Sixth-Order Compact Differencing with Staggered Boundary Schemes and 3(2) Bogacki-Shampine Pairs for Pricing Free-Boundary Options

TL;DR

This paper introduces a high-order, stable finite difference scheme with adaptive time stepping and novel boundary treatments for efficiently pricing American options with free boundaries, achieving high accuracy on coarse grids.

Contribution

It presents a sixth-order compact finite difference scheme combined with a new boundary scheme and adaptive time stepping for improved American option pricing.

Findings

Achieves sixth-order convergence rate.

Provides highly accurate results with coarse grids.

Efficiently handles free boundary problems in option pricing.

Abstract

We propose a stable sixth-order compact finite difference scheme with a dynamic fifth-order staggered boundary scheme and 3(2) R-K Bogacki and Shampine adaptive time stepping for pricing American style options. To locate, fix and compute the free-boundary simultaneously with option and delta sensitivity, we introduce a Landau transformation. Furthermore, we remove the convective term in the pricing model which could further introduce errors. Hence, an efficient sixth-order compact scheme can easily be 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 work, we introduce novel fifth- and sixth-order Dirichlet near-boundary schemes suitable for solving our model. The optimal exercise boundary and other boundary values are approximated using a high-order analytical approximation obtained from a novel fifth-order staggered boundary scheme. Furthermore, we investigate the smoothness of the first and second derivatives of the optimal exercise boundary which is obtained from this high-order analytical approximation. Coupled with the 3(2) RK-Bogacki and Shampine time integration method, the interior values are then approximated using the sixth order compact operator. The expected convergence rate is obtained, and our present numerical scheme is very fast and gives highly accurate approximations with very coarse grids.