A computational method to price with transaction costs under the nonlinear Black–Scholes model. (October 2019)

• Record Type: Journal Article
• Authors: Al–Zhour, Zeyad; Barfeie, Mahdiar; Soleymani, Fazlollah; Tohidi, Emran
• Is Part Of: Chaos, solitons and fractals. Volume 127(2019)
• Page Start: 291
• Page End: 301
• Publication Date: 2019-10
• Subjects: Nonlinear Black–Scholes equation -- Non–uniform grid -- Option pricing -- Transaction costs -- Time–varying system.; 91G60 -- 65M20; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2019.06.033
• Languages: English
• ISSNs: 0960-0779

Abstract

Highlights: To price under transaction costs, a new approach is presented to lead to sparse matrices of second order of convergence after a special semi–discretization. A stability theorem is given to demonstrate the stability criterion of the proposed scheme in option pricing. A non–uniform adaptive mesh points is used to reduce the computational burdensome of the arising nonlinear PDE models. Extensive simulations are carried out showing superior behavior of the proposed approach. Abstract: 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.