EM Estimation And Application Of GARCH-Jump Model Of Mixture Gaussian Distribution

As the core of asset pricing,derivatives design,and investment portfolios,the research on the rate of return and volatility of financial assets is particularly important.Under normal circumstances,the rate of return on financial assets presents a small and stable trend,but due to the impact of abnormal events,the rate of return on financial assets will also experience sudden upward or downward fluctuations in a short period of time.The large and sudden change characteristics of financial asset returns are called jumping or jumping behavior in financial measurement theory.The probability of jumping behavior is small,but once it occurs,the fluctuation range will be large,which will cause a huge impact on the financial market.Jumping behavior will not only have a significant impact on the rate of return on assets,but also affect many aspects such as market risks and risk premiums.Therefore,it is of great practical significance to choose a suitable model to identify and analyze the jumping behavior of financial asset yield.The recognition of jumping points by domestic and foreign scholars is more based on experience or non-parametric methods,so this paper tries to use parametric methods to identify jumping points.First of all,this article considers setting the residual term of the GARCH-Jump model to a mixed Gaussian model,that is,the residual term of the GARCH-Jump model obeys the non-jump term distribution with probability,and the probability obeys the jump term distribution,and uses the EM algorithm for parameters estimate.Therefore,the posterior probability of jumping items and non-jumping items at a certain time is calculated according to the model estimation results,and the threshold value is set to 0.5.When the posterior probability of jumping behavior at a certain time is greater than 0.5,it is considered to be at that point A jumping behavior occurred.Secondly,this paper uses the simulation method to study the effectiveness of the Gaussian mixture GARCH-Jump model based on the EM algorithm for jumping point recognition.The 3000 pieces of simulated data generated at one time were identified using the RV-BV method and the GARCH-Jump model of mixed Gaussian distribution respectively,and the experiment was repeated 100 times according to the same parameter characteristics.The comparison found that the recognition rate of the RV-BV method varies with the data The number of cuts increases,and the accuracy rate decreases with the increase of the number of data cuts.As the recognition rate increases,the accuracy drops faster,but at the same accuracy(87.60%),the new model has a higher recognition rate(69.80%).Then,comparing the recognition effect of jumping points based on EM algorithm and MLE algorithm,it is found that the recognition rate of jumping points by MLE algorithm is only 10.39%,which confirms the importance of EM algorithm for the recognition of jumping points by the new model.This paper also compares the jump recognition effect of the new model under different sample sizes of simulated data to illustrate the stability of the model for jump point recognition.This paper continues to explore the change rule of the jumping point recognition rate and accuracy rate of the new model by setting different conditional probability thresholds.It is found that as the threshold increases,the recognition rate rises and the accuracy rate decreases.Finally,this article empirically studies the Shanghai and Shenzhen 300 Index and the S&P 500 Index,so as to compare the differences in the jumping behavior of the Chinese and American stock markets.Through analysis,the daily logarithmic returns of the two stock indexes are both stable and non-white noise sequences,and both have the characteristics of left-bias,spikes,and thick tails,and the fluctuations are concentrated.The empirical study uses the RV-BV method and the GARCH-Jump model based on the mixed Gaussian distribution to identify jumps in the logarithmic return sequence of the two stock indexes.It is found that the RV-BV method needs to adjust the window width based on experience to improve the recognition rate of jump points.The GARCH-Jump model with mixed Gaussian distribution only needs to input data into the model to achieve a higher recognition rate.Then,through the analysis of jumping points identified by the GARCH-Jump model of the mixed Gaussian distribution,it is found that the Chinese stock market is more prone to jump behavior when it is in a bear market,and the Chinese stock market is more prone to jump behavior than the American stock market.