In this subsection, we present a high-order numerical scheme that enables grid refinement about the locality of the left corner point by introducing local mesh refinement results in a non-uniform grid. Hence, the conventional fourth-order compact scheme is no longer applicable. To this end, we introduce the non-equidistant fourth-order Hermitian differencing presented in the work of Shukla and Zhong [
16] and Shukla et al. [
17] for discretizing interior nodes as given below
with
Refining grids in the locality of the optimal exercise boundary point gives us some advantages, which will be described below. It allows us to use more nodal points very close to
when computing the optimal exercise boundary. We will exploit this feature in a more strategic way in this subsection.
For the near boundary scheme, i.e,
, we will consider fourth-order combined compact finite difference schemes for approximating the solution of the option value as given below
We point out that the fourth-order, one-sided, non-combined compact scheme given below
can also be used as the near boundary scheme for the option value as presented in the work of Zhao and Corless [
20] and Nwankwo and Dai [
21]. However, we observe that, for uniform implementation, the fourth-order combined compact scheme in (
33) provides more reasonable accuracy, although the discrepancy when compared with (
34) is not substantial. We imagine that the reason could be that we use fewer grid points for approximation. The availability of discrete solutions for hedge sensitivity also works in our favor here. Moreover, for the non-uniform grid, we are trying to avoid using more grid points to account for the near boundary scheme based on the slight uniformity we achieve near the left boundary. Any selected grid points for accounting for the near boundary value must fall within the local uniform domain, even though our space grid is globally non-uniform. Hence, in the
Section 3, we consider only (
33). Furthermore, for the delta sensitivity, we use a different fourth-order near boundary combined compact finite difference scheme by considering the following lemma
We observed after deriving this fourth-order combined compact scheme in (
35) that the latter had already been presented and used in the work of Abrahamsen and Fornberg [
22] and, as such, we recognized the need to acknowledge their work here. The high-order scheme in (
35) allows us to use the discrete value of the option value to approximate the near boundary value of delta, thereby reducing sensitivity. Moreover, fewer nodal values of the higher derivatives are used when deriving the near boundary scheme for the delta sensitivity, as shown in (
35). It is worth mentioning that, for the delta sensitivity, the novel near-boundary fourth-order combined compact scheme in (
35) is much more suitable and provides a more accurate result when compared with (
34). In the uniform case, the condition number of the discrete matrix system generated with (
34) is larger when compared with (
35). Similarly, the near boundary value for the gamma is approximated as follows
More importantly, we present below a high-order Robin boundary scheme for approximating the optimal exercise boundary and its derivative after each function evaluation stage with the high-order embedded time integration method. To this end, we consider the following lemma
The high-order scheme above will enable us to approximate the optimal exercise boundary and its first derivative in time with better precision. If we consider value matching, smooth pasting, and the implied second derivative boundary conditions
we obtain
For simplicity, let
Hence,
Equation (
50) approximates the optimal exercise boundary after each function evaluation based on Runge–Kutta embedded pairs that are described in the following subsection. Equation (
50) is also used to formulate an analytical approximation for initializing and further correcting the first derivative of the optimal exercise boundary. To this end, we take the derivative of (
48) with respect to time and obtain the following
Due to the time-dependent coefficient present in our model, the first derivative of the optimal exercise boundary needs to be computed with precision, especially very close to the expiration. Here, we derive a novel high-order analytical approximation for the initialization (prediction) of the first derivative of the optimal exercise boundary at each time level, which will be subsequently corrected from the solutions of the Runge–Kutta pairs at each stage of function evaluation using (
51). To this end, we recall the fourth-order combined compact finite difference presented as given below
For simplicity, let
Furthermore, considering the PDE governing the fixed free boundary American options, we obtain the following
Further simplification reveals that
with
It is important to observe that establishing a system of free boundary PDEs consisting of the option value and hedge sensitivity is very useful here. Not only does it enable us to deal with the convective term that could further introduce error and allow simple implementation of the fourth-order compact scheme, but we also use the discrete solution of the delta sensitivity for predicting the initial value of the first derivative of the optimal exercise boundary for each time level. Furthermore, computing the Greeks simultaneously with the option value and optimal exercise boundary using a fourth-order compact operator presents some benefits that could be substantial if the computational cost is very low. We hope to demonstrate extensively in the
Section 3 that our implementation holds such an advantage. For the rest of the interior points for the option value and delta sensitivity, we use the same fourth-order compact scheme presented in (
29). Hence,
with
,
,
,
,
,
, and
, given as:
With the discrete matrix system above, we then obtain the semi-discrete system for the option value and hedge sensitivity as follows