Variable Window Width Local Linear Estimation For Jump Diffusion Model And Its Application

It is well known that financial data is generally non-stationary,scholars and researchers usually use jump diffusion model to describe its motion trajectory,among which jump term can better explain the phenomenon of sudden jump of financial asset price.At present,there have been related non-parametric studies on the estimation of jump diffusion model parameters.On this basis,this thesis introduces variable window width factor to improve the local linear estimation.The core idea of the improvement is to select the optimal window width for each point traversal,and combine the theory of non-parametric estimation and regression analysis.The expressions of variable window width local linear estimators for drift coefficient and volatility coefficient of second order jump diffusion model are derived.Compared with traditional methods,the new estimator has smaller deviation and satisfies asymptotic normality under certain conditions.In order to verify the validity of the new estimator,subsequent simulation and empirical research were conducted respectively.In the simulation,in the first part of this thesis,the second order jump diffusion model is used to generate data,and then the traditional estimation methods(NW estimation,local linear estimation)and variable window width estimation are used to estimate the model parameters,and then the estimated values are compared with their real values.From the estimation graph we can intuitively and qualitatively conclude that the variable window width estimation is closer to the real value than the traditional fixed window width estimation.In addition,quantile bias is calculated to quantitatively evaluate the optimization effect of variable window width.The qualitative and quantitative results show that variable window width can improve the estimation performance.In the end,the normality of the estimator is tested and the results show that the estimator obtained by using variable window width local linear estimation obeies the normal distribution well.The second part to test and verify the stability of variable window width estimation method,this thesis uses the monte carlo algorithm to design the contrast experiment of multi-group volatility coefficient estimation,by setting the sampling time T=20,different sample size,variable window width and the fixed window width method,moreover combining three error evaluation index,the final results show that the variable window width do better than the traditional fixed window width method,that is to say the introduction of variable window width factor significantly improves the estimation performance of the nonparametric method.Because the central limit theorem of local linear estimation is difficult to obtain,the theoretical confidence interval can not be given,but the simulation results indirectly prove the asymptotic normality of the new estimators.In the empirical part,in order to verify the application of variable window width estimation based on jump diffusion model in the actual financial market,one representative stock in each of the five industry sectors,such as financial industry and construction industry,is selected for empirical analysis respectively.In particular,the empirical demonstration is made based on the adjusted closing price data of 600030 in the six years from 2015 to 2021.Before estimating,the jump test is carried out,and the test results show that there is indeed a jump phenomenon in the sequence.In addition,only the volatility coefficient is considered in this thesis,and the obtained estimators can obviously observe the "volatility smile" curve.Secondly,this thesis uses Shibor’s data of 8 term categories from 2006 to 2022.Then jump test,feature description and ADF stationarity test are performed.After that,local linear and variable window width methods are used for empirical estimation.Shibor estimation effects of different maturity categories are different,but the overall trend of "volatility smile" curve is presented.Furthermore,the results show that the deviation between the estimated value and the real value of the variable window width local linear method is significantly smaller than the deviation obtained by the local linear estimation method.