TY - JOUR
T1 - Sixth-Order Compact Differencing with Staggered Boundary Schemes and 3(2) Bogacki-Shampine Pairs for Pricing Free-Boundary Options
AU - Nwankwo , Chinonso
AU - Dai , Weizhong
JO - International Journal of Numerical Analysis and Modeling
VL - 6
SP - 793
EP - 821
PY - 2024
DA - 2024/10
SN - 21
DO - http://doi.org/10.4208/ijnam2024-1032
UR - https://global-sci.org/intro/article_detail/ijnam/23461.html
KW - Sixth-order compact finite difference, 3(2) Bogacki and Shampine pairs, Dirichlet and Neumann boundary conditions, options price, Delta sensitivity, optimal exercise boundary.
AB - <p style="text-align: justify;">We propose a stable sixth-order compact finite difference scheme coupled with a
fifth-order staggered boundary scheme and the Runge-Kutta adaptive time stepping based on
3(2) Bogacki-Shampine pairs for pricing American options. To compute the free-boundary simultaneously and precisely with the option value and Greeks, we introduce a logarithmic Landau
transformation and then remove the convective term in the pricing model by introducing the delta
sensitivity, so that an efficient sixth-order compact scheme can be easily implemented. The main
challenge in coupling the sixth order compact scheme in discrete form is to efficiently account
for the near-boundary scheme. In this study, we introduce novel fifth and sixth-order Dirichlet
near-boundary schemes that are suitable for solving our model. The optimal exercise boundary
and other boundary values are approximated using a high-order analytical approximation that is
obtained from a novel fifth-order staggered boundary scheme. Furthermore, we investigate the
smoothness of the first and second-order derivatives of the optimal exercise boundary which is
obtained from the high-order analytical approximation. Coupled with the adaptive time integration method, the interior values are then approximated using the sixth order compact schemes.
As such, the expected convergence rate is reasonably achieved, and the present numerical scheme
is very fast in computation and gives highly accurate solutions with very coarse grids.</p>