Analytical and numerical solutions for a special nonlinear equation

In the most of the option price dynamics in the financial markets, the transaction cost of option is ignored. Considering the transaction costs will lead to the emergence of models with nonlinear PDE. In this paper, transaction cost of option in the assumed market is considered, and the resulting dynamic is a nonlinear PDE, whose exact and numerical solutions have been computed in the present paper. To find the exact solution of the cited nonlinear equation, the Lie group algebra method has been used. The numerical solution has been given using the Chebyshev spectral method. In this method, the solution of the considered equation is approximated using Chebyshev polynomials. The convergence of the obtained polynomials to the solution of the differential equation has been shown, as well.