Integral Weighted Power Variation Estimation Of Instantaneous Volatility For Diffusion Process

The actual financial asset price process contains many known or unknown factors and is a very complicated process.The diffusion process is an important model that reflects the evolution of financial products to prices.The volatility parameters in the model can better describe the risk of financial market prices.Therefore,the use of diffusion process to study financial market risks has attracted more and more attention.As an important parameter in the process of diffusion,volatility is often used to measure the volatility of financial asset price volatility.It has a wide range of applications in portfolio and risk management.With the development of computer technology,high-frequency data is more easily acquired.Nowadays,people often use volatility to estimate and predict financial market risk under high-frequency data.Scholars often use the second moment of the return as the measure of volatility for the definition of volatility.For example,the Kristensen(2010)constructed kernel estimate is the second-order variation estimate.In addition,some scholars use r-order(high-order)variation to estimate volatility,as estimated by Barndorff-Nielsen and Shephard(2003).And Alvarez at el(2012)based on this estimate of the instantaneous volatility of the r step power variation estimation.Influenced by these documents,this paper constructs the r power variation estimation of instantaneous volatility using GM kernel regression method,which is called integral power exponential estimation.This estimate is a generalized form of the r power variation estimate for instantaneous volatility proposed by Alvarez at el(2012).Under appropriate conditions,we prove the weak consistency and asymptotic normality of the integral power difference estimation.Our theoretical results do not require the condition = 2 or ? 3 as required by Alvarez at el(2012),but only > 0.We give the central limit theorem of the integral power exponential estimation under different conditions.These central limit theorems can be used as the theoretical basis for constructing the confidence interval of volatility.In the fourth chapter,we use the numerical simulation method to show the estimation effect of the proposed power weight variation estimation.The numerical simulation results show that the higher the sampling frequency,the smaller the estimated error,and the closer the distribution is to the normal distribution.In the fifth chapter,we use the proposed integral power variation estimation to empirically analyze the volatility of the Shanghai Stock Index.We propose the estimation as the implemented measure to introduce the Realized GARCH model proposed by Hansen(2012),and the volatility of the Shanghai Stock Index was predicted.The empirical analysis shows that the value of r has an effect on the estimation effect.As the value of the order r is larger,the prediction effect of the realized volatility is worse.But in general,the prediction error is relatively small,which shows that our proposed integral power variation estimation is suitable for high-frequency data financial market.This estimation is an effective method to estimate the instantaneous volatility.The estimation we propose can be used as a reference for studying the instantaneous volatility.