Efficient Estimation of the Expected Shortfall

Abstract

We analyze three different methods that can approximate the expected shortfall of a financial portfolio in a nested simulation. In this simulation process, the outer simulation generates risk scenarios, and the inner simulation approximates the value of the financial portfolio under each risk scenario. The first method is the most standard one, and therefore we call it ’the standard Monte Carlo method’. This method uses the same amount of computational cost for each inner simulation. The second method adapts the computational cost of the inner simulation to the output of the outer simulation. Therefore, we call it ’the adaptive sampling method’. This technique has already been proven to work more efficiently than the standard Monte Carlo method. The third method is called ’the multilevel Monte Carlo method’ (MLMC) and is based on the technique that has been introduced by Giles. This method approximates the expected shortfall multiple times and with different levels of accuracy. These estimators are used to give an accurate overall approximation of the expected shortfall. To the best of our knowledge, it has never been examined how efficiently the expected shortfall can be approximated in this manner by the MLMC method. This thesis will thoroughly explain how each method can be applied in a nested simulation to approximate the expected shortfall. In addition, an analysis is given on how to ensure that each method is used as efficiently as possible in the nested simulation. We also perform a numerical experiment in which we simulate a simplified version of a financial portfolio using stochastic processes. With the help of the numerical experiment, we examine which method is most efficient to approximate the expected shortfall.