In this article, we develop new upper and lower bounds on American option prices
which improve the bounds by Broadie and Detemple. The main idea is the
consideration of doubly capped call options which have two cap prices. We present a
new option price approximation based on the two upper bounds. On average, our upper
bound extrapolation (named UBE) has an average accuracy better than a 1,000 time-
step binomial tree with a computation speed comparable to a 100 time-step binomial
tree. We also provide a new method of approximating the optimal exercise boundaries
of American options.

