AMFR-W numerical methods for solving high dimensional SABR/LIBOR PDE models

TL;DR

This paper introduces a high-order, efficient numerical method combining AMFR-W techniques with sparse grids and Neumann boundary conditions to solve high-dimensional PDEs for interest rate derivatives, significantly reducing computational time.

Contribution

The work develops a novel high-order in time AMFR-W method with sparse grid techniques and Neumann boundary conditions for high-dimensional PDEs in finance, improving efficiency and accuracy.

Findings

High-order AMFR-W methods enable larger time steps.

Sparse grid combination techniques improve computational efficiency.

Neumann boundary conditions mitigate boundary layer issues.

Abstract

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.