A Tempered Fractional White Noise Theory And Applications To Finance

Option pricing has always been an important research topic in the financial field.Black-Scholes formula is the most commonly used option pricing method.However,its pricing accuracy is difficult to meet the current financial market.One of the important reasons for the error of Black-Scholes formula is the independent increment of Brown-ian motion.The semi-long range dependence of tempered fractional Brownian motion improves this situation very well.In this paper,we develop a new theory for tempered fractional Brownian motion B?,?:with-1/2<?<0 and ?>0.Our results extend the contributions by[3],[4],[7]and others.First,we use the fundamental operator to rewrite the tempered fractional Brow-nian motion stochastic integral.Next,we define the Hida distribution space and prove that tempered fractional white noise is the derivative of the tempered fractional Brownian motion in this space.Then we generalize the Girsanov's theorem by using fundamental operator.We also define the directional derivative and quasi-conditional expectation of the tempered fractional Brownian motion and obtain the tempered fractional Ito isom-etry.Using these concepts and results,the tempered fractional Clark-Ocone theorem is proved.Finally,we prove that the tempered fractional Black-Scholes market is arbitrage-free and complete,and also present the tempered fractional Black-Scholes formula for every t ? 0,T].In practical applications,through the empirical analysis of the 50ETF index and its corresponding option,we find that the tempered fractional Black-Scholes formula greatly improves the accuracy of option pricing.By using sensitivity analysis,we find that the Hurst parameter,volatility and strike price are the parameters which have great influence on option price.