基于随机交互系统的金融波动模型的构造与分析
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摘要
本论文运用连续渗流、随机交互作用系统、伊辛模型、选举模型和Zipf定理等方法研究了证券市场中价格过程波动的统计特性,不同证券指数之间波动连锁反应的统计特性,上证和深证的宽尾现象和幂律分布.讨论了价格过程模型下的欧式未定权益的定价和套期保值问题.研究了二维W-R模型的双层随机相分离线的统计特性.本论文的组织结构如下.
第1章简单介绍了金融数学,特别是证券价格波动理论研究的发展背景、研究现状.列出了本论文的主要研究成果.
第2章通过连续渗流理论研究证券价格过程波动的统计特性.连续渗流方法被应用于建立金融模型,用以描述证券价格的行为,特别是描述金融市场里投资者的“羊群效应”.运用统计分析的随机方法,我们证明了价格过程的特征函数收敛于Black-Scholes模型相应的特征函数.
第3章考虑证券价格之间连锁反应的统计特性.应用交互作用系统和统计物理理论描述和研究证券市场的两种证券指数的波动,研究了二者之间的交互反应特性.本章中我们运用随机分析和双随机路径模型研究了证券指数之间连锁反应的概率分布,进一步,揭示出两种证券指数模型波动的概率测度的渐近性,通过连锁反应的单只证券指数的概率性质.对所建立的金融模型的有限维概率分布收敛性进行了讨论.
第4章运用随机过程理论及随机选举模型理论,我们建立了一个包含两种投资者类型的金融证券价格模型.我们运用该金融模型来描述证券市场的单只证券价格过程性质与波动.在该金融模型里,除了专业投资者,我们也考虑普通投资者或者说非专业投资者,这里停时理论和选举模型被用来建立数学模型以及研究非专业投资者投资的统计特性.讨论了该价格过程模型下的欧式未定权益的定价和套期保值问题.
第5章研究了二维Widom-Rowlinson模型的双层随机相分离线的统计特性.分离线把格点W-R模型的两个共存相分隔开来,当该模型的化学势μ足够大时,对描述相分离线波动的概率分布收敛性进行了研究.模型引入backbone的概念,分析并发展了与polymer权重对应的polymer-链及串展开方法.给出了二维W-R模型双层随机分离线的自由能的存在性.
第6章运用Zipf-图方法研究股票价格和交易量的波动特性,Zipf-图方法已被广泛地应用于物理科学领域.在本章的第一部分,分析了来自于上证综指和深证成指的股票价格和交易量数据,并研究了它们的统计特性.我们选取了中国股市2002-2006年度每天的数据,通过分析这些数据,我们讨论了宽尾现象的统计特性以及每天股票价格和交易量的幂律分布.在本章的第二部分,我们运用Zipf-图方法研究了2001-2006年期间上证和深证的宽尾现象和幂律分布.
第7章列出了一些与本研究密切相关的待解决的问题.这也是本人今后科研工作的目标之一.
By applying the theory of continuum percolation, interacting systems and statis-tical physics, Ising model, voter model and Zipf plot method, we mainly discuss the statistical properties of fluctuations of the stock price process, the properties of the in-teracting reaction of two stock indices, and the fat tails phenomena and the power law distributions of Shanghai Stock Exchange Index and Shenzhen Stock Exchange Index. We discuss the corresponding valuation and hedging of European contingent claims for the price process model. And we also investigate the statistical behaviors of two-layered random phase interfaces in two-dimensional W-R model. This thesis is organized as follows.
In Chapter2, we investigate the statistical properties of fluctuations of the stock price process in a stock market by continuum percolation theory. The methods of con-tinuum percolation are applied to construct a financial model that describes the behavior of a stock price, specifically the continuum percolation is used to describe the "herd effect" of investors in a financial market. By using the stochastic methods of statistical analysis, we show that the characteristic function of this stock price process converges to the corresponding characteristic function of Black-Scholes model.
In Chapter3, we consider the statistical properties of chain reaction of stock in-dices. The theory of interacting systems and statistical physics are applied to describe and study the fluctuations of two stock indices in a stock market, and the properties of the interacting reaction of the two indices are investigated in the present paper. In this work, stochastic analysis and the two random paths model are used to study the proba-bility distribution for the chain reaction of stock indices, further we show the asymptoti-cal behavior of probability measures of the fluctuations for the two stock indices model and the probability properties of one stock index by chain reaction. We discuss the convergence of the finite dimensional probability distributions for the financial model.
In Chapter4, by applying the theory of stochastic processes and interacting parti-cle systems and models, including stopping time theory and stochastic voter model, we model a financial stock price model that contains two types of investors. And we use this financial model to describe the behavior and fluctuations of a stock price process in a stock market. In the financial model, besides the professional investors, we also consider the general investors. Where the stopping time and the voter model are ap- plied to model and study the statistical properties of investment of the general investors. Further, we discuss the valuation and hedging of European contingent claims for this price process model.
In Chapter5, the statistical behaviors of two-layered random phase interfaces in two-dimensional Widom-Rowlinson model are investigated. The phase interfaces sep-arate two coexisting phases of the lattice Widom-Rowlinson model, when the chemical potential μ of the model is large enough, the convergence of the probability distribu-tions which describe the fluctuations of the phase interfaces is studied. The backbones of interfaces are introduced in the model, and the corresponding polymer chains and cluster expansions are developed and analyzed for the polymer weights. And the exis-tence of the free energy for two-layered random phase interfaces of the two-dimensional Widom-Rowlinson model is given.
In Chapter6, the fluctuations of stock prices and trade volumes are investigated by the method of Zipf plot, where Zipf plot technique is frequently used in physics science. In the first part, the data of stock prices and trade volumes in Shanghai Stock Exchange and Shenzhen Stock Exchange is analyzed, the statistical properties of stock prices and trade volumes are studied. We select the daily data for Chinese stock market during the years2002-2006, by analyzing the data, we discuss the statistical properties of fat tails phenomena and the power law distributions for the daily stock prices and trade volumes. In the second part, we consider the fat tails phenomena and the power law distributions of Shanghai Stock Exchange Index and Shenzhen Stock Exchange Index during the years2001-2006by Zipf plot method.
In Chapter7, some mathematical finance questions related to this thesis are listed.
第1章简单介绍了金融数学,特别是证券价格波动理论研究的发展背景、研究现状.列出了本论文的主要研究成果.
第2章通过连续渗流理论研究证券价格过程波动的统计特性.连续渗流方法被应用于建立金融模型,用以描述证券价格的行为,特别是描述金融市场里投资者的“羊群效应”.运用统计分析的随机方法,我们证明了价格过程的特征函数收敛于Black-Scholes模型相应的特征函数.
第3章考虑证券价格之间连锁反应的统计特性.应用交互作用系统和统计物理理论描述和研究证券市场的两种证券指数的波动,研究了二者之间的交互反应特性.本章中我们运用随机分析和双随机路径模型研究了证券指数之间连锁反应的概率分布,进一步,揭示出两种证券指数模型波动的概率测度的渐近性,通过连锁反应的单只证券指数的概率性质.对所建立的金融模型的有限维概率分布收敛性进行了讨论.
第4章运用随机过程理论及随机选举模型理论,我们建立了一个包含两种投资者类型的金融证券价格模型.我们运用该金融模型来描述证券市场的单只证券价格过程性质与波动.在该金融模型里,除了专业投资者,我们也考虑普通投资者或者说非专业投资者,这里停时理论和选举模型被用来建立数学模型以及研究非专业投资者投资的统计特性.讨论了该价格过程模型下的欧式未定权益的定价和套期保值问题.
第5章研究了二维Widom-Rowlinson模型的双层随机相分离线的统计特性.分离线把格点W-R模型的两个共存相分隔开来,当该模型的化学势μ足够大时,对描述相分离线波动的概率分布收敛性进行了研究.模型引入backbone的概念,分析并发展了与polymer权重对应的polymer-链及串展开方法.给出了二维W-R模型双层随机分离线的自由能的存在性.
第6章运用Zipf-图方法研究股票价格和交易量的波动特性,Zipf-图方法已被广泛地应用于物理科学领域.在本章的第一部分,分析了来自于上证综指和深证成指的股票价格和交易量数据,并研究了它们的统计特性.我们选取了中国股市2002-2006年度每天的数据,通过分析这些数据,我们讨论了宽尾现象的统计特性以及每天股票价格和交易量的幂律分布.在本章的第二部分,我们运用Zipf-图方法研究了2001-2006年期间上证和深证的宽尾现象和幂律分布.
第7章列出了一些与本研究密切相关的待解决的问题.这也是本人今后科研工作的目标之一.
By applying the theory of continuum percolation, interacting systems and statis-tical physics, Ising model, voter model and Zipf plot method, we mainly discuss the statistical properties of fluctuations of the stock price process, the properties of the in-teracting reaction of two stock indices, and the fat tails phenomena and the power law distributions of Shanghai Stock Exchange Index and Shenzhen Stock Exchange Index. We discuss the corresponding valuation and hedging of European contingent claims for the price process model. And we also investigate the statistical behaviors of two-layered random phase interfaces in two-dimensional W-R model. This thesis is organized as follows.
In Chapter2, we investigate the statistical properties of fluctuations of the stock price process in a stock market by continuum percolation theory. The methods of con-tinuum percolation are applied to construct a financial model that describes the behavior of a stock price, specifically the continuum percolation is used to describe the "herd effect" of investors in a financial market. By using the stochastic methods of statistical analysis, we show that the characteristic function of this stock price process converges to the corresponding characteristic function of Black-Scholes model.
In Chapter3, we consider the statistical properties of chain reaction of stock in-dices. The theory of interacting systems and statistical physics are applied to describe and study the fluctuations of two stock indices in a stock market, and the properties of the interacting reaction of the two indices are investigated in the present paper. In this work, stochastic analysis and the two random paths model are used to study the proba-bility distribution for the chain reaction of stock indices, further we show the asymptoti-cal behavior of probability measures of the fluctuations for the two stock indices model and the probability properties of one stock index by chain reaction. We discuss the convergence of the finite dimensional probability distributions for the financial model.
In Chapter4, by applying the theory of stochastic processes and interacting parti-cle systems and models, including stopping time theory and stochastic voter model, we model a financial stock price model that contains two types of investors. And we use this financial model to describe the behavior and fluctuations of a stock price process in a stock market. In the financial model, besides the professional investors, we also consider the general investors. Where the stopping time and the voter model are ap- plied to model and study the statistical properties of investment of the general investors. Further, we discuss the valuation and hedging of European contingent claims for this price process model.
In Chapter5, the statistical behaviors of two-layered random phase interfaces in two-dimensional Widom-Rowlinson model are investigated. The phase interfaces sep-arate two coexisting phases of the lattice Widom-Rowlinson model, when the chemical potential μ of the model is large enough, the convergence of the probability distribu-tions which describe the fluctuations of the phase interfaces is studied. The backbones of interfaces are introduced in the model, and the corresponding polymer chains and cluster expansions are developed and analyzed for the polymer weights. And the exis-tence of the free energy for two-layered random phase interfaces of the two-dimensional Widom-Rowlinson model is given.
In Chapter6, the fluctuations of stock prices and trade volumes are investigated by the method of Zipf plot, where Zipf plot technique is frequently used in physics science. In the first part, the data of stock prices and trade volumes in Shanghai Stock Exchange and Shenzhen Stock Exchange is analyzed, the statistical properties of stock prices and trade volumes are studied. We select the daily data for Chinese stock market during the years2002-2006, by analyzing the data, we discuss the statistical properties of fat tails phenomena and the power law distributions for the daily stock prices and trade volumes. In the second part, we consider the fat tails phenomena and the power law distributions of Shanghai Stock Exchange Index and Shenzhen Stock Exchange Index during the years2001-2006by Zipf plot method.
In Chapter7, some mathematical finance questions related to this thesis are listed.
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