摘要：As the American early exercise results in a free boundary problem, in this article weadd a penalty term to obtain a partial differential equation, and we also focus on an improveddefinition of the penalty term for American options. We replace the constant penalty parameter witha time-dependent function. The novelty and advantage of our approach consists in introducingabounded, time-dependentpenaltyfunction, enablingustoconstructanefficient, stable,andadaptivenumericalapproximationscheme, whileincontrast, theexistingstandardapproachtothepenalisationof the American put option-free boundary problem involves a constant penalty parameter. To gaininsight into the accuracy of our proposed extension, we compare the solution of the extension tostandard reference solutions from the literature. This illustrates the improvement of using a penaltyfunction instead of a penalising constant.
