The Research Of European And Exchange Options Pricing Under The Environment Of Mixed Fractional Brownian Motion

In 1973, the formula of the Black-Scholes came into being. The research of the option pricing has been a hot topic in the financial markets. What’s more, with the long-time development of financial markets and the constant improvement of some related areas, the distribution which the price of the underlying asset belongs to is constantly corrected and improved. In 1991, Peters, an American scientist, put forward the definition of fractal market at the first time, at the same time, he put forward the idea that it could be more accurate to describe the volatility of the financial market by using the Fractional Brownian Motion. As we all know, the price of the underlying asset is a time series, so some related time series and some amounts should be changed necessarily when time changes.This passage introduces some important achievements of some famous scholars, and we know that some scholars only discussed and studied the option pricing model of the one-dimensional Fractional Brown Motion. This passage will do a little further research on the research of European and exchange options pricing under the environment of Mixed Fractional Brownian Motion. There are five chapters in the passage as follow.The first chapter, the introduction. It mainly gives a general introduction about the subject of this passage. It contains the source and the meaning of the subject, the development and history of the option pricing model, the main argument work and the structure arrangement of this passage.The second chapter, preparation knowledge. This chapter mainly discusses some related basic knowledge about the subject, that is to say, some option theories and some main knowledge.The third chapter, this chapter mainly discusses the problem:linearly combine several Fractional Brownian Motion and a standard Brownian Motion, and we will get the European option pricing model of the price of risk security under the mixed Fractional Brownian Motion. At first, suppose, the risk security is no dividend payment if it is influenced by a linear combination which is combined with several Fractional Brownian Motions and a standard Brownian Motion. What’s more, we get the price risk security which is no dividend payment.The risk neutral probability measure P has an effect on the risk security. Then, suppose, all the discussion about the risk security is a dividend payment, with the same impact as before, we can also get the price of risk security. Besides, it is a dividend payment. At last, by citing theorems lemmas and the price of risk security which is a dividend payment, we can get the main discussion about the European option pricing model under the mixed Fractional Brownian Motion.The fourth chapter, this chapter mainly discusses the stock price under the mixed Fractional Brownian Motion. By using the principle of fair premium pricing to discuss the problem of the exchange option pricing. And finally, we can get two conclusions, they are the formula of the exchange option pricing and the formula of the option pricing which is a dividend payment under the Standard Brownian Motion.The fifth chapter, a general summary about all the work we have done before, and the expectation about the future research work.