In this work, we develop a novel efficient quadrature and sparse grid based polynomial interpolation method to price American options with multiple underlying assets. The approach is based on first formulating the pricing of American options using dynamic programming, and then employing static sparse grids to interpolate the continuation value function at each time step. To achieve 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 of the resulting multivariate function over $(-1,1)^d$ by a bubble function and simultaneously, to significantly reduce the number of interpolation points. We rigorously establish that with a proper choice of the bubble function, the resulting function has bounded mixed derivatives up to a certain order, which provides theoretical underpinnings for the use of sparse grids. Numerical experiments for American arithmetic and geometric basket put options with the number of underlying assets up to 16 are presented to validate the effectiveness of the approach.

翻译：本文提出了一种基于高效求积与稀疏网格的多项式插值新方法，用于定价具有多个标的资产的美式期权。该方法首先通过动态规划构建美式期权定价公式，随后在每个时间步采用静态稀疏网格对继续价值函数进行插值。为实现高效计算，我们首先通过缩放双曲正切映射将定义域从$\mathbb{R}^d$变换至$(-1,1)^d$，再利用泡函数消除$(-1,1)^d$上多元函数的边界奇异性，同时显著减少插值点数量。严格的理论分析表明：通过合理选择泡函数，所得函数具有有界混合偏导数（直至特定阶数），为稀疏网格的应用提供了理论基础。针对标的资产数量达16个的美式算术/几何篮子看跌期权的数值实验，验证了该方法的有效性。