The Maximum Likelihood Estimation Of Jump-diffusion Process Under The Domination Of Jump Threshold

In 1973,the BS option pricing model was proposed and widely used,in which the stock price was based on the traditional geometric Brownian motion model.Later,this model gradually developed,and the diffusion model with jump became the main-stream.The frequency of such model jumps may be subject to different processes,such as the Poisson counting process,and the magnitude of the jump may also be subject to different distributions,such as uniform distribution,normal distribution,and double exponential distribution.This paper mainly studies the parameter estimation of Mer-ton jump diffusion model and double exponential jump diffusion model,and provides a parameter basis for option pricing.By using the Ito formula,we can solve the stochastic differential equations with jump diffusion process and then find the expression of the logarithmic rate of return.For the probability distribution obeyed by the logarithmic rate of return,if the tradi-tional maximum likelihood estimation method is used for parameter estimation,due to the excessive model parameters and the uncertainty of the hopping frequency,the infinite term and the two in the likelihood function will be caused.The dimension tran-scends the integral,even if the numerical method is used to solve the extremum of the likelihood function is extremely complicated,and the calculation is extremely difficult.Therefore,this paper proposes a concept of the jump threshold,which is used to screen out the stock price of the jump,and then discretize the time,that is,the time interval of the data is extremely short.Combined with the differential definition of the Poisson process,the Poisson is first estimated.The parameters of the process,which greatly sim-plifies the likelihood function,make it easy to solve the remaining parameter estimates.At the end of the paper,Monte Carlo's method is used to simulate the data to verify the rationality and feasibility of the method.Since different thresholds correspond to differ-ent estimation parameters,we will find the optimal estimation parameters by constantly changing the threshold,and the corresponding threshold is the optimal threshold.