Accurate numerical solution of Black-Scholes option pricing equations

We discuss the accurate numerical solution of Black-Scholes differential equations. We check that the stochastic part of the equation could convert small round-off or truncation errors in big errors. However, the numerical method used are low order even in the non-stochastic case due to the complexity of their development. So if we cannot increase the order the numerical method should mimic the differential equation. Finally, we found that the numerical methods of the type 'exponential fitting' are the better choice when we are integrating ordinary Black-Scholes type equations.