基于随机交互系统的金融波动模型的构造与分析

发布时间：2018-03-03 19:55

本文选题：连续渗流　切入点：价格过程　出处：《北京交通大学》2012年博士论文　论文类型：学位论文

【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.
[Abstract]:This paper uses continuous seepage, random interaction system, Ising model, election model and the Zipf theorem and other methods to study the statistical characteristics of the price volatility in the stock market, the stock index fluctuation between different statistical characteristics of chain reaction, Shanghai and Shenzhen wide tail phenomenon and the power-law distribution. The price of European contingent claim process model the pricing and hedging problem. To study the statistical properties of the two-dimensional W-R model of double random phase separation line. This thesis is organized as follows.
The first chapter briefly introduces the development background of the research on financial mathematics, especially the theory of stock price fluctuation, and lists the main research results of this paper.
The second chapter through the statistical characteristics of the continuous flow theory of stock price volatility. The continuous flow process method is applied to establish a financial model, used to describe the stock price behavior, especially in the financial market investors "herd behavior". By using the stochastic method of statistical analysis, we prove that the characteristic function of the price process in convergence the characteristic function of the Black-Scholes model.
The third chapter consider the statistical characteristics of stock price between the chain reaction. The two stock index system and application of interaction of statistical physics to describe and study the stock market volatility on the interaction characteristics between the two probability distribution. In this chapter, we use stochastic analysis and double random path model to study the stock index between the chain reaction further, to reveal the asymptotic probability measure two stock index volatility model, the probabilistic properties of the single chain reaction of stock index. The financial model established by finite dimensional probability distribution of convergence is discussed.
The fourth chapter uses the theory of stochastic process and stochastic election model theory, we established a model of financial securities price includes two types of investors. We use this model to describe the financial securities market of individual securities price fluctuation. In the process of property and financial model, in addition to professional investors, we also consider ordinary investors or non professional investors, here the stopping time theory and the election model is used to establish the mathematical model and the study of non professional investors. Discuss the statistical characteristics of pricing and hedging of European contingent claims the price process model.
The fifth chapter studies the statistical properties of the two-dimensional Widom-Rowlinson model of double random phase separation line offline. The two coexist in the lattice W-R model is separated, when the chemical potential of the model is large enough, the probability distribution of convergence to describe the phase fluctuation offline was studied. The concept of backbone model is introduced. Analysis and development of the corresponding polymer weight polymer- chain and series expansion method. The existence of the 2D W-R model of double randompaths free energy.
The sixth chapter uses the fluctuation characteristics of stock price and trading volume of Zipf- diagram, Zipf- diagram method has been widely used in the physical sciences. In the first part of this chapter, the analysis from the Shanghai and Shenzhen stock stock price and trading volume data, and its statistical properties are studied. We selected the stock market Chinese 2002-2006 annual daily data, through the analysis of these data, we discuss the statistical characteristics of the wide tail phenomenon and power-law day stock price and trading volume distribution. In the second part of this chapter, we use the Zipf- method to study for 2001-2006 years during the Shanghai and Shenzhen wide tail phenomenon and the power-law distribution.
The seventh chapter lists some problems to be solved closely related to this study. This is also one of the goals of my future research work.

【学位授予单位】：北京交通大学
【学位级别】：博士
【学位授予年份】：2012
【分类号】：F224;F832.51

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