hard · Quantitative Finance

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 errors scale as O(M^-1/2), since more paths reduce discretization bias as well; thus only M matters for accuracy.
3. Statistical error is O(M^-1/2) and the Euler bias is O(N^-1/2) (the strong order), so the bias dominates and the RMSE is O(N^-1/2).
4. The Euler scheme is bias-free for expectations of smooth payoffs, so the only error is the O(M^-1/2) statistical term regardless of N.

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