基于复杂系统理论的金融市场动力学研究
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摘要
随着世界经济贸易的日益频繁,金融市场正发挥着越来越重要的作用,对金融市场的研究也引起了各领域的关注。另一方面,复杂系统理论研究了当相互之间存在复杂非线性相互作用的个体自组织形成一个整体时所涌现出的宏观规律,特别地,当个体具有智能体的特性时,该复杂系统将呈现各种有趣的性质,这样的系统有金融市场、生物学系统等。利用复杂系统理论、统计物理、应用数学、非线性科学以及理论物理等手段来研究金融市场中的物理规律正发展成为一门新兴研究热点,被称为金融物理学。
本研究采用复杂系统以及统计物理等理论,根据研究方法从以下三个方面研究了金融市场的动力学演化特性：
1.采用湍流理论中的SL层次结构模型研究股票价格序列的层次结构和多重分形特性。
SL层次结构理论最初用于解释湍流中发现的非线性标度律,其优点在于仅用间歇参数β、最奇异标度指数h0、余维数C三个参数就能够较好地描述和解释多重分形非线性标度行为。由于湍流和金融时间序列的描述存在相似之处,我们用SL层次结构理论对股票价格序列进行分析后发现,金融时间序列不仅具有多重分形性,而且存在SL层次结构。用三个参数分别对发达和新兴的金融市场中的非线性标度行为进行量化后比较得出,证券市场的多重分形程度与经济发展水平和地理位置等因素有关,且层次结构会随着市场的演化而变化。本文的研究成果能够帮助我们更好地理解金融市场的多重分形性质,且可以利用湍流中与SL层次结构相关联的多变量级联过程来解释金融市场中的动态演化。
2.将金融市场看作一个由多个交易者和投资产品所组成的复杂网络,利用复杂网络的理论对证券市场的资产回报序列进行建模,研究网络的静态拓扑结构和动态演化。
采用阈值和滑动窗技术,我们分别构造了基于动态阈值和静态阈值的动态全局金融网络,并进一步计算了网络的三个全局特征参数平均度、最短路径长度和平均聚类系数分别随时间演化的曲线,从而研究网络拓扑结构的动态变化规律及其与历史上所发生的金融事件之间的联系。我们发现,三个参数都在金融风暴发生期间出现了异常的波动。该研究结果表明,金融危机与证券市场价格走势的统计性质之间存在着因果关系,因而研究动态的金融网络有助于我们更深入地理解经济危机的发生和传播机制。
3.构建一个包含投资者和投资产品的人工金融市场,考察市场的动态演化过程。我们提出的市场生态学模型不仅考虑投资者的动力学演化,而且考虑投资产品的动力学行为,在该模型中,投资者根据投资策略的不同分为两种类型：主动投资者掌握较多的市场信息,投资策略更为优化,因而选择优良资产的能力较强；被动投资者由于只能获取少量信息因而投资能力弱于主动投资者。这与实际市场的情况相符合,因为现实中每个投资者都只能掌握市场的部分信息。另一方面,投资产品根据它们的品质可以分为优良资产和劣质资产,优良资产的品质高于劣质资产,更能吸引投资者进行投资,但是优质产品的成本要高于劣质产品。模拟结果发现,在没有任何外界调控的情况下,系统依靠投资者的投资策略和与投资产品的相互作用,运行到足够长时间能够自组织地到达一个准静态,在准静态系统的交易者和投资产品的数量都维持着一个动态的平衡,且不同的影响因素能够带来不同的准静态性质,这与生态系统有着类似的性质,对实际金融市场的动态演化和金融危机的发生和恢复有一定的指导意义。
The influence of financial markets is becoming prominent since a continually-growing amount of economic activities trading all over the global, and different fields of scientists have been engaged on the study of financial markets besides economists. On the other hand, the theory of complex systems, investigate the macroscopic laws emerged when numerous individuals self-organize to an ensemble through non-linear interactions between them. Financial market is a typical complex system. Therefore, it is available to study financial markets based on the theory of complex systems, statistical physics, applied mathematics, etc, and this research field has blossomed into a branch of learning, called econophysics. In this study, we aim our research on the dynamic properties of financial markets using theory of complex systems and statistical physics. It consists of three aspects:
1. We study the stock price fluctuations of7developed and emerging financial markets, adopting the measurement of She-Leveque (SL) hierarchy from turbulence.
The SL hierarchy, which was projected to characterize the anomalous scaling law in velocity fluctuations in fluid turbulence, provides a concise measurement to describe multi-fractality and nonlinear scaling law, with only three parameters:the intermittent parameter β, the most high intensity scaling exponent ho and the codimension C. Since the financial market and turbulence have been broadly compared on account of the same quantitative methods and several common stylized facts they share, the SL hierarchical structures model is applied here to study and quantify the hierarchical structure of stock price fluctuations. Several interesting results are observed:(i) The hierarchical structure related to multifractal scaling generally presents in all the7stock price fluctuations;(ii) The quantitatively statistical parameters that describe SL hierarchy are different between developed financial markets and emerging markets, distinctively;(iii) For the high-frequency stock price fluctuation, the hierarchical structure varies with different time periods. These results increase the analogies between the turbulence and financial market dynamics, and provide a profound understanding beyond the phenomenological description of multifractal scaling in stock price fluctuations, and thus may help us to better model the dynamic evolution of financial markets based on multiple cascade processes.
2. Regarding the financial markets as a complex network, in which the nodes represent the stocks and corresponding companies, we study the static topological structure and dynamical evolution of the networks to obtain the relationship between the stock price fluctuations and financial crisis. A series of dynamic correlation-based financial networks are constructed by winner-take-all approach and sliding window technology, with both dynamic and static threshold values. By analyzing the three global parameters, average degree, average shortest path length, and average cluster coefficient evolving in a14-year period, we find that the financial networks show a robust small-world property. Furthermore, the irregularities of curves indicating the dynamic evolution of financial network highly associate with the economic crashes. These results may provide a novel view of complex network science to deeply understand the origin of financial crisis.
3. To study the dynamic process and the statistical properties of financial markets, we build up a simplified artificial financial markets model including investors and investments. In our model, the investors, according to their selection capabilities, are regarded as active or passive, considering the fact that every investor can only perceive partial information to make right decisions. Meanwhile, the investments can be good or bad defined by their qualities. The good investments have a larger probability to attract investors, with higher cost yet. An interesting result is derived that without any external influence, the system can self-organized evolve to a quasi-stationary stable by the interaction between investors and investments according to their own strategies. Moreover, the partial information asymmetry of financial market and various qualities of investments commonly result in a diversity of investors'and investments'dynamic behaviors. This study verifies that the dynamics of a stock market is comparable with that of an evolutionary ecology such as the population of biological species, and that the financial market can be referred to as the ecological system, thus may give vital clues to investigate the financial market ecology.
本研究采用复杂系统以及统计物理等理论,根据研究方法从以下三个方面研究了金融市场的动力学演化特性：
1.采用湍流理论中的SL层次结构模型研究股票价格序列的层次结构和多重分形特性。
SL层次结构理论最初用于解释湍流中发现的非线性标度律,其优点在于仅用间歇参数β、最奇异标度指数h0、余维数C三个参数就能够较好地描述和解释多重分形非线性标度行为。由于湍流和金融时间序列的描述存在相似之处,我们用SL层次结构理论对股票价格序列进行分析后发现,金融时间序列不仅具有多重分形性,而且存在SL层次结构。用三个参数分别对发达和新兴的金融市场中的非线性标度行为进行量化后比较得出,证券市场的多重分形程度与经济发展水平和地理位置等因素有关,且层次结构会随着市场的演化而变化。本文的研究成果能够帮助我们更好地理解金融市场的多重分形性质,且可以利用湍流中与SL层次结构相关联的多变量级联过程来解释金融市场中的动态演化。
2.将金融市场看作一个由多个交易者和投资产品所组成的复杂网络,利用复杂网络的理论对证券市场的资产回报序列进行建模,研究网络的静态拓扑结构和动态演化。
采用阈值和滑动窗技术,我们分别构造了基于动态阈值和静态阈值的动态全局金融网络,并进一步计算了网络的三个全局特征参数平均度、最短路径长度和平均聚类系数分别随时间演化的曲线,从而研究网络拓扑结构的动态变化规律及其与历史上所发生的金融事件之间的联系。我们发现,三个参数都在金融风暴发生期间出现了异常的波动。该研究结果表明,金融危机与证券市场价格走势的统计性质之间存在着因果关系,因而研究动态的金融网络有助于我们更深入地理解经济危机的发生和传播机制。
3.构建一个包含投资者和投资产品的人工金融市场,考察市场的动态演化过程。我们提出的市场生态学模型不仅考虑投资者的动力学演化,而且考虑投资产品的动力学行为,在该模型中,投资者根据投资策略的不同分为两种类型：主动投资者掌握较多的市场信息,投资策略更为优化,因而选择优良资产的能力较强；被动投资者由于只能获取少量信息因而投资能力弱于主动投资者。这与实际市场的情况相符合,因为现实中每个投资者都只能掌握市场的部分信息。另一方面,投资产品根据它们的品质可以分为优良资产和劣质资产,优良资产的品质高于劣质资产,更能吸引投资者进行投资,但是优质产品的成本要高于劣质产品。模拟结果发现,在没有任何外界调控的情况下,系统依靠投资者的投资策略和与投资产品的相互作用,运行到足够长时间能够自组织地到达一个准静态,在准静态系统的交易者和投资产品的数量都维持着一个动态的平衡,且不同的影响因素能够带来不同的准静态性质,这与生态系统有着类似的性质,对实际金融市场的动态演化和金融危机的发生和恢复有一定的指导意义。
The influence of financial markets is becoming prominent since a continually-growing amount of economic activities trading all over the global, and different fields of scientists have been engaged on the study of financial markets besides economists. On the other hand, the theory of complex systems, investigate the macroscopic laws emerged when numerous individuals self-organize to an ensemble through non-linear interactions between them. Financial market is a typical complex system. Therefore, it is available to study financial markets based on the theory of complex systems, statistical physics, applied mathematics, etc, and this research field has blossomed into a branch of learning, called econophysics. In this study, we aim our research on the dynamic properties of financial markets using theory of complex systems and statistical physics. It consists of three aspects:
1. We study the stock price fluctuations of7developed and emerging financial markets, adopting the measurement of She-Leveque (SL) hierarchy from turbulence.
The SL hierarchy, which was projected to characterize the anomalous scaling law in velocity fluctuations in fluid turbulence, provides a concise measurement to describe multi-fractality and nonlinear scaling law, with only three parameters:the intermittent parameter β, the most high intensity scaling exponent ho and the codimension C. Since the financial market and turbulence have been broadly compared on account of the same quantitative methods and several common stylized facts they share, the SL hierarchical structures model is applied here to study and quantify the hierarchical structure of stock price fluctuations. Several interesting results are observed:(i) The hierarchical structure related to multifractal scaling generally presents in all the7stock price fluctuations;(ii) The quantitatively statistical parameters that describe SL hierarchy are different between developed financial markets and emerging markets, distinctively;(iii) For the high-frequency stock price fluctuation, the hierarchical structure varies with different time periods. These results increase the analogies between the turbulence and financial market dynamics, and provide a profound understanding beyond the phenomenological description of multifractal scaling in stock price fluctuations, and thus may help us to better model the dynamic evolution of financial markets based on multiple cascade processes.
2. Regarding the financial markets as a complex network, in which the nodes represent the stocks and corresponding companies, we study the static topological structure and dynamical evolution of the networks to obtain the relationship between the stock price fluctuations and financial crisis. A series of dynamic correlation-based financial networks are constructed by winner-take-all approach and sliding window technology, with both dynamic and static threshold values. By analyzing the three global parameters, average degree, average shortest path length, and average cluster coefficient evolving in a14-year period, we find that the financial networks show a robust small-world property. Furthermore, the irregularities of curves indicating the dynamic evolution of financial network highly associate with the economic crashes. These results may provide a novel view of complex network science to deeply understand the origin of financial crisis.
3. To study the dynamic process and the statistical properties of financial markets, we build up a simplified artificial financial markets model including investors and investments. In our model, the investors, according to their selection capabilities, are regarded as active or passive, considering the fact that every investor can only perceive partial information to make right decisions. Meanwhile, the investments can be good or bad defined by their qualities. The good investments have a larger probability to attract investors, with higher cost yet. An interesting result is derived that without any external influence, the system can self-organized evolve to a quasi-stationary stable by the interaction between investors and investments according to their own strategies. Moreover, the partial information asymmetry of financial market and various qualities of investments commonly result in a diversity of investors'and investments'dynamic behaviors. This study verifies that the dynamics of a stock market is comparable with that of an evolutionary ecology such as the population of biological species, and that the financial market can be referred to as the ecological system, thus may give vital clues to investigate the financial market ecology.
引文
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