آنتروپی شانون چند مقیاسی و کاربرد آن در بازار سهام Multiscale Shannon entropy and its application in the stock market

Efficient market hypothesis (EMH) proposed by Fama [1] in 1960s is the cornerstone of modern financial research. It states that if all the information available in the past can be reflected in the stock price, the market is efficient. In this way, the stock price would follow a random walk behavior and is unpredictable. However, it has been well documented that EMH cannot be established in real market, implying that stock price is predictable to some extent. Entropy is an important notion in nonlinear science, it is a measurement for the uncertainty and complexity of dynamic system. Recently, entropy has been used to study the predictability of stock market. For instance, Maasoumi et al. [2] used metric entropy to detect the predictability of stock market, and found that compared with traditional predicting method, metric entropy can capture more nonlinear relations. Eom et al. [3] used metric entropy and Hurst index to study the predictability of several stock indices, they found that the predictability of a stock index is positively related to the Hurst index but negatively related to the value of metric entropy. Shannon entropy is a measurement of information contained in a system. The greater value of Shannon entropy indicates the more information is needed for people to understand this system. Caraiani [4] introduced the singular value decomposition entropy based on the correlated coefficient matrix, and the entropy turns out to have predictive power for the Dow Jones Industrial Average Index. Following Caraiani, Gu et al. [5] studied the predictive power of singular value decomposition entropy on Chinese Shenzhen stock market. They found that the predictive power is affected by the structural breaks in the market, and it only works on the Shenzhen component index after the reform of non-tradable shares. This is an interesting result that differ from Caraiani [4] to some degree. Behavioral finance theory holds that irrational investors exist in the stock market. Both the useful information and noise is the fundamentals of investors’ decision making. Though the traditional singular value decomposition technique can effectively distinguish different information, the singular value decomposition entropy proposed by Caraiani [4] is a composite measurement for all kinds of information. Then, the more efficient measurement of the different information and further analysis of its predictive power for stock market index is an available topic that can promote a deeper understanding of stock market. In this paper we introduce a new notion of entropy, the multi-scale Shannon entropy, and apply it on the Dow Jones Industrial Average Index to detect the predictive power of the singular value decomposition multi-scale entropy for the index. Caraiani [6] suggests that what has been observed in terms of entropy as a systemic measure is also observable in terms of local properties. We contribute in this direction by analyzing the implications of entropy for different local scales for stock markets. The remainder of paper is organized as follows: Section 2 is dedicated to an introduction about the singular value decomposition multi-scale entropy. Section 3 is the multi-scale entropy analysis for the Dow Jones Industrial Average Index. The predictive power of the singular value decomposition multi-scale entropy for the index is presented in Section 4. Section 5 is a brief conclusion.