As the simplest example to price an option we consider the
approach of Cox, Ross and Rubinstein (CRR) which is based on the assumption of a binomial model, and which can be interpreted as a numerical
method to solve the Black-Scholes equation. We treat exclusively
European options and assume for the time being that the underlying
pays no dividends within the time to maturity. Again, we
discretize time and consider solely the points in time
with
The binomial model proceeds from the assumption
that the discrete time stock price process
follows a
geometric random walk (see Chapter 4), which is
the discrete analogue of the geometric Brownian motion. The
binomial model has the feature that at any point of time the stock
price has only two possibilities to move:
- either the price moves at rate
and with probability
in one direction (for example it
moves
p)
- or the price moves at rate
and with probability
in another direction (for example
it moves
own).
Using the notation introduced above, if the stock price in time
is equal to
then in time
it can take only the values
and
The probabilities
and
are independent of
All other probabilities
associated to
and
are
In order to approximate the Black-Scholes differential equation
by means of the Cox-Ross-Rubinstein approach, the probabilities
as well as the rates
have to be chosen such that in
the limit
the binomial model converges to
a geometric Brownian motion. That is, arguing as in (6.23)
the conditional distribution of
given
must be
asympotically a normal distribution with expectation parameter
and variance
parameter
However, the conditional
distribution of
given
implied by the binomial
model is determined by
and
their associated probabilities
and
We set the parameters
of the geometric random walk such that the conditional
expectations and variances implied by the binomial model are equal
to their asymptotic values for
Taking
into account that
we obtain three equations for the four
unknown variables
and
Due to the first equation, the current stock price
disappears from the remaining
equations. By substituting
into the latter two equations, we obtain, after some
rearrangements, two equations and three unknown variables:
To solve this nonlinear system of equations we introduce a further condition
i.e. if the stock price moves up and subsequently down, or down
and subsequently up, then it takes its initial value two steps
later. This recombining feature is more than only intuitively
appealing. It simplifies the price tree significantly. At time
there are only
possible values the stock price
can take. More precisely, given the starting value
at
time
the set of possible prices at time
is
because it holds
and
In the
general case there would be
possible states since the not only the number of
up and down movements would determine the final state but also the order of up and down
movements.
Solving the system of three equations for
and neglecting terms being small
compared to
it holds approximatively:
 |
(8.2) |
For the option price at time
and a stock price
we use the shortcut
As in equation (7.1) we obtain the option price at
time
by discounting the conditional expectation of the option price at time
 |
(8.3) |
At maturity
the option price is known. In case of a European option we have
 |
(8.4) |
Beginning with equation (7.1) and applying equation (7.3)
recursively all option values
can be
determined.
Example 8.1
An example of a call option is given in Table
7.1.
First the tree of stock prices is computed. Since

it follows from equation (
7.2) that

Given the current stock price

the stock
can either increase to

or decrease to

after the first time step. After the
second time step, proceeding from state

the stock
price can take the values

or

, proceeding from

it can move to

or

and so on. At maturity, after 5 time
steps, the stock price

can take the following
six
values
Following, given the tree of stock prices, we compute the option price at maturity
applying equation (7.4), for example
or
, since
Equation (7.2) implies
, since the cost of carry
are equal to the risk free interest rate
when no dividends are paid. Proceeding from the options' intrinsic values at maturity we
compute recursively the option values at preceding points of time by means of equation
(7.3). With
we obtain the option value
at time
corresponding to a stock price
by substituting the known values of
Analogously we obtain the
option value
at time
and current stock price
by
means of equation (7.3) and the time
option values
Using only 5 time steps
is just a rough approximation to
the theoretical call value. However, comparing prices implied by
the Black-Scholes formula (6.24) to prices implied by the
Cox-Ross-Rubinstein approach for different time steps
the
convergence of the numerical binomial model solution to the
Black-Scholes solution for increasing
is evident (see Table
7.2).
Table 7.1:
Evolution of option prices (no dividend paying underlying)
Current stock price  |
230.00 |
Exercise price  |
210.00 |
Time to maturity  |
0.50 |
Volatility  |
0.25 |
Risk free rate  |
0.04545 |
Dividend |
none |
Time steps |
5 |
Option type |
European call |
Stock prices |
Option prices |
341.50558 |
|
|
|
|
|
131.506 |
315.54682 |
|
|
|
|
106.497 |
|
291.56126 |
|
|
|
83.457 |
|
81.561 |
269.39890 |
|
|
62.237 |
|
60.349 |
|
248.92117 |
|
44.328 |
|
40.818 |
|
38.921 |
230.00000 |
30.378 |
|
26.175 |
|
20.951 |
|
212.51708 |
|
16.200 |
|
11.238 |
|
2.517 |
196.36309 |
|
|
6.010 |
|
1.275 |
|
181.43700 |
|
|
|
0.646 |
|
0.000 |
167.64549 |
|
|
|
|
0.000 |
|
154.90230 |
|
|
|
|
|
0.000 |
Time |
0.00 |
0.10 |
0.20 |
0.30 |
0.40 |
0.50 |
SFEBiTree.xpl
|
Table 7.2:
Convergence of the price implied by the binomial model to the price implied by
the Black-Scholes formula
Time steps |
5 |
10 |
20 |
50 |
100 |
150 |
Black-Scholes |
Option value |
30.378 |
30.817 |
30.724 |
30.751 |
30.769 |
30.740 |
30.741 |
|
The numerical procedure to price an option described above does not change if the
underlying pays a continuous dividend at rate
It is sufficient to set
instead of
for the cost of carry. Dividends paid at discrete points of time,
however, require substantial modifications in the recursive option price computation,
which we going to discuss in the following section.
Example 8.2
We consider a call on US-Dollar with a time to maturity of

months, i.e.

years, a current exchange rate of

EUR/USD and an exercise price

EUR/USD. The continuous dividend yield, which corresponds to the US interest rate,
is assumed to be

, and the domestic interest rate is

. It follows that the cost
of carry being the difference between the domestic and the foreign interest rate is equal
to

Table
7.3 gives as in the previous example the
option prices implied by the binomial model.
Table 7.3:
Evolution of option prices (with continuous dividends)
Current EUR/ USD-price |
1.50 |
Exercise price |
1.50 |
Time to maturity |
0.33 |
Volatility |
0.20 |
Risk free interest rate |
0.09 |
Continuous dividend  |
0.01 |
Time steps |
6 |
Option type |
European call |
Price |
Option prices |
1.99034 |
|
|
|
|
|
|
0.490 |
1.89869 |
|
|
|
|
|
0.405 |
|
1.81127 |
|
|
|
|
0.324 |
|
0.311 |
1.72786 |
|
|
|
0.247 |
|
0.234 |
|
1.64830 |
|
|
0.180 |
|
0.161 |
|
0.148 |
1.57240 |
|
0.127 |
|
0.105 |
|
0.079 |
|
1.50000 |
0.087 |
|
0.067 |
|
0.042 |
|
0.000 |
1.43093 |
|
0.041 |
|
0.022 |
|
0.000 |
|
1.36504 |
|
|
0.012 |
|
0.000 |
|
0.000 |
1.30219 |
|
|
|
0.000 |
|
0.000 |
|
1.24223 |
|
|
|
|
0.000 |
|
0.000 |
1.18503 |
|
|
|
|
|
0.000 |
|
1.13046 |
|
|
|
|
|
|
0.000 |
Time |
0.00 |
0.06 |
0.11 |
0.17 |
0.22 |
0.28 |
0.33 |
SFEBiTree.xpl
|