The Threshold Re-Weighted Estimation Of The Jump Diffusion Model Based On Empirical Likelihood

The jump diffusion model determined by stochastic differential equation is widely used in the financial field,which can deal with the impact of "abnormal" sudden events in the financial market.Compared with the nonparametric estimation of drift coefficient in jump diffusion model,the research on volatility coefficient is more significant.Because volatility is an important parameter for modeling changes in asset prices,exchange rates and other financial derivatives.This paper introduces the nonparametric estimation of drift coefficient,and focuses on the nonparametric estimation of volatility coefficient in jump diffusion model.The first part of this paper mainly discusses the existing nonparametric statistical inference of drift coefficient and volatility coefficient in jump diffusion model.On the basis of the N-W estimator based on the standard symmetric kernel,the asymmetric kernel estimator and the local linear estimator are applied to the jump diffusion model respectively,considering the boundary effect problem.In the second part of this paper,diffusion volatility and jump volatility are identified and estimated respectively.First,considering that the local linear estimation method may assign negative weights to some sample points,the estimator of volatility coefficient under limited samples may produce negative values.This paper proposes a re-weighted estimator using empirical likelihood weights.The new estimator retains the advantage that the boundary deviation of the local linear estimator is small,and can ensure that the volatility coefficient is nonnegative under the limited sample statistics.Secondly,combined with the threshold estimation,the threshold re-weighted estimation based on empirical likelihood is further proposed,and the large sample properties of the new estimators are given respectively.In addition,the optimal window width of the new estimator is obtained by balancing the bias and variance of the new estimator.For the selection of threshold function,this paper considers the time-varying random case.The third part of this paper tests the theoretical results of the second part through Monte Carlo simulation.First,when the time span T and sampling interval " are fixed,the threshold re-weighted estimator proposed in this paper is compared with N-W estimator,asymmetric kernel estimator and local linear estimator.According to the simulation results,it is verified that the other three estimators are closer to the true value than the N-W estimator in the case of limited samples,no matter at the inner point or at the boundary point,and the deviation is reduced.Secondly,under different window widths and measurement criteria,the threshold re-weighted estimator performs better than the other three estimators under limited samples.Finally,the asymptotic behavior of the threshold re-weighted estimator is tested by changing the sample time span T and the sampling interval ".The results show that the integral mean error of diffusion volatility will decrease with " → 0 or % → ∞,while the integral mean error of jump volatility only decreases when % → ∞,which well verifies that the two new estimators have different asymptotic properties.The fourth part of this paper is to obtain the 5-minute high-frequency data of the closing price of the Shanghai Stock Exchange Index from August 2017 to July 2022 through the Jukuan platform,take the logarithmic return rate of stocks as the research object,establish a jump diffusion model,estimate the diffusion volatility and jump volatility in the model using the new estimators proposed in this paper,and draw the estimation chart of volatility.