Application Of Fractional Brownian Motion In Quantitantive Investment

Starting from the Markowitz(1952)Portfolio selection problem,after several decades of development,a large number of modern financial theories emerged,such as CAPM of Sharpe(1964)and Linter(1965),Market efficiency hypothesis proposed by Fama(1965),consumption and portfolio selection theory proposed by Merton(1973),study on predictability of stock return by Fama and Macbeth(1973)and the APT arbitrage pricing theory proposed by Ross(1976)and so on.These theories promote the development of quantitative trading,which provides a theoretical basis for quantitative trading,and the essence of quantitative trading is how to correctly understand and describe financial markets.A large number of studies on financial markets have found that market effectiveness hypotheses can be improved,such as the existence of large long-term memory and self-similarity in financial markets(Lo and Mac Kinlay,1988;Cont,2001).Peter(1994)put forward the fractal market hypothesis,and made a reunderstanding of the financial market.Therefore,in the market with fractal features,the dynamic process of the prices can be described by the fractional Brownian motion,this paper investigate some key parts of quantitative trading,such as portfolio selection,the cross-sectional returns of the stocks and the forecasting of the finance time series.Firstly,the problem of portfolio selection is analyzed under fractional Brownian motion enviroment,and the optimal risk asset investment ratio is deduced under the framework of continuous Kelly Criterion and Merton(1973)method,and it is found that the long-term memory and investor's investment horizon will play an important role in investment decision-making,empirical study shows the Kelly strategy under fractional Brownian motion can beat the strategy under regular Brownian motion,this new model can offer another portfolio seletion method to fund manager.In this paper,the cross-sectionn of stock returns is also studied,based on the Hurst parameters of fractional Brownian motion to measure the long-term memory,through the group test and Fama-macbeth cross-sectional regression,it is found that the long-term memory is not priced to the stock return rate and cannot be explained by other common factors,the long-short strategy based on Hurst parameter can offer investors 12% annaulized return;Finally,in a finance market which can be described by the fractional Brownian motion,based on the fractal feature and self-similarity,this paper utilizes the pattern recognition theory and nonparametric statistics to design the similarity prediction model and compares this model with the classic time series model,the similarity prediction model is found to have a good performance on the CSI 300 Index forecasting.