Efficient stratified sampling for financial risk simulation

Abstract

Monte Carlo simulation is frequently the only method available for computing nancial risk, particularly under the realistic and complex portfolio models. The naive simulation generally leads to large con dence intervals on typical risk measures. Thus, to enhance the e ciency of the estimates, the necessity of variance reduction techniques becomes apparent. In this thesis, we discuss the e cient implementation of strati ed sampling technique for Monte Carlo simulation problems. As the application eld, we consider the risk evaluation of a linear asset portfolio. For given portfolio and the loss threshold, tail loss probability and conditional excess values are essential. To understand the general risk situation, one needs to e ciently estimate these values for multiple threshold levels in a single simulation. Strati ed sampling is especially useful for such a task as the allocation fractions can be used as decision variables to minimize the overall error of all estimates. We develop an e cient simulation algorithm that combines optimal strati cation and importance sampling to estimate multiple tail loss probabilities and conditional excess values for linear asset portfolios under the t-copula model. Two di erent classes of objective functions are proposed to represent the overall error. The rst, including the mean squared and the mean squared relative error, allows for a simple closed-form solution. For the second class of error functions, including the maximal absolute and the maximal absolute relative error, a simple and fast heuristic is proposed. The application of the new method, called \OASIS: optimal allocation strati cation and importance sampling", is explained for linear asset portfolios under the t-copula model and its performance is demonstrated with numerical examples.