G-M Integrated Type Instantaneous Volatility Estimation

Diffusion process is an important model reflecting the logarithmic price evolution process of financial products,which plays an important role in the study of the operation law of financial markets.In the diffusion process,the volatility is an important measure to reflect the volatility of financial products' logarithmic price.The volatility is an important characteristic of financial market.It is the great theoretical and practical significance to study the law of volatility in financial markets for preventing and evading financial risks.Therefore,it is very important to estimate volatility using observed actual data.In recent years,with the rapid development of computer and communication technology,it is possible to record,collect,store and process a large number of real-time data,which makes the modeling theory and empirical research of high-frequency data become a hot topic in the field of financial statistics.However,the traditional low-frequency data can not meet the needs of high-frequency traders in today's market,and the common models in the traditional low-frequency field are no longer applicable to the high-frequency field.In order to avoid errors caused by setting any model,more and more scholars consider using non-parametric methods to estimate volatility in high-frequency data.Recently,Kristensen(2010)uses the idea of rolling window in Foster and Nelson(1996)and Nadaraya-Watson kernel regression estimation to construct the kernel estimator of the instantaneous volatility for diffusion process,and proved the asymptotic properties and asymptotic normality of the estimate.Inspired by these literature,this paper proposes a Gasser-Muller integral instantaneous volatility estimation based on the idea of Gasser-M"u ller kernel regression estimation,which is derived from integral weighting and is different from the kernel spot volatility estimator discussed by Kristensen(2010).In this paper,the large-sample properties of the proposed G-M integral instantaneous volatil-ity estimator are proved under appropriate conditions.The asymptotic unbiasedness of G-M in-tegral instantaneous volatility estimator is proved firstly;secondly,the asymptotic variance of the estimator is derived to obtain the mean square consistency of the estimator;finally,the asymp-totic normality of the estimator is proved.These asymptotic properties theoretically guarantee that the G-M integral instantaneous volatility estimation has good convergence properties,and the asymptotic normality also provides the theoretical basis for constructing the confidence interval of instantaneous volatility.In the fourth chapter,the estimation effect of this estimation is studied by numerical simu-lation.The results show that the estimation effect of this integrated type instantaneous volatility estimator is the same as that of the kernel weighted instantaneous volatility estimator proposed by Kristensen(2010),and both of them are good estimators under high frequency financial da-ta environment.Therefore,the G-M integral instantaneous volatility estimation proposed in this paper provides a selection method for instantaneous volatility estimation in high frequency data environment.