Parameter Estimation Of Stochastic Volatility Model Based On Bayesian Estimation And MCMC Method

The study of financial time series has been the focus of research in the field of finance.In the past,Black-Scholes(B-S)model provided a theoretical basis for asset pricing theory.However,it was found that the assumption of constant volatility in the Black-Scholes model does not correspond to the variability of actual financial markets.Researchers then further proposed to model volatility,which are called conditional heteroskedasticity models.Conditional heteroskedasticity models are divided into two categories.the first is to portray the change in volatility with a deterministic function,such as the autoregressive conditional heteroskedasticity model,and the second is to portray the change in volatility with a stochastic equation,such as the stochastic volatility model(SV)proposed.The SV model can better characterize the actual financial data.On the basis of this,Bates took into account the phenomenon of asset price jumps and added the process of jumps to the Heston model,proposing the stochastic volatility model with jumps(SVJ).When estimating the unknown parameters in the stochastic volatility model,the high-dimensional integrals involved are difficult to obtain analytical solutions,so it is difficult to estimate the unknown parameters in the model.Therefore,the Markov Chain Monte Carlo(MCMC)method is used in this thesis to solve the parameters estimation problem in the models.There are many ways to construct a Markov chain in the MCMC method.Two of the most widely used algorithms are the Gibbs sampling method and the Metropolis-Hastings sampling method.The Gibbs sampling method relies on the posterior distributions of the parameters to be estimated,while the Metropolis-Hastings sampling method does not rely on the posterior distributions of the parameters to be estimated and has a smaller estimation error,however its computational efficiency is relatively low.In this thesis,the posterior distribution of each parameter is mathematically derived from the known prior distribution,and it is found that the Gibbs samplingmethod cannot be used for the estimation of the volatility parameters,then different sampling methods are selected for different parameters in this thesis.The combination of the two sampling methods ensures the accuracy of the parameters estimation and also has a faster sampling speed.In this thesis we take the Shanghai Composite Index as an example and select five years of daily trading data as a sample for parameters estimation of the stochastic volatility model and draws the following conclusions:First,the Shanghai Composite Index data has the statistical characteristics of volatility clustering and spikes and fat-tails,which are more suitable for application in stochastic volatility models:Second,we find from the parameters estimation results that the improved MCMC algorithm is more stable for the estimation of each parameter in the model,while the variance is smaller and the results are valid;Finally,the parameters estimateion of the Bates model are more accurate than those of the Heston model.In summary,the improved combination algorithm has improved the parameter estimation of both the Heston model and the Bates model,and for the Shanghai Composite Index data,the Bates model has better estimation results.