We use the ideas developed by Madan and Milne (1994. Mathematical Finance 3
, 223-745), Lacoste (1996. Mathematical Finance 6, 197-213) to explore the
optimality of polynomial approximations in pricing securities. In particula
r, we look at the approximations for security payoffs as well as the associ
ated pricing formula in a L-2 framework. We apply these ideas to two exampl
es, one where the state variable follows an Ornstein-Uhlenbeck process and
one based on Brownian motion with reflecting barriers, to illustrate the st
rengths and weaknesses of the approach. (C) 2000 Elsevier Science B.V. All
rights reserved. JEL classification: C10; C63; G12.