  In the paper, we propose a new calculation scheme for American options in the
framework of a forward backward stochastic differential equation (FBSDE). The
well-known decomposition of an American option price with that of a European
option of the same maturity and the remaining early exercise premium can be
cast into the form of a decoupled non-linear FBSDE. We numerically solve the
FBSDE by applying an interacting particle method recently proposed by Fujii and
Takahashi (2012d), which allows one to perform a Monte Carlo simulation in a
fully forward-looking manner. We perform the fourth-order analysis for the
Black-Scholes (BS) model and the third-order analysis for the Heston model. The
comparison to those obtained from existing tree algorithms shows the
effectiveness of the particle method.
