Laplace Transform And Nonparametric Estimation Of Volatility In Jump Diffusion Models

In recent years,with the diversification of financial products and the continuous emergence of financial derivatives,the construction of financial stochastic models has become more perfect,accompanied by more complicated models.Therefore,the classical estimation method has failed in the calculation of some model parameters.To this end,Todorov and Tauchen(2012a)[1]proposed a new measure based on highfrequency data,the Realized Laplace Transform of Volatility(RLTV)which is easy to calculate,so it provides a nonparametric estimation method for volatility within a given time interval.Because RLTV is a mapping from real high-frequency data to a stochastic process,while synthetic volatility maps data to only one random variable,this statistic contains more information about volatility.Todorov and Tauchen(2012a)[1]simultaneously established a one-dimensional central limit theorem that provides a corresponding theoretical basis for nonparametric estimation of potential volatility.Starting from this perspective,this thesis mainly studies two parts.First,we establish a central limit theorem for RLTV in the multidimensional case by defining the joint Laplace measure and give a detailed proof process.Moreover,we apply RLTV to the financial random jump-diffusion model,use the regularized RLTV inversion function given by Todorov and Tauchen(2012b)[2]to estimate the volatility process under the jump diffusion model which is different from the estimation result of parameter estimation,and obtain the distribution function of the volatility under equal interval sampling in the model to get more information.This thesis uses the data of Shanxi Fenjiu(600809)from 2020 to 2021 for empirical analysis,and obtains the density function and distribution function of the volatility,given appropriate regularization parameters.The parameter estimation method we use for comparison is first to use the jump test statistic that obeys the t distribution to check point by point,find the jump points that exist in the data and eliminate them,so as to separate the diffusion process from the jump process,and then use the great likelihood method for parameter estimation.This provides an estimation method of RLTV for research that addresses issues such as pricing of actual financial products and their derivatives.