The American option price can only be determined numerically. Similar to the European options, the binomial model after Cox-Ross-Rubinstein can be used. In this section we introduce a less complex but numerically efficient approach based on trinomial trees, see Dewynne et al. (1993). It is related to the classical numerical procedures for solving partial differential equations, which are also used to solve the Black-Scholes differential equations.
The trinomial model (see Section 4.2) follows
the procedure of the binomial model whereby the price at each time
point
can change to three instead of
two directions with
, see Figure
8.4. The value
at time
can reach to
the values
at
, where
are the suitable parameters
of the model. The probability with which the price moves from
to
is represented as
The price process
in discrete time is also a
trinomial process, i.e. the logarithms of the price
is an ordinary trinomial process with possible
increments
As in the binomial model three conditions must be fulfilled: The
sum of the probabilities
is one, the expectation
and variance of the logarithms increments
must be the same
as those of the logarithms of the geometric Brownian motion over
the time interval
. From these conditions we get three
equations:
a.) The first approach requires that a time step of in the trinomial model corresponds to two time steps in the
binomial model:
represents two upwards increments,
two
downwards increments and
one upward and one downward
increment (or reversed). The binomial model fulfills the
recombination condition
. Since now the length of the
time step is
, it holds following Section
7.1
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American options differ from the European in that the options can
be exercised at any time
. Consequently the
value of a call falls back to the intrinsic value if it is early
exercised:
Mathematically we have solved the free boundary problem, which is only numerically possible.
denotes the option value at time
if the spot price
of stock is
. As in the binomial model on European
options we use
to denote the discounted expectation that
is calculated from the prices attainable in the next time step,
and
. Different from
the European options, the expectation of American options may not
fall under the intrinsic value. The recursion of American call
price is thus:
The American put is on the other hand more valuable than the
European. With the parameters from Table 8.3 one gets
and
.
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b.) In the second approach the trinomial parameters
and
are certainly decided through additional conditions.
Here a certain upwards trend is shown in the whole price tree
since we replace the condition
by
Furthermore we assume and receive therefore together
with the four above-mentioned conditions:
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We consider now a European option. Here the trinomial model
delivers the following recursion for the possible option value
dependent on the probabilities and the change rates
:
We consider
for all
and we put
as well as
The recursion (8.21) for the option value becomes
then
This is the explicit difference approximation of the parabolic
differential equation (6.15), see Samaskij (1984). The
condition (8.20) corresponds to the well-known stability
requirement for explicit difference operations. Compared to the
previously discussed approach, the probabilities in the
variant of the trinomial models and the calculation in
(8.21) are not dependent on the volatility. The recursion
(8.21) depends only on the starting condition, i.e.
at the exercise moment as an observation depends on
from the geometric Brownian motion.