Brownian motion,
(6) dP / P  rdt  dz
where dz is an element of a standard Wiener process. That is, the price
P(),  t, given price P at time t, is distributed log-normally as


2
(7) log P( ) ~ n log P  (r 
)(  t ),  2 (  t ) .
2


where n denotes the normal probability density function.
The following property of the log-normal distribution is of help. *When
logx~n(g,s2) and m is a parameter then
x0
(8)
x
m
f ( x)dx  N (u  ms)e
mg  m 2 S 2
2


(9)
x
m
f ( x)dx  1  N (u  ms)e
mg  m 2 S 2
2
x0
where N denotes the cumulative standard normal distribution and u is
defined as
(10) u=(logX0-g)/s.
Using equations (8) – (10) equation (4) yields the value of a call option,
(11) V ( P, t )  PN ( T  t  u )  e  r (T t ) cN (u )
where u is given by


(12) u  log( c / P)  (r   2 )(T  t ) /  T  t ,
2
which is the Black-Scholes formula. Similarly, equation (5) gives the BlackScholes solution for a put option as,
(13) V ( P, t )  e  r (T t ) cN (u )  PN (u   T  t ).
*
See Aitchison and Brown (1957) or Johnston and Kotz (1970).