Faster comparison of stopping times by nested conditional Monte Carlo

We show that deliberately introducing a nested simulation stage can lead to significant variance reductions when comparing two stopping times by Monte Carlo. We derive the optimal number of nested simulations and prove that the algorithm is remarkably robust to misspecifications of this number. The method is applied to several problems related to Bermudan/American options. In these applications, our method allows us to substantially increase the efficiency of other variance reduction techniques, namely, quasi-control variates and multilevel Monte Carlo.