Research And Application Of Multi-parameter Jump Diffusion Model Based On Interval Probability Optimization

The jump diffusion models have achieved remarkable results in recent years,while most of the researches focused on the assumptions of the jump amplitude and jump interval,and only few experts investigate the real feature by the means of statistics.In present research,the jump interval was mostly assumed as exponential distribution,and the power-law distribution was also used to describe the feature of jump interval.On the basis of the exponential distribution hypothesis,the jump time would be showed as a Poisson process.However,the empirical research found that the exponential distribution hypothesis firmly underestimates the probability of the long interval jumps.Compared to the exponential distribution,the power-law distribution hypothesis can describe the time variation and fatter tail better.But the power-law distribution hypothesis would underestimate the probability of the short interval jumps.As a whole,the exponential distribution and the power-law distribution have their advantages and shortages respectively.To address the problems above,based on the jump and volatility theory,this paper have introduced the multi-parameter exponential distribution in order to describe the feature of jump interval,and trying to provide the most suitable probability distribution hypothesis of jump interval,and construct the corresponding jump diffusion model.First,this paper have summarized the features of the jump interval,namely,time variation,degression,sudden drawdown and heavy tail.Next,utilizing the multi-parameter exponential distribution to measure the jump interval,then a multi parameter jump diffusion model is been built.Additionally,the measure degree of jump interval among exponential distribution,power-law distribution and multi-parameter exponential are compared,also the precision of asset pricing between multi-parameter jump diffusion model and general jump diffusion model is presented.Finally,the empirical research between multi-parameter jump diffusion model and general jump diffusion model has also presented.The results show that the multi-parameter exponential distribution can measure the four properties of jump interval much better than exponential distribution and power-law distribution,themulti-parameter jump diffusion model have the higher pricing ability than the general jump diffusion model.Finally,according to the results,this paper make the suggestions that the jump diffusion model should assume the probability distribution of jump interval as multi-parameter exponential distribution rather than exponential distribution and pow er-law distribution.