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

Machine Learning for Pricing American Options in High-Dimensional Markovian and non-Markovian models