Efficient Numerical Schemes for Computations of European Options with Transaction Costs

Abstract

This paper aims to find numerical solutions of the non-linear Black-Scholes partial differential equation (PDE), which often appears in financial markets, for European option pricing in the appearance of the transaction costs. Here we exploit the transformations for the computational purpose of a non-linear Black-Scholes PDE to modify as a non-linear parabolic type PDE with reliable initial and boundary conditions for call and put options. Several schemes are derived rigorously using the finite volume method (FVM) and finite difference method (FDM), which is the novelty of this paper. Stability and consistency analysis assure the convergence of these schemes. We apply these schemes to various volatility models, such as the Leland, Boyle and Vorst, Barles and Soner, and Risk-adjusted pricing methodology (RAPM). All the schemes are tested numerically. The convergence of the obtained results is observed, and we find that they are also reliable. Finally, we display all the approximate results together with the exact values through graphical and tabular representations.