his is a short summary of
the technique presented in
[Longstaff]
.
We would like to calculate the
quantity
where
is a stochastic process in
holding all the state variables, the
is some deterministic function representing the interest rate term structure,
is the known payoff function depending on the path
up to the moment of exercise
.
The functional dependence of the moment of exercise
on the state variables
is the subject of optimization.
Suppose the
is the result of MonteCarlo simulation of the stochastic process
with
being the time index,
being the simulation index and
being the dimension index,
is the result of immediate exercise
,
is the discount factor between neighbor indexes
.
Introduce an array
and a set of functions
acting
.
Start with
For
do the
following:
The answer is
.
We do not proceed to step k=0 because the cross sectional information
collapses to a point at this step. The obtained this way value is biased high
because this is a forward looking procedure. If we continue the MC simulation
on the obtained strategy
then we get a biased low value because the strategy is suboptimal.
The motivation for the steps above is the following. The
is the value of the quantity in question at time
given the information
. Hence, the starting condition is obvious. The sum
is used to construct the function that depends only on available information
and best approximates the Y. Hence, we discount the Y from the previous step
with (a), find the best approximation in (b), chose the best strategy and
calculate the new Y in (c) and (d).
The step (b) may be performed using the Normal Equations technique presented
in the next section.
The step (b) is unstable if the time step is small and
is close to the origin. In such situation the
does not contain much
cross
information because all the
originate from a single point
and did not have time to evolve. For this reason the procedure is not
effective if the exercise is immediately possible.
