In
this section we are going to use Black-Scholes' equation to
compute the price of European options. We keep the notation
introduced in the previous chapter. That is, we denote
the value of a European call respectively put option with exercise
price
and maturity date
at time
where the
underlying, for example a stock, at time
has a value of
The value of a call option thus satisfies for all prices
with
the differential equation
The first boundary condition (6.18) follows directly from
the definition of a call option, which will only be exercised if
thereby procuring the gain
The definition of
Brownian motion implies that the process is absorbed by zero. In
other words, if
for one
it follows
That is
the call will not be exercised, which is formulated in the first
part of condition (6.19). Whereas the second part of
(6.19) results from the reflection that the probability
that the Brownian motion falls below
is fairly small if it
attained a level significantly above the exercise price. If
for a
then it holds with a high probability that
The call will be, thus, exercised and procures the cash
flow
.
The differential equation (6.17) subject to boundary
conditions (6.18). (6.19) can be solved
analytically. To achieve this, we transform it into a differential
equation known from the literature. First of all, we substitute
the time variable
for the time to maturity
By
doing this, the problem with final condition (6.18) in
changes to a problem subject to an initial condition in
Following, we multiply (6.17) by
and substitute the parameters
for
as well as the variables
for
While for the original parameters hold
for now the new parameters it holds
Finally, we set
and obtain the new differential equation
 |
(7.20) |
with the initial condition
 |
(7.21) |
Problems with initial conditions of this kind are well known from
the literature on partial differential equations. They appear, for
example, in modelling heat conduction and diffusion processes. The
solution is given by
The option price can be obtained by undoing
the above variables and parameter substitutions. In the following
we denote, as in Chapter 2, by
the
call option price being a function of the time to maturity
instead of time
Then it holds
Substituting
we obtain the original terminal condition
Furthermore, replacing
and
by the variables
and
we obtain
In the case of Brownian motion
is lognormally
distributed, i.e.
is normally distributed with
parameters
and
The
conditional distribution of
given
is therefore
lognormal as well but with parameters
and
However, the
integrant in equation (6.22) is except for the term
the density of the latter distribution. Thus, we can
interpret the price of a call as the discounted expected option
payoff
which is the terminal condition, given
the current stock price
![$\displaystyle C(S,\tau) = e^{-r\tau} \mathop{\text{\rm\sf E}}[ \max ( 0,S_T-K) \, \vert \, S_t = S] .$](sfehtmlimg941.gif) |
(7.23) |
This property is useful when deriving numerical methods to compute
option prices. But before doing that, we exploit the fact that
equation (6.22) contains an integral with respect to the
density of the lognormal distribution to further simplify the
equation. By means of a suitable substitution we transform the
term in an integral with respect to the density of the normal
distribution and we obtain
 |
(7.24) |
where we use
as a shortcut for
 |
(7.25) |
denotes the standard normal distribution
Equations (6.24) and (6.25) are called the Black-Scholes Formulae. For the
limit cases
and
it holds:
- If
then
and thus
It follows that the value of a call option on a non dividend paying stock,
approaches
That is, it can be approximated by the current stock price minus the discounted
exercise price.
- If
then
and therefore
Thus the option is worthless:
The corresponding Black-Scholes Formula for the price
of a European put option can be derived by solving Black-Scholes
differential equation subject to suitable boundary conditions.
However, using the put-call parity (Theorem 2.3) is more
convenient:
From this and equation (6.24) we obtain
 |
(7.26) |
As we see the value of European put and call options can be
computed by explicit formulae. The terms in equation
(6.24) for, say the value of a call option, can be
interpreted in the following way. Restricting to the case of a non
dividend paying stock,
the first term,
represents the value of the stock which the option
holder obtains when he decides to exercise the option. The other
term,
represents the value of the exercise
price. The quotient
influences both terms via the variable
Deriving Black-Scholes' differential equation we saw in particular that the value of a
call option had been duplicated by means of bonds and stocks. The amount of money
invested in stocks was
with
being the hedge ratio. This ratio, also called delta, determines the relation of
bonds and stocks necessary to hedge the option position. Computing the first derivative
of Black-Scholes' formula in equation (6.24) with respect to S we obtain
Thus the first term in equation (6.24) reflects the amount of money of the
duplicating portfolio invested in stocks, the second term the amount invested in bonds.
Since the standard normal distribution can be evaluated only numerically, the
implementation of Black-Scholes' formula depending on the standard normal distribution
requires an approximation of the latter. This approximation can have an impact on the
computed option value. To illustrate we consider several approximation formulae (see for
example Hastings (1955))
a.) The normal distribution can be approximated in the following way:

where
The approximating error is independently of
of size
.
SFENormalApprox1.xpl
b.)

where
The error of this approximation is of size
.
SFENormalApprox2.xpl
c.) An approximation of the normal distribution, with error
size
is given by:

where
SFENormalApprox3.xpl
d.) Finally we present the Taylor expansion:
By means of this series the normal distribution can be approximated arbitrarily close
depending on the number of terms used in the summation. Increasing the number of terms
increases as well the number of arithmetic operations.
SFENormalApprox4.xpl
Table 6.1 compares all four approximation formulae. The
Taylor series was truncated after the first term whose absolute
value is smaller than
The last column shows the number
of terms used.
Table 6.1:
Several approximations to the normal distribution
 |
norm-a |
norm-b |
norm-c |
norm-d |
iter |
1.0000 |
0.8413517179 |
0.8413447362 |
0.8413516627 |
0.8413441191 |
6 |
1.1000 |
0.8643435425 |
0.8643338948 |
0.8643375717 |
0.8643341004 |
7 |
1.2000 |
0.8849409364 |
0.8849302650 |
0.8849298369 |
0.8849309179 |
7 |
1.3000 |
0.9032095757 |
0.9031994476 |
0.9031951398 |
0.9031993341 |
8 |
1.4000 |
0.9192515822 |
0.9192432862 |
0.9192361959 |
0.9192427095 |
8 |
1.5000 |
0.9331983332 |
0.9331927690 |
0.9331845052 |
0.9331930259 |
9 |
1.6000 |
0.9452030611 |
0.9452007087 |
0.9451929907 |
0.9452014728 |
9 |
1.7000 |
0.9554336171 |
0.9554345667 |
0.9554288709 |
0.9554342221 |
10 |
1.8000 |
0.9640657107 |
0.9640697332 |
0.9640670474 |
0.9640686479 |
10 |
1.9000 |
0.9712768696 |
0.9712835061 |
0.9712842148 |
0.9712839202 |
11 |
2.0000 |
0.9772412821 |
0.9772499371 |
0.9772538334 |
0.9772496294 |
12 |
|
Table:
Prices of a European call option for different approximations of the normal
distribution
SFEBSCopt1.xpl
Stock price  |
|
230.00 |
EUR |
|
Exercise price  |
|
210.00 |
EUR |
|
Time to maturity
 |
|
0.50000 |
|
|
Continuous interest rate  |
|
0.04545 |
|
|
Volatility  |
|
0.25000 |
|
|
No dividends |
|
|
|
|
|
norm-a |
norm-b |
norm-c |
norm-d |
Option prices |
30.74262 |
30.74158 |
30.74352 |
30.74157 |
|
Table 6.2 shows the price of a particular European call option computed by
means of the four approximations presented above.