Research And Application Of Nonparametric Estimation Of Diffusion Coefficient In Second Order Diffusion Models

The diffusion model and the diffusion model with jumps often play an important role in financial mathematics and financial statistics.The estimation and statistical inference of the diffusion model have become one of the most popular research fields of financial statistics in modern.Generally,in most cases,diffuse-on model requires financial data to meet stationarity.However,many empirical studies have found that financial data is often nonstationary,so diffusion model is no longer suitable for modeling nonstationary financial data.Therefore,Nicolau(2007)systematically proposed for the first time that second-order diffusion model could be used to model nonstationary financial data.The second-order diffusion model establishes integrated and differentiated processes,which are used in economics,finance and engineering.In this paper,we study the problem of statistical inference for second-order diffusion model.Based on the idea of bias adjustment,we propose a new nonparametric kernel type estimator for diffusion coefficient in the second-order diffusion model,that is,based on the existing local linear kernel type estimator for the diffusion coefficient,we introduce the drift coefficient or its local linear type estimator in the diffusion estimator to correct the asymptotic bias of the estimation.By a Monte Carlo experiment,under the condition of discretized sampling,we compare the finite-sampling performance between the local constant kernel type estimator(NW)proposed in Wang,Tang and Lin(2017)and the local linear kernel type estimator(LL)proposed in this paper.Simulation results show that LL kernel type estimator performance is better than NW kernel type estimator in many aspects.We also indirectly verify asymptotic normality of LL kernel type estimator through simulation study.Finally,this paper selects the stock index of Shanghai Stock Exchange 5-minute high-frequency data for the empirical analysis.We first analyze the statistical characteristics of stock index price,then establish the second-order diffusion model of stock index price.The new estimation method proposed in this paper is used to estimate the volatility coefficient of the model.We draw the stock index volatility estimation chart,which shows "volatility smile",indicating that higher absolute values of log-price increments correspond to higher volatility.