Finite Element Analysis of the Black-Scholes PIDE Under CGMY Jump Process with Infinite Activity and Transaction Costs Enabled

Abstract

In this work, I considered the Black-Scholes Partial Integral Differential Equation (PIDE) under the CGMY process with added transaction costs. Firstly, the integral discretisation approach invented by [12] is explained. Secondly, the finite element formulation of the whole equation was written. The Theta method was used for the discretisation of the time derivative, and FEM for the discretisation of the space derivative. As a result, we obtained similar results to the paper by [12] with only a difference in a near boundary region. Furthermore, in Appendix C, we established the existence and uniqueness of the viscosity solution for the discretised equation using Perron’s method and comparison principle.