Study On Exchange Options Pricing Under The Fractional Jump-Diffusions

With the rapid development of financial markets,financial derivatives have the function of avoiding risks and protecting assets,so the research of it is necessary.In this paper,firstly,we assuming that the underlying asset is subject to fractional Brownian motion;secondly,the fluctuation due to uncertainty in the market is taken into account.Therefore,the fractional Brownian motion and jump-diffusion model are combined to study the pricing of exchange options.The pricing formula of exchange option under the fractional jump diffusion model is obtained by using the mathematical tools of stochastic analysis and martingale theory,this is a pricing formula with practical financial significance.The first chapter mainly introduces the concept of options and the history of the development process.Then it introduces the present research situation of exchange options.The second chapter focuses on the theoretical knowledge of stochastic analysis,advanced probability and financial mathematics.The knowledge of random process,martingale theory,Brownian motion,Ito integral,Ito process,Ito formula and so on are introduced one by one.In the third chapter,the market model of fractional Brownian motion is introduced.Secondly,the stochastic integral method is used to discuss the European option pricing problem when the underlying asset price is subject to the dividend Brownian motion.Thirdly,the stochastic integral method is used to discuss the pricing of European options when the underlying asset price is subject to the dividend Brownian motion.Finally,the pricing formula of exchange options under fractional Brownian motion with dividend payment is obtained.The fourth chapter is the core content of the paper,the fractional jump diffusion model can better describe the changes of stock price.So when the stock price follows the fractional jump diffusion model,the pricing model of exchange option is studied by the risk neutral pricing theory.And then the pricing formula of exchange option is generalized.Finally,using the method of actuarial pricing,when the underlying asset price satisfies the fractional jump diffusion model,the pricing model of exchange options is deduced.