Scenario Tree Reduction Algorithm In Complex Situations And Its Application In Portfolio Problems

With the development of stochastic optimization theory and the improvement of computer technology,the idea and method of stochastic programming have been widely used in describing the uncertainty of asset returns.The core of this method is to depict the future return of assets by generating a large number of scenarios,so as to build a scenario tree as the input of portfolio model,and then get an optimal investment decision,and provide guidance and suggestions for investors.However,in this method,the quality and size of the scenario tree will directly affect the performance of optimal portfolio decision and the computational burden of the optimization model.Moreover,in order to accurately describe the uncertainty of asset returns,it is usually necessary to generate a large number of scenarios,which will greatly reduce the solution speed of the portfolio model,and thus affect the advantages of this method in practical application.Therefore,the scenario reduction method came into being.The purpose of this method is to select some representative scenarios and delete other scenarios without affecting the optimal solution of the optimization model as much as possible,so as to reduce the scale of the scenario tree and the complexity of the model.At present,although there are many researches on scenario reduction methods at home and abroad,there is still a lack of related research on how to get the most accurate investment decision with the smallest scenario set.Especially,some recent studies have shown that the optimal investment decision is closely related to the distribution characteristics of asset returns.Moreover,the downside risk measurement is the most commonly used method to measure the investment risk at present,and its remarkable feature is that it is only related to the tail of the return distribution.Based on this,this paper takes the portfolio problem as the main background,and makes a deeper theoretical and empirical research on scenario reduction method.The main research work and innovations of this paper are summarized as follows:Firstly,considering the influence of the high-order moments of asset returns on the optimal portfolio,a mixed integer linear programming model is constructed,which objective is to minimize the error of the first four order moments of asset returns before and after scenario reduction.The conditions for the existence of optimal solution of this model are also proved.Most of the existing methods aim to minimize a certain of probability distance between the initial scenario set and the reduced scenario set,and often ignore the change of the distribution characteristics of asset returns.Therefore,this paper proposes a new scenario reduction method by considering the first four order moment error caused by scenario reduction.By solving this model,we can find an optimal subset of scenarios and the corresponding probability of each scenario,which has great flexibility in practical application.In addition,in order to solve the model effectively,we also propose an improved Benders decomposition algorithm.Finally,this method is applied to a single-stage international portfolio model.The empirical analysis shows that,compared with the existing scenario reduction methods,our method can better maintain the statistical characteristics of asset returns and obtain a more accurate optimal portfolio.Secondly,a scenario reduction method and a multi-stage scenario generation method considering tail mean information are proposed.Because some unnecessary scenarios are deleted,the scenario reduction method will inevitably lead to the loss of distribution information of asset returns,especially the tail information,which has great influence on the value of the lower risk measurement that is usually used to measure the portfolio risk.Therefore,in this paper,the tail mean feature is added to the traditional moment matching model,and a scenario reduction method with high efficiency is proposed.In addition,in order to describe the dynamic asymmetric tail dependence between asset returns,we also use VineCopula model and sampling simulation method to propose a multi-stage scenario generation method to obtain a suitable multi-stage scenario tree.Finally,the effectiveness of the proposed scenario reduction method and multi-stage scenario generation method is analyzed by numerical experiments.Thirdly,a new scenario reduction method is designed by using the concepts of effective scenario and ineffective scenario for a kind of portfolio model,which objective is to minimize a class of higher order downside risk measures.Most of the existing methods are to select some representative scenarios by traversing all the original scenarios without considering the characteristics of the portfolio model,that is,the downside risk only depends on those scenarios corresponding to high losses or high costs.Therefore,this paper first derives the dual representation of the downside risk measures,and transforms the original portfolio model into a min-max optimization problem.Then,when the optimal solution of the portfolio model is known,the identification methods of ineffective scenarios and effective scenarios are given,based on which a new scenario reduction method is designed.Finally,an empirical study is carried out with the historical return data of 9 stock indices.The results show that the method proposed in this paper can not only obtain a smaller reduced scenario set,but also achieve a more accurate optimal solution and optimal value.Fourthly,a multi-stage scenario reduction method based on the clustering idea is proposed.In view of the low efficiency of the existing multi-stage scenario reduction methods,we divide the child nodes with the same parent node into several clusters,and then take a new single node to replace each cluster,based on which some new multi-stage scenario reduction methods are proposed.In order to obtain better reduction results,we also use the single-stage moment matching model and the multi-stage moment matching model respectively,and consider the statistical characteristics error between the reduced scenario tree and the initial scenario tree.Finally,by considering a multi-stage portfolio problem,the effectiveness of the proposed method is empirically analyzed,and the differences between the single-stage moment matching model and the multi-stage moment matching model are compared and analyzed,as well as the results obtained when the portfolio models have different objective functions.