Nonparametric Statistical Inference Of Volatility For Diffusion Models

Statistical inference of the diffusion processes is widely used in fields of natural science and social science,such as engineering,economics,military and etc.In financial market anal-ysis,the correct descriptions of the dynamic rules for basic variables(such as stock price and portfolio value)are essential.The dynamics of these fundamental variables are often described by diffusion(with jump)type models.Along with the increasing perfection of financial market and the rapid development of information technology,obtaining high frequency financial data(intraday data,hour data,minute data,even real time data)is becoming easier and easier.Ex-ploring appropriate models and effective approach to analyze these high frequency data has become an important issue that some mathematical scholars,statistical scholars and economet-ric scholars have to face and solve.This dissertation focusses on the nonparametric volatility statistical inference for diffusion models with high frequency data.The main research works of the dissertation include the following aspects.Part 1:Nonparametric estimation of spot volatility for level-dependent diffusion models is considered and a spot volatility estimator based on continuous sampling kernel function is proposed.The kernel weighted function is first assumed to be continuous sampling and then discretized.A nonparametric estimation procedure for volatility based on sample interpolation is also considered.Under some certain regularity conditions,the proposed volatility estimator is consistent in probability and asymptotically follows a normal distribution.By theoretical in-ference,the precision of the estimator is better than that of the conventional estimator based on realized volatility.Part 2:Nonparametric estimation of spot volatility for time-varying diffusion models is considered by a two-step smoothing method.Firstly,a rough estimator of spot volatility is given by adopting local mean of the square of return.Secondly,the volatility function is re-estimated from the crude estimator by using the usual non-parametric kernel smoothing method.When sample frequency is increased,given some conditions,consistency and asymptotic normality of the estimator is considered,and it is also obtained that the convergence rate of the proposed estimation is better than that of one step smoothing.Part 3:Nonparametric estimation of spot volatility for continuous diffusion models is stud-ied by using range thchnique.Using ranges instead of increments in traditional methods,high frequency data can be processed with low frequency techniques.Under some weak conditions,the proposed estimator is proved to be convergence in probability.A central limit theorem is also proved with some necessary conditions.With high frequency data in hand,the estimator is more precise than those pure realized spot volatility ones.Part 4:By extending range method to level-dependent diffusion models with jump and combining it with threshold technique,spot volatility estimation is considered.Meanwile,a range-based threshold spot volatility estimator with high frequency discrete observations is pro-posed.Under some weak conditions,the consistency and asymptotic normality of proposed estimator are provided.The precision of the statistic is frve times greater than those of pure threshold estimators.Part 5:A framework for estimating the quadratic variation of discontinuous semi-martin-gales with intra-day high-low statistics is developed.Restricting the realized range-based vari-ance smaller than a suitably defined threshold,an integrated volatility estimator is considered and its consistency and asymptotic normality is proposed under a set of weak conditions.It is found that the precision of our statistics is about five times greater than that of realized variance purely restricted by threshold.Simulation results illustrate the good finite sample properties of the proposed estimator.Part 6:In general,the economic environment is changing,but sometimes may keep local stability in a certain stage.In view of this,in this part,a concept of Cox-Ingersoll-Ross(CIR for short)model with time-varying parameters is proposed,and the model is used to model short-term interest rate and exchange rate.The mehtold of generalized residual goodness of fit test is employed to test time-varying characteristic of the CIR model.Using numerical simula-tion and empirical analysis,feasibility and rationality of CIR model with segmented time-var-ying parameters modeling interest and exchange rate are certified.Numerical simulation results show that if two groups of data in accordance with CIR model are combined together,they may not conform to this model again.According to the data analysis for the weekly 6-month U.S.Treasury bill rate and the Canada/U.S.daily foreign exchange rates,it is found that the CIR model with segmented time-varying parameters is more reasonable to describe the short-term interest(exchange)rate than the general constant coefficient CIR model.