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

Financial derivatives plays an increasingly important role as a kind of financial innovation tool in the international financial market. Options as One of its4categories are attracted attention because they can be used for copying other financial derivatives through the form of combination. Option pricing problem has been the focus of research in the field of modern finance. Considering the real financial products of the complexity of the environment, all kinds of option pricing research in recent years has become a hot topic in the research field of the option.Since option pricing of B-S model was published, financial derivatives pricing problem is becoming more and more important in different assumptions in the financial circles, it is finally confirmed that the price of stock market cannot be described by briefly using original B-S pricing formula but a long-term dependence and self-similarity through constantly improved the model under the different hypothesis, and the capital market is the persistent time sequence, which requires a process with a long-term memory to describe the structural characteristics of the market. Thereby, it imported the fractional Brownian motion as random variables which can be more accurately depicted financial market volatility, and more in accordance with the actual situation. This paper is mainly discussed that the research of European and American option pricing under the environment of fractional Brownian motion. Then introduced hybrid fractional Brownian motion, and given the European and American option pricing formula under the environment of the mixed fractional Brownian motion.The first chapter, introduced the background significance of the option research under the fractional Brownian motion environment, early option pricing theory, the research and development of option pricing problem after the classic B-S option pricing model, and the mainly introduces of the content.The second chapter, the introduction of relevant basic knowledge:random process and related martingale theory, got the quasi conditional mathematical expectation to be used the martingale transformation theory; leaded to the fractional Brownian motion, and reached the price of the option through applying score type risk neutral measure.The third chapter, deduced the pricing formula of European bidirectional option and two kinds of assets and a maximum of more than two assets under fractional Brownian motion environment to use quasi conditional mathematical expectation. And expanded to the mixed fractional brown motion environment by a linear combination of the geometric Brownian motion and multidimensional fractional Brownian motion, and got the pricing formula of maximum option. Finally, it discussed and analyzed five kinds of risk aversion parameters in the price model which can influence the option price.The forth chapter, applied numerical method to solve the unification partial differential equation under the fractional brown motion environment which financial derivatives meet, got the American option pricing formula of bonus. And given the American option pricing formula under the mixed fractional brown motion environment.The fifth chapter, summarized the conclusion of the article, and concluded that this paper corresponding problems should be paid attention to improve in the future.