Error bound of Monte Carlo is $\sigma ^2/N$?

Fallacy 3   The error bound of a Monte Carlo is $\sigma ^2/N$ where N is the number of scenarios and $\sigma$ is the standard deviation of some output like loss.

Counter Statement 3   This is the expected error and is not an error bound. The actual error in a Monte Carlo simulation can be greater than this. If the output variable is unbounded, the error in any set of Monte Carlo Scenarios is unbounded as well.

The Monte Carlo may technically be bounded because of the finite size of the generator of pseudo-random numbers where those are used. A perfect random number generator results in no bound on the error when the loss or output variable is unbounded. An example is a call option on a stock. Its payoff is unbounded. Monte Carlo estimation of its average payoff or even its price has an error that is not bounded prior to the simulation.