We propose an efficient and easy-to-implement gradient-enhanced least squares Monte Carlo method for computing price and Greeks (i.e., derivatives of the price function) of high-dimensional American options. It employs the sparse Hermite polynomial expansion as a surrogate model for the continuation value function, and essentially exploits the fast evaluation of gradients. The expansion coefficients are computed by solving a linear least squares problem that is enhanced by gradient information of simulated paths. We analyze the convergence of the proposed method, and establish an error estimate in terms of the best approximation error in the weighted $H^1$ space, the statistical error of solving discrete least squares problems, and the time step size. We present comprehensive numerical experiments to illustrate the performance of the proposed method. The results show that it outperforms the state-of-the-art least squares Monte Carlo method with more accurate price, Greeks, and optimal exercise strategies in high dimensions but with nearly identical computational cost, and it can deliver comparable results with recent neural network-based methods up to dimension 100.

翻译：我们提出了一种高效且易于实现的梯度增强最小二乘蒙特卡洛方法，用于计算高维美式期权的价格和希腊值（即价格函数的导数）。该方法采用稀疏埃尔米特多项式展开作为延续价值函数的代理模型，并充分利用梯度快速计算的优势。展开系数通过求解一个线性最小二乘问题获得，该问题利用了模拟路径的梯度信息进行增强。我们分析了所提方法的收敛性，并建立了基于加权$H^1$空间中的最佳逼近误差、离散最小二乘问题求解的统计误差以及时间步长大小的误差估计。我们通过全面的数值实验展示了所提方法的性能。结果表明，在计算成本几乎相同的情况下，该方法在高维场景中比现有最先进的最小二乘蒙特卡洛方法具有更精确的价格、希腊值和最优执行策略，并且能够与近期基于神经网络的方法在维度高达100的情况下取得可比较的结果。