Information Technology Reference
In-Depth Information
Chapter 5
American Options
Pricing American contracts requires, due to the early exercise feature of such con-
tracts, the solution of optimal stopping problems for the price process. Similar to
the pricing of European contracts, the solutions of these problems have a deter-
ministic characterization. Unlike in the European case, the pricing function of an
American option does not satisfy a partial differential equation, but a partial differ-
ential inequality, or to be more precise, a system of inequalities. We consider the
discretization of this inequality both by the finite difference and the finite element
method where the latter is approximating the solutions of variational inequalities.
The discretization in both cases leads to a sequence of linear complementarity prob-
lems (LCPs). These LCPs are then solved iteratively by the PSOR algorithm. Thus,
from an algorithmic point of view, the pricing of an American option differs from
the pricing of a European option only as in the latter we have to solve linear systems,
whereas in the former we need to solve linear complementarity problems. The cal-
culation of the stiffness matrix is the same for both options since the matrix depends
on the model and not on the contract.
We assume that the dynamics of the stock price is modeled by a geometric Brow-
nian motion and that no dividends are paid. Under this assumption, the value of an
American call contract is equal to the value of the corresponding European call op-
tion. Therefore, we focus on put options in the following.
5.1 Optimal Stopping Problem
Recall that a
stopping time τ
for a given filtration
F
t
is a random variable taking
values in
(
0
,
∞
)
and satisfying
{
τ
≤
t
}∈
F
t
,
∀
t
≥
0
.
Search WWH ::
Custom Search