Black-Scholes Approximation Formula For Options With Fractional Ornstein-Uhlenbeck Stochastic Volatility

Financial asset pricing is the core of finance,and financial derivatives pricing is its main research content.The emergence of the Black-Scholes formula has opened the door to the issue of option pricing,but the assumptions on which the derivation of the classical Black-Scholes formula is based are too ideal,with obvious limitations,and are not in line with the actual situation of financial market operation.The change from the original constant volatility to stochastic volatility is an often discussed direction of improvement.Fractional Brownian motion with long memory(H>1/2)can better describe the volatility process of risky assets in financial markets,so it is reasonable to be introduced to price the option with stochastic volatility.In this thesis,the fractional Ornstein-Uhlenbeck process is used to describe the volatility process of risky assets,so the model becomes:where B_t and W_t^H are independent.Based on the above assumptions,the firstorder approximation of the Black-Scholes formula about ? is derived by using the Gisanov's theorem and the It?'s formula.Especially for ?t=?+?ZtH,the fractional It?'s formula and the method of hedging are used to deduce the partial differential equation satisfied by the coefficient of the first-order approximation.The derivation of the first-order approximation is instructive for the further study of option pricing formulas under fractional volatility,and provides an idea for solving partial differential equations involving fractional Brownian motion.