Kernel Weighted R-order Variation Estimation Of Instantaneous Volatility For Stochastic Diffusion Model

As an important quantitative index to measure the risk and uncertainty of financial assets,volatility has been widely used in investment portfolio,asset pricing and risk management.With the deepening of financial market globalization in recent years,financial market volatility is in-creasingly frequent.How to measure the volatility more accurately has become a hot topic in the financial field in recent years.Compared with the low-frequency data,the high-frequency data of-ten contains more market information.Therefore,based on the high-frequency data,the parameter or non parameter method is used to construct more accurate volatility estimation,which is more suitable for the actual needs of the contemporary financial market.As a more precise intra day risk measurement method than the traditional risk measurement,the accurate estimation and prediction of the instantaneous volatility is of great theoretical and practical significance for both the law of financial market volatility and the risk management of financial assets.Andreou and Bollerslev(1998)uses nonparametric estimation method for high-frequency da-ta,and puts forward the concept of realized volatility(RV).It uses the sum of the square of the yield over a period of time to estimate the integral volatility over that period of time.It has the ad-vantages of independent model,simple calculation and no need for complex parameter estimation.After that,more and more scholars continue to perfect and improve the post integration volatili-ty.Brandorff and Shephard(2002)proves that the realized volatility is the consistent estimation of the integral volatility in the asset price diffusion model,and gives the central limit theorem.Manci-ni(2004)proposes the threshold realized volatility.Hansen(2008)deals with the realized volatility by Ma filter,and proves that the obtained estimate is still the uniform estimate of the integrated volatility,and gives the asymptotic distribution of the estimate.Kristensen(2010)has proposed the kernel-weighted bipower estimators of instantaneous volatility for stochastic diffusion model.In this paper,we will proposed kernel-weighted esti-mators of instantaneous volatility with r(r>0)-order variation for stochastic diffusion model,and then we will prove under weaker assumptions that it's consistent,asymptotically unbiased and asymptotically normal.A simulation study and empirical analysis examine the finite sample prop-erties of the estimators.It is shown that(1)The shorter the sampling time interval is(i.e.the larger the sample size is),the less the deviation of the estimation is.(2)As the value of order r getting bigger,jump behaviours are relatively stronger,which means that the value of order r can serve as an amplifier to magnify the amplitude of jump behaviours,so that jump behaviours are easier to be seen from the graphs.So it is of interest to study the different order variation estimation of instantaneous volatility.