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In this work, we develop a novel efficient quadrature and sparse grid basedpolynomial interpolation method to price American options with multipleunderlying assets. The approach is based on first formulating the pricing ofAmerican options using dynamic programming, and then employing static sparsegrids to interpolate the continuation value function at each time step. Toachieve high efficiency, we first transform the domain from $\mathbb{R}^d$ to$(-1,1)^d$ via a scaled tanh map, and then remove the boundary singularity ofthe resulting multivariate function over $(-1,1)^d$ by a bubble function andsimultaneously, to significantly reduce the number of interpolation points. Werigorously establish that with a proper choice of the bubble function, theresulting function has bounded mixed derivatives up to a certain order, whichprovides theoretical underpinnings for the use of sparse grids. Numericalexperiments for American arithmetic and geometric basket put options with thenumber of underlying assets up to 16 are presented to validate theeffectiveness of the approach.

On Sparse Grid Interpolation for American Option Pricing with Multiple Underlying Assets