Numerical methods for nonlinear partial differential equations and inequalities arising from option valuation under transaction costs

[Truncated abstract] This thesis develops the numerical methods and their mathematical analysis for solving nonlinear partial and integral-partial differential equations and inequalities arising from the valuation of European and American option with transaction costs. The models can hardly be solvable analytically. Therefore, in practice, approximate solutions to such a model are always sought. In this thesis, we discuss two models for the asset price movements: the geometric Brownian motion and jump diffusion process. For the valuation of European options with transaction costs when the underlying asset price follows a geometric Brownian motion, the classical Black-Scholes model becomes a nonlinear partial differential equation. To approximately solve this, we use an upwind finite difference scheme for the spatial discretization and a fully implicit time-stepping scheme.
We prove that the system matrix from this scheme is an M-matrix and that the approximate solution converges unconditionally to the exact one by proving that the scheme is consistent, monotone and unconditionally stable. The discretized nonlinear system is then solved using a Newton iterative algorithm.