  In this paper we propose two efficient techniques which allow one to compute
the price of American basket options. In particular, we consider a basket of
assets that follow a multi-dimensional Black-Scholes dynamics. The proposed
techniques, called GPR Tree (GRP-Tree) and GPR Exact Integration (GPR-EI), are
both based on Machine Learning, exploited together with binomial trees or with
a closed formula for integration. Moreover, these two methods solve the
backward dynamic programming problem considering a Bermudan approximation of
the American option. On the exercise dates, the value of the option is first
computed as the maximum between the exercise value and the continuation value
and then approximated by means of Gaussian Process Regression. The two methods
mainly differ in the approach used to compute the continuation value: a single
step of binomial tree or integration according to the probability density of
the process. Numerical results show that these two methods are accurate and
reliable in handling American options on very large baskets of assets. Moreover
we also consider the rough Bergomi model, which provides stochastic volatility
with memory. Despite this model is only bidimensional, the whole history of the
process impacts on the price, and handling all this information is not obvious
at all. To this aim, we present how to adapt the GPR-Tree and GPR-EI methods
and we focus on pricing American options in this non-Markovian framework.
