Sub-fractional Poisson Process And Its Preliminary Application In Mathematical Finance

Black Scholes option pricing model is a classical model in option pricing research,which assumes that the stock price follows geometric Brownian motion.However,more and more scholars find that the distribution of stock price returns has the characteristics of self-similarity,long memory,high kurtosis,thick tail and so on.This thesis first proposes a new non-Gaussian process,called sub-fractional Poisson process,which has the characteristics of long memory and high kurtosis,and can be used to simulate the return and volatility of risky assets.In particular,we prove that the process has self-similarity in a broad sense,and as t?+?,the distribution of(?)is non-normal;E(PH(t+?t)-PH(t))3?0.In addition,in order to apply sub-fractional Poisson process to finance,this thesis gives the definition of mixed sub fractional Poisson process,which also has high kurtosis characteristics.Secondly,this thesis gives the application of mixed fractional Poisson process in finance.By using the mixed sub fractional Poisson process,we establish a new non arbitrage long-memory stock price model,which can be used to describe the long memory characteristics of risky asset returns.Moreover,the statistical analysis of the new model is given.By simulating the sample path,analyzing the probability density function and calculating the value-at-risk,we get the following conclusions(1)The sample path of the difference between the absolute value of mixed sub-fractional Poisson process and Brownian motion is generally low,which indicates that the mixed fractional Poisson process has the characteristics of high kurtosis.(2)Compared with Brownian motion,the probability density function of mixed fractional Poisson process shows the characteristics of high kurtosis,and the kurtosis and skewness decrease with the increase of time.(3)Under the same confidence level,the value at risk of the new stock price model based on mixed sub fractional Brownian motion is smaller than that of Brownian motion,which indicates that the mixed sub-fractional Poisson process does not have thick tail characteristics.Finally,a new fractional long memory stochastic volatility model is proposed.Under the risk neutral probability measure,the closed form solution of the corresponding European option price at t=0 is obtained.The numerical simulation results show that the model can simulate the phenomenon of " smile" with implied volatility.With the increase of Hurst index,the smile curve of implied volatility becomes smoother.