Research On The Two-Component Transformed Gram Charlier Mixtures And Its Applications

Practice and experience tell us that it is very necessary and important to measure financial risk scientifically and reasonably.Currently,the commonly used financial risk measurement methods are Value at Risk(Va R)and Expected Shortfall(ES).Va R provides the measurement method of maximum loss,while ES further considers the scale and likelihood of loss exceeding the confidence level and comprehensively measures risk.However,the estimation of its value depends on the choice of the model and the distribution of the loss random variable.A large number of empirical analysis found that many financial return data have a peak and thick tail.Scholars began to use the methods of skewing,mixing,polynomial expansion and so on to make the distribution non-normal to describe the financial asset data.The probability density function based on polynomial expansion is widely used in the fitting of financial asset yield data.Among them,the Gram Charlie(GC)distribution has attracted much attention due to its parameters directly corresponding to skewness and kurtosis.Based on this,this paper proposes a new distribution with the characteristics of peak and thick tail,the Gram Charlie distribution of mixed two-component transformation(M2TGC),and studies its parameter properties,including moment,skewness,kurtosis and cumulative distribution function.Then,using the M2 TGC distribution as the error distribution of GARCH(1,1),a financial data asset yield model,namely M2TGC-GARCH(1,1),is established.The theoretical Va R and ES values under this model are given,and the general method of backtesting is explored.In order to explore the influence of sample size and extreme value of mixed parameters on parameter estimation,a random simulation was carried out.The results showed that EM algorithm performed well in tail fitting,which is of great significance for calculating and backtesting Va R and ES.Finally,the M2TGC-GARCH,Normal-GARCH,TGC-GARCH,and M2GC-GARCH models are applied to the exchange rate yield data,and the parameters are estimated and back-tested,and the advantages and disadvantages of the models are compared and analyzed.The conclusion is that the M2TGC-GARCH(1,1)model is more suitable for the modeling and risk quantification of the exchange rate yield data.In this paper,the research on the two-component mixed transformed Gram Charlier distribution has enriched the theory of probability distribution to a certain extent,and combined it with time series models to establish a data model of financial asset returns,expanding the measurement tools of financial risk analysis,providing a reference for estimating risk in financial markets,especially exchange rates.