  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.
