In this section, we consider the example for pricing American put options from

Nielsen et al. (

2002). All results were computed on an Intel

^{®} Core™ i7-5557U CPU with 3.10 GHz. We chose

${x}_{\mathrm{min}}=-4$,

${x}_{\mathrm{max}}=4$,

$M=5000$, and used the parameter sets from

Table 1. To facilitate the optimization, we summarised

$\Delta \tau {p}^{j}$ as

where

$\overline{a}=\Delta \tau \xb7k\xb7a$. Through the initial guess, the only unknown parameter is

$\tilde{a}$. Since a deterministic expression for

$\tilde{a}$ is a goal of our future research, the penalty parameter

$\tilde{a}$ was obtained by an optimization. The optimization was done by minimizing the mean square error (MSE) of the solution corresponding to

$\tilde{a}$. The MSE is given by

where

${P}^{\mathrm{PSOR}}$ is the solution obtained by the projected SOR algorithm and

${P}^{\mathrm{Pen}}$ is the maximum of the solution of the penalised system and the payoff function. The maximum was used to gain comparable results to the PSOR algorithm. Since the parameter set 2 is widely used in research, we compared the free boundary value of the parameter set 2 with the free boundary solution of

Nielsen et al. (

2002),

Fazio et al. (

2019), and

Company et al. (

2014). The value for the free boundary solution obtained by Nielson is 0.8622, Fazio obtained 0.86274, and the free boundary value of

Company et al. (

2014) is 0.8628, where our approach gives 0.86269 with a finer optimization and

$\tilde{a}=10.11\times {10}^{-4}$. This comparison illustrates the high accuracy of this method in comparison with related research.

Since the PSOR method does not consider a penalty term, we also computed the MSE of the solution of the penalised PDE

using a well-known penalty term (

Günther and Jüngel 2010) with

$\widehat{\delta}=1\times {10}^{-4}$ and

$N=1000$. The results are presented in

Table 2, and the comparison of the numerical results illustrate the accuracy of the method. As we included

$\Delta \tau $ in

$\overline{a}$, the obtained values for

$\overline{a}$ are small, and the corresponding values for

$\tilde{a}$ are larger than 1. The best results were obtained by the sample sets with little volatility and short time maturity. The observation of the short time maturity is based on the fact that the number of points were different. The dependence on the volatility was caused by the simplification of the term

p, since we cancelled out

${\sigma}^{2}/2$ and included

$2/{\sigma}^{2}$ in

$\tilde{a}$. We observed that the differences were in the range between an estimated free boundary value and the final free boundary value. They were caused by the time-dependent movement of the free boundary position. There are several ways to analyse this approach in detail. Since a sensitive point of the presented method is the choice of the initial free boundary value, an interesting approach for future research is a detailed analysis of the effect of the choice of the approximation formulas to the solution. As approximation formulas for the initial guess, one can choose the formulas in (

Evans et al. 2002;

Stamicar et al. 1999;

Zhu 2006). Another idea is the consideration of an iterative scheme. In this idea, the free boundary value of the obtained solution is used as an initial guess for a second iteration of solving the penalised American Put problem.