We introduce a novel numerical framework for pricing American options in high
dimensions. Our scheme manages to alleviate the problem of dimension scaling
through the use of adaptive sparse grids. We approximate the value function with a
low number of points and recursively apply fast approximations of the expectation
operator from an exercise period to the previous period. Given that available option
databases gather several thousands of prices, there is a clear need for fast
approaches in empirical work. Our method processes an entire cross section of
options in a single execution and offers an immediate solution to the estimation of
hedging coefficients through finite differences. It thereby brings valuable advantages
over Monte Carlo simulations, which are usually considered to be the tool of
choice in high dimensions, and satisfies the need for fast computation in empirical
work with current databases containing thousands of prices. We benchmark our algorithm
under the canonical model of Black and Scholes and the stochastic volatility
model of Heston, the latter in the presence of discrete dividends. We illustrate the
massive improvement of complexity scaling over dense grids with a basket option
study including up to eight underlying assets. We show how the high degree of parallelism
of our scheme makes it suitable for deployment on massively parallel computing
units to scale to higher dimensions or further speed up the solution process.