There exists a big zoo of options of which the boundary conditions
of the Black-Scholes differential equation are too complex to
solve analytically. An example is the American option. For this
reason one has to rely on numerical price computation. The best
known methods approximate the stock price process by a discrete
time stochastic process, or, as in the approach followed by Cox,
Ross, Rubinstein, model the stock price process as a discrete time
process from the start. By doing this, the options time to
maturity
is decomposed into
equidistant time steps of
length
We consider therefore the discrete time points
By
we denote the stock price at time
At the same
time, we discretize the set of values the stock price can take
such that it takes on the finitely many values
with
denoting the point of time and
representing the value. If the stock price is in time
equal
to
then it can jump in the next time step to one of
new states
The
probabilities associated to these movements are denoted by
:
with
If we know the stock price at the current time, we can build up a
tree of possible stock prices up to a certain point of time, for
example the maturity date
Such a tree is also called stock price tree. Should the option price
be known at the final point of time
of the stock price tree,
for example by means of the options intrinsic value, the option
value at time
can be computed (according to
(6.24)) as the discounted conditional expectation of the
corresponding option prices at time
given the stock price at
time
again denotes the option value at time
if the underlying has a price of
Repeating this step for the remaining time steps
the option
prices up to time
can be approximated.