We investigate numerical solution of finite difference approximations to Am
erican option pricing problems, using a new direct numerical method: simple
x solution of a linear programming formulation. This approach is based on a
n extension to the parabolic case of the equivalence between linear order c
omplementarity problems and abstract linear programs known for certain elli
ptic operators. We test this method empirically, comparing simplex and inte
rior point algorithms with the projected successive overrelaxation (PSOR) a
lgorithm applied to the American vanilla and lookback puts. We conclude tha
t simplex is roughly comparable with projected SOR on average (faster for f
ine discretizations, slower for coarse), but is more desirable for robustne
ss of solution time under changes in parameters. Furthermore, significant s
peedups over the results given here have been achieved and are published el
sewhere.