Equivalent Martingale Measure Of Non-standard Options And Portfolio Research

Exotic options, a more complicated derivatives than plain vanilla products, have played a very important role in the financial market. Exotic options can satisfy various investor, so the reaserch for the exotic options is very necessary. Investment portfolio originate from the classical work of Markowitz, and many scholars improve on the theory in the following years. The model changes from one-period time to discrete time and continuous time, from fixed coefficient to stochastic coefficient. And a lot of methods are created. All have made the theory more approach the actuality. Importantly, the application of the martingale accelerates and ameliorates the reaserch about the financial theory to a great extent.The paper is divided into four parts:The first part introduces the background and some important results about the options and investment portfolio; the second part enumerates the basic knowledge and essential theorem; the third part studies the pricing of several exotic options such as the digital power-options and the chooser-compound options. They are both improved derivatives:the digital power-option is the digital form of power-option, and it can be used for speculation and hedging; chooser-compound option is the derivative of compound, that can satisfy more investors. At the end of the part, the article simulates the results by the method of Monte Carlo; the last part is an improvement about the optimal investment/consumption portfolio based on utility function.