Kernel-Weighted Volatility Estimation For Stochastic Diffusion Model With Jumps

Volatility is an important characteristic of the financial market.It is directly related to the uncertainty and risk of the market,and it is an effective index to reflect the quality and efficien-cy of the financial market.How to effectively describe the dynamic behavior of financial market volatility has always been a hot topic in financial market research.Especially in the financial and economic fluctuations,the financial market globalization,the further study of the nature and regu-larity of volatility is of great theoretical and practical significance to the prevention and avoidance of financial risks.Volatility of stochastic diffusion model is an important measure of asset price volatility.The s-tochastic diffusion model has become a convenient tool for financial research,effectiveness of op-tion pricing models and related dynamic hedging strategies is critical.Barndorff-Nielsen and Shep-hard(2002)prove that when the asset price process obeys stochastic volatility,the realized volatility is the consistent estimates of the integral volatility,and the central limit theorem is given.In or-der to describe the change of asset prices better,a researcher adds price jump components.When the asset price process contains a jump behavior,Barndorff-Nielson(2004,2006,2007)proposed the process of bipower variation and multipower variation and a robust integrated volatility esti-mate for jumps was obtained.Christensen and Posolskij(2006)combined with Barndorff-Nielson's idea to propose robust range-based estimation of quadratic variance.Mancini(2004,2009)pro-posed non—parametric threshold estimation for models with stochastic diffusion coefficient and jumps?Bandi and Nguyen(2003)and Johannes(2004)obtained kernel estimation of the price-dependent spread function under the assumption that the jump process has finite activity and boundedness.Li and Guo(2018)proposed a new estimator of the integral volatility under the con-dition that the asset price satisfies the jump-diffusion model and in the presence of market mi-crostructural noise.There are also a large number of studies that have achieved volatility estimates,such as Jing and kong(2015)and Li,xie and zheng(2016).Inspired by these research papers,this paper constructs the kernel weight smoothing volatility estimation and the kernel weight spot volatility estimation using the Bipower variation under the jump-diffusion model,and proves the weak consistency and asymptotic normality of the kernel weight smoothing volatility estimation,as well as the weak consistency of the kernel weight spot volatility estimation.These theoretical results show that:The convergence rate of the weak consis-tency of the kernel weight smoothing volatility estimate is n-1/2+n-r.The convergence rate of the weak consistency of the kernel spot volatility estimator is n-1/2 +n-r+(nhn2)-1+hrn.Therefore,the convergence rate of weak consistency of the kernel weight smoothing volatility estimation is faster than that of the kernel weight spot volatility estimation.Finally,we investigate the finite sample properties of the kernel weight spot volatility estima-tor by numerical simulation.The numerical simulation results show that:The jump size has an effect on the accuracy of the kernel weight spot volatility estimation.However,the higher the sam-pling frequency,the higher the accuracy of the kernel weight spot volatility estimation.This shows that the the kernel weight spot volatility estimation is a good estimation under high frequency data.