Stochastic Volatility Model Based On HMC Algorithm

It is necessary to prevent and control the potential risks in the economic and financial system,and the volatility characteristics of China's stock market can reflect the stability of China's economic and financial system to a certain extent.Therefore,models and methods to describe the stock market volatility have got a wide range of attention and study by scholars.More importantly,new methods and models should be used to study the volatility of Shanghai Stock Index in a more accurate and intensive way which is a useful guidance for investors make investment decisions and stock selection,also a profound reference for goverment to make effective measures in control of risks in stock market.The volatility model is an important tool for the analysis of potential risks of economic and financial systems.Many domestic and foreign empirical researches show that traditional volatility model can not objectively describe the characteristics of financial time series such as heteroscedasticity and time-varying.At present,there are two main categories of volatility models:stochastic volatility model(SV)and ARCH model.SV model introduces the potential random variable in its variance equation,which is more suitable to describe the volatility of stock market returns than ARCH family model.However,the likelihood function of the SV model is a high-dimensional integral which is difficult to be sovled.According to previous reseaches,Markov Chain Monte Carlo(MCMC)simulation has been proposed and developed to overcome this difficulty.However,the traditional MCMC method has unavoidable random walk behavior,which makes the Markov chain fall into the local optimum in the iterative process,and the convergence effect is not ideal.The Hamiltonian Monte Carlo method is a new algorithm which combines Hamiltonian Dynamics System and the Metropolis criterion.The HMC algorithm introduces virtual dynamic variables into the Hamiltonian system,the Hamiltonian system's internal physical properties and leapfrog techniques are used to accomplish the state updating.The conservation of the Hamiltonian makes the dynamics with higher acceptance probability,and the reversibility and volume preservation also help to reduce the random walk behavior of the traditional MCMC method,to a certain extent,during the Metropolis updates,which improve the Markov Chain validity and ensure the algorithm converges rapidly.The HMC algorithm fully considers the sensitive factors of state space,explores the trajectory of the target distribution,and especially applies to the case when the target distribution is high-dimensional or strong correlation bewteen variables of the state space.The HMC method is more efficient in solving high-dimensional integrals due to its global iterative update system,and it is often used in astrophysics,artificial intelligence and tracking dynamic objects at home and abroad.However,there are few researches on the application of HMC algorithm in domestic financial market,especially on the analysis of stock return volatility.Moreover,as a MCMC method,empirical research about HMC algorithm compared with other traditional MCMC method is also worthy of further attention.Therefore,this paper begins with the study of volatility of China's stock market returns,the characteristics of stock market can be obtained by the available data and appropriate model been established after analysis.This paper again starts with parameter estimation and focuses on the HMC algorithm which been used to solve the model established above,and its convergence of the Markov Chain have been compared with that of the traditional MCMC method.This article takes the Shanghai Stock Market Index as the representative research object,and chooses the daily closing price of Shanghai Composite Index from March 29,2013 to March 31,2016 as the sample.Firstly,the descriptive statistical analysis shows that volatility clustering and fat tail exists in the return of the Shanghai Composite Index,which is suitable for the establishment of the thick-tailed stochastic volatility(SV-T)model.Secondly,the Gibbs sampling and Metropolis-Hasting(MH)algorithm are performed to estimate parameters of SV-T model.In order to ensure the accuracy of the parameter estimation,this paper,after judging the convergence of each parameter Markov chain,has carried out 120000 effective iterations after burning the non-convergent part of the two algorithms separately.Thirdly,the HMC algorithm is applied to estimate parameters of the SV-T model.Under the HMC algorithm,the first 2000 samples are discarded as burn-in process,and parameters are been obtained after other 8000 effective iterations.Finally,through qualitative observation and horizontal comparison,the advantages and disadvantages of the three algorithms are been considered.The volitility of China's stock market has been analyzed in this paper under the best algorithm.The results show that the efficiency of HMC algorithm in solving SV-T model is much higher than that of traditional MCMC method,which can best describe the volatility of Shanghai stock market.The comparison of algorithms mentioned above has been performed under the same 10000 iterations.The iteration trajectories and autocorrelation plots clearly indicate that the correlation between the sampling samples under the HMC algorithm is smaller and the autocorrelation decay rate of the samples is faster,which means that the HMC algorithm samples the volatility variables more effectively than the traditional MCMC algorithms.Therefore,HMC algorithm is the best algorithm in the MCMC algorithms above.By analyzing the parameters under the optimal HMC algorithm,we can see that the volatility of the return in Shanghai stock market will not be very large but with a strong persistence at least in a short time in the future.Therefore,the government and relevant departments need to prevent risks in advance and investors should be cautious when selecting shares.