Numerical Schemes and Monte Carlo Method for Black and Scholes Partial Differential Equation: A Comparative Note

This paper comparatively investigates some iterative methods and Monte Carlo simulation technique for the dynamics underlying the celebrated Black and Scholes (BS) model. In particular we attempt to answer the question: 'How many Monte Carlo replications can yield prices, for plain vanilla type European derivatives on a stock, which are similar to those obtained by solving the BS PDE using iterative numerical schemes?' We confine to three frequently referred iterative schemes such as Successive over Relaxation (SOR), Gauss-Seidel (GS) and Jacobi (JC). This information together with the information of 'differences in time requirements' will help to guess the similar trade-offs for complex derivatives(exotic) pricing for which there are no analytic pricing formulas.