Efficient risk estimation via nested multilevel quasi-Monte Carlo simulation

We consider the problem of estimating the probability of a large loss from a financial portfolio, where the future loss is expressed as a conditional expectation. Since the conditional expectation is intractable in most cases, one may resort to nested simulation. To reduce the complexity of nested simulation, we present an improved multilevel Monte Carlo (MLMC) method by using quasi-Monte Carlo (QMC) to estimate the portfolio loss in each financial scenario generated via Monte Carlo. We prove that using QMC can accelerate the convergence rates in both the crude nested simulation and the multilevel nested simulation. Under certain conditions, the complexity of the proposed MLMC method can be reduced to O(ϵ−2(logϵ)2). On the other hand, we find that using QMC in MLMC encounters a high-kurtosis phenomenon due to the existence of indicator functions. To remedy this, we propose a smoothed method which uses logistic sigmoid functions to approximate indicator functions. Numerical results show that the optimal MLMC complexity O(ϵ−2) is almost attained even in moderate high dimensions.