Monte Carlo pricing under fast mean-reverting stochastic volatility: the multi-scale limit ${ε\to 0}$

We compute $\E[(S_T-K)^+]$ by Monte Carlo for a scalar stochastic-volatility model with a fast mean-reverting factor of time scale $\eps$, for $\eps$ ranging from $1$ down to $10^{-3}$. A conditional (mixing) estimator gives finite variance, whereas the direct estimator has infinite variance for thi