| With the development of economy,the financial market is also booming and growing,and the financial derivatives in the market emerge in endlessly.How to price the financial derivatives reasonably has become the main research problem in the financial market.Black-scholes option pricing model,which was introduced in 1973,was used for a long time afterwards.In this model,it is assumed that the price of the underlying asset is wandering and subject to geometric Brownian motion,but empirical research shows that in most cases,the underlying asset does not conform to geometric Brownian motion.Based on this,this paper carries out the extension research on the option pricing problem,studies the option pricing problem of the underlying asset in different environments,focuses on the pricing problem of European option and commercial option under fractional Brownian motion and jump diffusion environment,and gives the pricing theory and method,the full text is as follows.In chapter 1,we mainly introduce the background of this study,the significance of studying this problem and the research status at home and abroad.In chapter 2,the theory of fractional Brown motion is introduced,and some properties of fractional Brown motion are listed,and the important fraction ITO lemma is cited to lay a theoretical foundation for the related research in Chapter 3 and chapter 4.In chapter 3,we study the fractional Brown motion and jump diffusion process respectively,using the risk-neutral pricing theory,fractional ITO lemma,quasi-martingale theory and Girsanov lemma,and finally give the European call option pricing formula with dividend payment under fractional Brown motion and jump diffusion process.In chapter 4,we study the pricing of dividend payout under fractional Brown motion and jump diffusion process respectively.By introducing fractional ITO lemma and risk-neutral pricing theory,we give the pricing formula of commercial option and dividend payout under two conditions.Finally,this paper summarizes the work done,reflects on the shortcomings of the work,and gives the prospect of this paper. |
