Bifractional Poisson Process And Its Application In Finance

The Black-Scholes option pricing model is a classic one in the study of stock price models,which assumes that stock prices obey geometric Brownian motion.Empirical studies show that the stock price return distribution is characterized by self-similarity,long-memory,non-smooth increments,leptokurtosis and heavy tail.Based on the bifractional Brownian motion and fractional Poisson process,the bifractional Poisson process is proposed in this paper.The bifractional Poisson process has the same covariance function as the bifractional Brownian motion,but its distribution has leptokurtosis characteristics.In addition,the bifractional Poisson process has self-similar in the wide sense and long-memory.First,this paper proposes the definition of the bifractional Poisson process and prove its basic properties.The expressions of skewness and kurtosis of this process are calculated by its characteristic function,and the asymptotic distribution of the process as t?+? is discussed.Considering the demand of practical financial market application,this paper gives the definitions of two different forms of mixed bifractional Poisson process.They all have the leptokurtosis characteristics.In particular,the bifractional Poisson process M2(t)converges in distribution to Gaussian form as t?+? or ??+?.Secondly,this paper discusses the application of bifractional Brownian motion in finance and proposes two new stock price models based on fractional Poisson process and mixed bifractional Poisson process.We find that both stock price models can be used to describe the stock price reversal effect and the momentum effect.What's more,this paper presents the statistical analysis of the stock price model based on a mixed bifractioanl Poisson process,and the following conclusions are obtained,(1).Within a period of time,the fluctuation of the sample path of M2(t)at 0<HK<1/2 is significantly more drastic than that of the sample path of M2 t)at 1/2<HK<1;(2).The density function images of M2(t)exhibit leptokurtosis;(3).As the t and ? values increase,the skewness and kurtosis of M2(t)decrease and converge to 0;(4).The new stock price model based on a mixed bifractioanl Poisson process M2(t)has a lower VaR than the Black-Scholes stock price model at different confidence levels.Finally,a long-memory stochastic volatility model based on the bifractional Poisson process has been proposed.In particular,the closed form solution of the corresponding European option price C(0,S0)has been obtained under the risk-neutral probability measure.