Hi, I am reading "Multilevel Monte Carlo path simulation" by Mike Giles, and I don't get why this method helps in calculating non path dependent option prices (for example european call in one of the applications in the paper). He uses the following estimatorwhere \hat{P_l} is the f(S(T)) payoff using time step equal to M^{-l}T. In the approximation of E[\hat{P}_{l}-\hat{P}_{l-1}] the ith draw \hat{P}^{i}_{l}-\hat{P}^{i}_{l-1} comes from two discrete approximations with different timesteps but the same Brownian path. I think \hat{P}^{i}_{l}-\hat{P}^{i}_{l-1} should be zero as the payoff only depends on the underlying atmaturity and that is the same regardless the number of timesteps as we are using the same Brownian path. In this case the whole procedure collapses to a simple monte carlo estimate as the second term is zero in the estimator.What am I doing wrong ?Thanks in advance!

Basically I think you are assuming you are dealing with a normal process ( eg log of stock under Geometric Brownian Motion ) whereas he is considering more general processes. Just working with stock under gbm will produce differences and is easy to identify performance because you have closed form.