Research On A Class Of Generalized Error Distribution And Its Application In Financial Risk Measurement

As one of the core of modern finance problems,risk management has attracted a great deal of attention.Risk measurement is an indispensable part of risk management.The Value at Risk(VaR)and Expected Shortfall(ES)methods are the current mainstream risk measures.Both of these metrics rely on the specific distributions that the loss random variables are subject to;under different distribution assumptions,the estimates obtained tend to vary widely.Therefore,choosing a distribution that adequately fits the data has practical implications for the accuracy of VaR and ES estimates.Firstly,this paper defines a new symmetric heavy-tailed distribution—scale mixture of generalized error distribution,by scaling the generalized error distribution.We focus on the mixture distribution with gamma distribution as the scaled distribution,namely gamma mixture of generalized error distribution(GGED).And its probability density function,k moment,skewness and kurtosis are derived.After tedious calculation,we also give the observed Fisher information matrix,moment estimators and maximum likelihood estimation for the parameters of GGED distribution.At the same time,we make a classified discussion according to the parameters are given or not,and a large number of random simulations were carried out,and the estimation,standard deviation and mean square error of the parameters were calculated.The results show that the parameters of the GGED distribution have good robustness under any circumstances.Secondly,the GGED distribution is applied to to fit the log-return data of Apple's(AAPL)stock price,and compared with the normal distribution,F-S skewed standardized t distribution and generalized error distribution.the results show that the GGED distribution under the AIC criterion has a good fitting effect on the data.Finally,we review several classical VaR estimation and ES estimation models.The Cornish-Fisher expansion method and the GGED distribution are applied to the calculation of VaR values,and the calculation of ES under GGED distribution fitting.on this basis,we also apply these models to the AAPL yield data as an example analysis.A comparison of the results reveals that the model based on GGED distribution can also obtain a reasonable VaR estimation and ES estimation.In this paper,based on the study of generalized error distribution,our research results enrich the probability distribution theory and develop the scale mixture distribution to some extent.We apply the proposed new distribution to the risk measurement model to provide help and reference for financial data analysis.