Application Of Nonlinear Expectation In Risk Measure

Since the 17th century,financial crisis has been frequent,which has had a profound impact on the world economy.It has become the focus of financial research.J.P.Morgan puts forward the value at risk index under the classical probability framework to measure the risk of financial products.The subprime financial crisis in 2007-2009 has once again spread to the world.Extreme events repeatedly challenge the original model of risk prediction,and also substitute Knight uncertainty into the researchers' view.It is found that the guidance of calculating VaR value under the classical probabilistic space framework is not significant,so it is necessary to rethink the distribution hypothesis of the model.Compared with the calculation method under the framework of linear expectation theory,the nonlinear expectation theory can better describe the highly dynamic phenomenon in the real world.Therefore,this paper uses the G-expectation theory under the framework of nonlinear expectation theory to calculate the VaR value.The G-normal model in the existing literature assumes that the upper and lower mean values are constant 0,which eliminates the influence of mean uncertainty on the model and can not solve the problem of Knight uncertainty.Based on this,this paper proposes a G-normal model with uncertain mean value.At the same time,the definition of upper variance parameter is redefined and solved by nonlinear regression approximation algorithm.In this paper,the most representative financial markets in China and the United States,the Shanghai and Shenzhen 300 index and the SP 500 index,are selected as the main research objects.When modeling the logarithmic return,the classical ARMA model is used to describe the mean part,and the G-normal distribution with Knight uncertainty is used to fit the residual part.On this basis,the performance of the improved mean model and the benchmark model in the calculation of G-VaR is compared by using the accuracy of violation and deviation.The results show that the deviation of the improved mean model is lower,and the accuracy of violation is closer to the confidence level.If the mean partial model is the same,the G-VaR value calculated by the improved variance model will be lower than that calculated by the original model,and then the number of violations will be reduced and the deviation.