A computational method to price with transaction costs under the nonlinear Black-Scholes model

More realistic models in option pricing are based on nonlinear modifications of the well-known Black-Scholes PDE due to considering other factors such as transaction costs and risks from an unprotected portfolio. The aim of this research is to price a nonlinear volatility model. The new approach leads to sparse matrices of second order of convergence after a special semi-discretization. The resulting system of equations is time-varying. Accordingly, an implicit time-stepping method is applied with quadratical accuracy, which is not as step-size sensitive as the commonly-used explicit ones. It is discussed that under what conditions the overall scheme is time-stable. Numerical results are given to verify the robustness and usefulness of our method in contrast to the commonly-used methods of the literature for this task. (C) 2019 Elsevier Ltd. All rights reserved.