In this work we mainly develop a new numerical methodology to solve a PDE model recently proposed in the literature for pricing interest rate derivatives. More precisely, we use high order in time AMFR-W methods, which belong to a class of W-methods based on Approximate Matrix Factorization (AMF) and are especially suitable in the presence of mixed spatial derivatives. High-order convergence in time allows larger time steps which combined with the splitting of the involved operators, highly reduces the computational time for a given accuracy. Moreover, the consideration of a large number of underlying forward rates makes the PDE problem high dimensional in space, so the use of AMFR-W methods with a sparse grids combination technique represents another innovative aspect, making AMFR-W more efficient than with full grids and opening the possibility of parallelization. Also the consideration of new homogeneous Neumann boundary conditions provides another original feature to avoid the difficulties associated to the presence of boundary layers when using Dirichlet ones, especially in advection-dominated regimes. These Neumann boundary conditions motivate the introduction of a modified combination technique to overcome a decrease in the accuracy of the standard combination technique.

翻译：本文主要发展了一种新的数值方法，用于求解近期文献中提出的利率衍生品定价PDE模型。具体而言，我们采用时间高阶的AMFR-W方法，该方法属于一类基于近似矩阵分解（AMF）的W方法，特别适用于存在混合空间导数的情况。时间方向的高阶收敛性允许采用更大的时间步长，结合所涉及算子的分裂，可在给定精度下大幅减少计算时间。此外，由于考虑了大量的底层远期利率，使得该PDE问题在空间上呈高维特性，因此采用稀疏网格组合技术的AMFR-W方法代表了另一个创新方面，这使得AMFR-W比使用全网格时更高效，并开启了并行化的可能性。同时，采用新的齐次Neumann边界条件是另一个原创性特征，以避免在使用Dirichlet边界条件时，尤其是在对流占优区域中，因边界层存在而带来的困难。这些Neumann边界条件促使我们引入一种改进的组合技术，以克服标准组合技术精度下降的问题。