  This work addresses the problem of pricing American basket options in a
multivariate setting, which includes among others, the Bachelier and the
Black-Scholes models. In high dimensions, nonlinear partial differential
equation methods for solving the problem become prohibitively costly due to the
curse of dimensionality. Instead, this work proposes to use a stopping rule
that depends on the dynamics of a low-dimensional Markovian projection of the
given basket of assets. It is shown that the ability to approximate the
original value function by a lower-dimensional approximation is a feature of
the dynamics of the system and is unaffected by the path-dependent nature of
the American basket option. Assuming that we know the density of the forward
process and using the Laplace approximation, we first efficiently evaluate the
diffusion coefficient corresponding to the low-dimensional Markovian projection
of the basket. Then, we approximate the optimal early-exercise boundary of the
option by solving a Hamilton-Jacobi-Bellman partial differential equation in
the projected, low-dimensional space. The resulting near-optimal early-exercise
boundary is used to produce an exercise strategy for the high-dimensional
option, thereby providing a lower bound for the price of the American basket
option. A corresponding upper bound is also provided. These bounds allow to
assess the accuracy of the proposed pricing method. Indeed, our approximate
early-exercise strategy provides a straightforward lower bound for the American
basket option price. Following a duality argument due to Rogers, we derive a
corresponding upper bound solving only the low-dimensional optimal control
problem. Numerically, we show the feasibility of the method using baskets with
dimensions up to fifty. In these examples, the resulting option price relative
errors are only of the order of few percent.
