Black-scholes Equation Driven By Fractional Brownian Motion

The Black-Scholes equation driven by fractional Brownian motion is a classical equation for pricing research in the economic market.The non-martingale non-Markov property of fractional Brownian motion leads to the further study of the dynamic properties of the Black-Scholes equation driven by fractional Brownian motion,such as the stability problem.In addition,the long-term dependence of the fractional Brownian motion coincides with the dependence of asset price changes in the economic market.Therefore,it is necessary to study the stability of the Black-Scholes equation driven by fractional Brownian motion and its application in option pricing,which can not only meet the needs of theoretical development,but also solve the investment decision-making,financial analysis,debt management and other problems in the economy.The stability and stochastic bifurcation of Black-Scholes equations with time-varying parameters driven by fractional Brownian motion and mixed fractional Brownian motion and their application in option pricing are studied in this paper.The first chapter introduces the research background and significance of this paper and the research status at home and abroad.The second chapter describes the definitions and properties of fractional Brownian motion and mixed fractional Brownian motion,the definitions of stochastic stability,the definitions of actuarial price of reload option,reset option and quanto option,and some common lemmas used in this paper.The third chapter studies the stability of the Black-Scholes equation driven by fractional Brownian motion.By using the largest Lyapunov exponent method and the pth moment Lyapunov exponent method,the conditions of almost sure exponential stability and pth moment exponential stability of the Black-Scholes equation driven by fractional Brownian motion with time-varying expected return function and volatility are obtained.Then,numerical examples are used to verify the correctness of the above stability results.Finally,the conditions of stochastic bifurcation are given.The fourth chapter considers the pricing of European option under fractional Brownian motion environment.Based on the theory of stochastic analysis,the pricing formulas of European call options and put options with time-varying expected return function and volatility are obtained by actuarial method.The fifth chapter studies the stability of Black-Scholes equation driven by mixed fractional Brownian motion.By using the method similar to that in chapter 3,the sufficient conditions for almost sure exponential stability and the pth moment exponential stability of this system are obtained.Then,the correctness of the result is verified by numerical examples.Finally,the conditions of stochastic bifurcation of this system are obtained.The sixth chapter considers the new option pricing problem under the mixed fractional Brownian motion environment.The pricing of three new options-reload option,reset option and quanto option are obtained using actuarial method and stochastic analysis theory,and the option pricing model under fractional Brownian motion is improved and extended.The seventh chapter summarizes the research work of this paper,and prospects the future research direction.