Multi-level Monte Carlo simulations with importance sampling

We present an application of importance sampling in a Monte Carlo simulation for multi-asset options and in a Multi-Level Monte Carlo simulation. We demonstrate that applying importance sampling only on the first level of the Multi-Level Monte Carlo significantly improves its effective performance. We extend the Likelihood Ratio Method Based on Characteristic Function to estimate the Greeks of multi-asset options and in a Multi-Level Monte Carlo in a computationally efficient manner. Moreover, we combine it with the importance sampling to reduce the variance of the Greeks. Finally, we study the impact of the skew on the effective performance of importance sampling