Convergence of the two point flux approximation method and the fitted two point flux approximation method for options pricing with local volatility function (Q6604201)

In this article, the authors deal with numerical approximations for solving the Black-Scholes partial differential equation (PDE) for European and American options pricing with local volatility. More precisely, the authors give a rigorous convergence proof of a fully discretized scheme using the classical two-point flux approximation (TPFA) method for degenerated Black-Scholes operator in one dimension with local volatility function. Furthermore, the fitted-TPFA method for the degenerated Black-Scholes PDE by combining the classical TPFA method and the fitted finite volume method is derived and provide rigorous convergence proof of the fully discretized scheme where the time discretization is performed using the classical Euler methods. The fitted finite volume method is used when the stock price approaches zero with the goal to handle the degeneracy of the PDE while the TPFA method is used on the rest of space domain. Numerical experiments to support theoretical results for European and American options are provided in Section 5.