Portfolio Based On The Variance - Optimal Martingale Measure

Optimal investment problem is one central element in the study of micro-finance. "Portfolio Selection" published in 1952, which is called a modern landmark in the history of micro-finance theory. In the paper, he studied the portfolio problem with the mean-variance analysis and proposed to use quantitative methods to determine the basic model of optimal portfolio. Then, the study of optimal investment problem has made some progress in Markowitz mean-variance sense. In recent years, some scholars assume that the prices of assets subject to the geometric Brownian motion, in continuous time model to study the optimal portfolio problem in the incomplete market. In this paper, we use B-variance optimal martingale measure to discuss the optimal portfolio problem with the price of the risky assets obey jump-diffusion processes, fractional Brownian motion.The article is divided into five parts. The first part briefly introduces the research background and current situation of portfolio selection. The second part gives details of the relevant theoretical basis of financial economics and mathematical finance. The third part of the paper describes the Markowitz mean-variance model, and an empirical study was carried out by Matlab to prove the validity of the model with the historical data of five stocks form the Shanghai and Shen zhen stock market. Part four focuses on the problem of dynamic asset allocation that the prices of stock follow jump-diffusion process in an incomplete market; the dynamic mean-variance efficient strategies and the efficient frontiers are explicitly derived, and the mean-variance efficient frontier is proved to be a straight line in the mean-standard-deviation diagram. These results improve the finding under the assumption that the stock prices obey the Brownian motion in the incomplete market. The last part further improved the result which under the assumption that the stock prices obey the Brownian motion in the incomplete market. The paper discusses the problem of dynamic asset allocation under the assumption that the stock price subject to the fractional Brownian motion by determining the B-variance optimal martingale measure.In this paper, the further study of dynamic portfolio was carried out by B-variance optimal martingale measure. This method is very simple and the result accord with actual financial market.