hard · Quantitative Finance numerical

Which statement correctly characterizes the two distinct error sources and how the total root-mean-square error scales?

1. Statistical (sampling) error scales as O(M^-1/2) while the Euler discretization bias scales as O(N^-1) (weak order one); the RMSE combines as O(M^-1/2+N^-1), so both must be refined together for an efficient estimator.
2. Both statistical and discretization errors scale identically as O(M^-1/2), since increasing the number of simulated Monte Carlo paths also reduces the Euler discretization bias, so only M matters for overall accuracy
3. Statistical sampling error scales as O(M^-1/2) and the Euler discretization bias equals O(N^-1/2), the scheme's strong pathwise order, so this bias term always dominates and the total RMSE becomes O(N^-1/2) instead
4. The Euler-Maruyama discretization scheme is essentially bias-free for expectations of smooth terminal payoffs, so the only error present in the estimator is the O(M^-1/2) statistical term, regardless of how coarse N is

Sign up free to see the explanation and track your rank →