American option, Black and Scholes inequality, discontinuous Galerkin method, option pricing, penalty method.
Option pricing is one of the classical problems in financial engineering. Since exact solutions in analytical form are available for simple option contracts in particular, a numerical approach is desirable due to the fact that relaxed standard assumptions do not allow the construction of such solutions. In this paper, we consider the problem of pricing American-style options in the classical Black–Scholes framework; that is, we admit the early exercise feature. This constraint can be viewed as an additional non-linear source term in the option-pricing partial differential equation. The contribution of the paper lies in the proposal of a numerical scheme to solve this pricing equation and in the relationship of the presented technique with the existing pricing approaches. The numerical approach is based on the modification of the discontinuous Galerkin method incorporating a penalty term that handles the early exercise constraint. The capabilities of the scheme derived are documented using reference experiments and compared with the standard finite difference method.