Pricing Of Bi-direction European Option Under Fractional Brownion Motion With Time-varying Parameters

Option pricing is one of the core problems in financial mathematics. In the classical option pricing theory the asset price was supposed to follow standard geometric Brownion motion. However, financial evidence shows that the asset price and the yield rate don’t obey normal distribution, the process of the asset price has long range dependence, the volatilities of the asset price and the yield rate have self-similar and fractional properties, thus using fractional Brownion motion to describe the asset price is more reasonable. Some scholars have studied the price of European option under fractional Brownion mo-tion since Hu and Oksendal introduced fractional Ito’s integral. Most of them supposed the volatility a of the asset is constant. However, in the practical life, the volatility is always changing with time t. In this paper, we study the price of bi-direction European option under fractional Brownion motion with time-varying parameters.Firstly, we give the properties of Ito’s integral and fractional Ito’s integral, market hypothesis and related lemmas.Secondly, we suppose the asset price S(t) follows the geometric fractional Brownion motion here r(t), q(t) and σ(t) of the asset are certain functions on t, BH(t) is the fractional Brownion motion with Hurst parameter H∈(1/2,1). Using the properties of fractional Ito’s integral we obtain the pricing formulas of bi-direction European option. The results extend the related results.Lastly, we suppose the asset price S(t) follows the geometric fractional Brownion motion (1) and the riskless interest rate r(t) follows fractional Vasicek model here a(t), b(t),σ2(t) are certain functions on t, ZH1(t) is the fractional Brownion motion with Hurst parameter H1∈[1/2,1), we study the price of bi-direction European option in the three cases. We get the pricing formulas by the help of multi-dimension Girsanov theorem and the change of measure.