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

J. G. López-Salas, S. Pérez-Rodríguez, C. Vázquez

TL;DR

The paper develops a high-order, AMFR-W based time integrator for the SABR-LMM PDE pricing problem, addressing the curse of dimensionality with a sparse-grid spatial discretization and a modified combination technique to handle Neumann boundaries. By exploiting operator splitting and AMF, the AMFR-W2 scheme achieves third-order accuracy in time while solving many small, efficiently structured linear systems, outperforming traditional θ-method approaches in high dimensions. Numerical experiments on caplets and swaptions demonstrate substantial speedups over full grids and favorable accuracy, with the modified sparse-grid strategy enabling reliable results in up to 6 or more spatial dimensions. The work offers a scalable, parallelizable framework for high-dimensional PDEs in finance and other fields, with potential extensions to basket options and XVA calculations.

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.

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

TL;DR

The paper develops a high-order, AMFR-W based time integrator for the SABR-LMM PDE pricing problem, addressing the curse of dimensionality with a sparse-grid spatial discretization and a modified combination technique to handle Neumann boundaries. By exploiting operator splitting and AMF, the AMFR-W2 scheme achieves third-order accuracy in time while solving many small, efficiently structured linear systems, outperforming traditional θ-method approaches in high dimensions. Numerical experiments on caplets and swaptions demonstrate substantial speedups over full grids and favorable accuracy, with the modified sparse-grid strategy enabling reliable results in up to 6 or more spatial dimensions. The work offers a scalable, parallelizable framework for high-dimensional PDEs in finance and other fields, with potential extensions to basket options and XVA calculations.

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.
Paper Structure (15 sections, 3 theorems, 53 equations, 13 tables, 1 algorithm)

This paper contains 15 sections, 3 theorems, 53 equations, 13 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. PDE formulation of the SABR-LMM model
  3. Space discretization with finite differences
  4. Time discretization
  5. AMFR-W-methods
  6. $\theta-$method + Gauss-Seidel as an $W$-method
  7. Sparse grids in space
  8. Standard sparse grid combination technique
  9. Modified sparse grid combination technique
  10. Numerical results
  11. Numerical results with full grids
  12. Numerical results with the standard sparse grid combination technique
  13. Numerical results with the modified sparse grid combination technique
  14. Conclusions
  15. Appendix

Key Result

Lemma 3.1

\newlabellemmabij0 Given $N$ pairs of integers $m_i \le M_i$ for $i=1,\dots,N$, let us define the set ${\@fontswitch{}{\mathcal{}} I}_N=\{ \mathbf{j}=(j_1,\dots,j_N) \,\,|\,\, m_i\le j_i \le M_i,\,\,\forall i=1,\dots,N \}$. Moreover, for $M_T=\prod_{k=1}^N (M_k-m_k+1)$, define the map $\vartheta: Then, the map $\vartheta$ is bijective.

Theorems & Definitions (3)

  • Lemma 3.1
  • Lemma A.1
  • Lemma A.2