论文标题
通过嵌套多级准蒙特卡洛模拟进行有效的风险估计
Efficient risk estimation via nested multilevel quasi-Monte Carlo simulation
论文作者
论文摘要
我们考虑了估计金融投资组合造成的巨大损失的可能性的问题，金融投资组合将未来的损失表示为有条件的期望。由于条件期望在大多数情况下是棘手的，因此可能会诉诸嵌套模拟。为了降低嵌套模拟的复杂性，我们提出了一种结合多级蒙特卡洛（MLMC）和Quasi-Monte Carlo（QMC）的方法。在外部模拟中，我们使用蒙特卡洛来生成财务场景。在内部模拟中，我们使用QMC在每种情况下估计投资组合损失。我们证明，使用QMC可以在粗嵌套模拟和多层嵌套模拟中加速收敛速率。在某些情况下，通过合并QMC，可以将MLMC的复杂性降低至$ O（ε^{ - 2}（\logε）^2）$。另一方面，我们发现MLMC由于存在指标功能而遇到灾难性耦合问题。为了解决这个问题，我们提出了一种平滑的MLMC方法，该方法使用Logistic Sigmoid函数近似指示函数。数值结果表明，当在MLMC和平滑MLMC中使用QMC方法时，最佳复杂性$ O（ε^{ - 2}）$即使在中等高尺寸中也是如此。
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 a method that combines multilevel Monte Carlo (MLMC) and quasi-Monte Carlo (QMC). In the outer simulation, we use Monte Carlo to generate financial scenarios. In the inner simulation, we use QMC to estimate the portfolio loss in each scenario. 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 MLMC can be reduced to $O(ε^{-2}(\log ε)^2)$ by incorporating QMC. On the other hand, we find that MLMC encounters catastrophic coupling problem due to the existence of indicator functions. To remedy this, we propose a smoothed MLMC method which uses logistic sigmoid functions to approximate indicator functions. Numerical results show that the optimal complexity $O(ε^{-2})$ is almost attained when using QMC methods in both MLMC and smoothed MLMC, even in moderate high dimensions.
