A robust numerical method for single and multi-asset option pricing

In this work, we numerically solve some different single and multi-asset European options with the finite difference method (FDM) and take the advantages of the antithetic variate method in Monte Carlo simulation (AMC) as a variance reduction technique in comparison to the standard Monte Carlo simulation (MC) in the end point of the domain, and the linear boundary condition has been implemented in other boundaries. We also apply the grid stretching transformation to make a non-equidistance discretization with more nodal points around the strike price (K) which is the non-smooth point in the payoff function to reduce the numerical errors around this point and have more accurate results. Superiority of our method (GS&AMC) will be demonstrated by comparison with the finite difference scheme with the equidistance discretization and the linear boundary conditions (Equi&L), the grid stretching discretization around K with linear boundary conditions (GS&L) and also the equidistance discretization with combination of the standard Monte Carlo simulation at the end point of the domain (Equi&MC). Furthermore, the root mean square errors (RMSE) of these four schemes in the whole region and the most interesting region which is around the strike price, have been compared.