Research On Options Under Fractional Brownian Motion In Random Environment

A key problem of applying the mathematical theory into the financialfield is to solve the pricing of risk assets and their derivatives. Optionpricing is an important content and Brownian motion is commonly used tocharacterize asset price changes. Compared with the Brownian motion,Fractional brownian motion has the property that the increments arecorrelated; it can reflect the distribution of stock returns with spike andthick tail characteristics. Researches on fractional brownian motion haveextensive and far-reaching impact on financial mathematics and capitalmarkets.In this paper, foreign and domestic researches on fractional brownianmotion have been stated. On the basis of the former researches, some morestochastic assumptions have been introduced in this paper and thecorresponding pricing formulas of the both situations have been obtained.The main achievements can be summarized into two points:1. ExtendedVasick model and fractional exponents O-U process have been combined,European options and reset options under fractional O-U process have beenstudied and the pricing formula has been obtained;2. In a general randomenvironment the pricing of European options under fractional brownianmotion have been studied, by the equivalent martingale measure, Europeanoptions with stochastic expected rate of return and interest rate, changeablestrike price have been studied.