金融波动模型及其在中国股市的应用
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摘要
金融风险的规避与防范一直是投资理论与投资实践的中心课题。自从Markwitz 于1952 年提出均值——方差模型以来,对金融风险及其它金融问题的研究开始呈现数量化的趋势,各种模型与理论不断出现,其中影响最大、在今天仍有影响的有:有效市场假说、证券组合理论、资本资产定价模型、套利定价理论、期权定价方程、资产结构理论,这些理论或模型除有效市场理论与资产结构理论外都与波动有着直接的不可分割的联系,这正是本文研究的理论背景。本文研究的实际背景则是金融市场波动所呈现出来的各种特征。
通过对大量金融时间序列的研究,人们发现时间序列的波动(也即不确定性)呈现出时变性,即波动不是固定不变的,而是随时间变化的。描述时变波动的模型一般有两类,即自回归条件方差(ARCH)模型和随机波动(SV)模型。本文着重对后者的建模、估计及其在中国股市的应用进行了研究,其主要内容包括:
1. 本文在广泛阅读文献的基础上,从如何把离散数据与连续模型相统一的角度,系统概括了随机波动建模问题的研究进展。首先在一些概念与术语的基础上,讨论了最为一般的所谓随机自回归随机波动(SARV)模型,这类模型包含了迄今文献上出现的所有随机波动模型;接下来,讨论了离散时间随机波动的有关建模问题,并介绍了离散时间SV 模型的各种扩展形式。
2. 模型的应用必然要涉及到参数的估计问题。对于SV 模型,学者们提出各种各样的估计方法,其中最简单、最常用的方法是伪极大似然估计(QML),这就必然涉及到函数的优化问题。由于SV 模型的(伪)似然函数的导数不能直接求得,所以其优化只能借助于不使用导数的方法,而传统方法优化结果的好坏直接依赖于初点的选择,且在实现过程中存在种种问题。为此,本文提出了一种基于禁忌遗传算法(TSGA)的QML 估计,简称为TSGA-QML。Monte Carlo试验表明TSGA-QML 方法在参数估计与波动估计上都有着很好的效果。同时,我们还用所提出的方法,对上海股市综合指数、个股、股票组合的日收益序列进行了实证分析,结果发现几乎所有的序列都呈现出较高的波动持续性,传统的投资组合理论不能消除这种持续性。
3. 对于波动持续性的描述可以有两种方式:一是用波动方程中的单位根或近单位根来描述;另一种则是在波动方程中引入分数差分算子得到长记忆随机波动(LMSV) 模型。本文第四章就是从第一个角度出发来研究SV 模型的波动持续性,即对波动过程进行单位根检验。在此部分,我们在介绍单位根检验的一般方法基础上,研究了SV 模型的单位根检验问题,并以之对上海股市收益波动的持续性进行了分析,结果发现:我们所选的所有序列的收益波动都拒绝了单位根假设,
Avoiding financial risk is always the main subject of investment theories and practice. Since Markwitz put forward mean-variance model in 1952, research on financial risk and other related issues have begun to depend on mathematical methods and all kinds of mathematical models and theories have come into being, the most famous ones of which are Efficient Market Hypothesis (EMH), Portfolio Theory, Capital Asset Pricing Model (CAPM), Arbitrage Pricing Theory (APT), Option Pricing Equation and Capital Structure Theory and all of them except EMH and Capital Structure Theory are associated with volatility, which is the theory background of this dissertation. The empirical backgrounds of this dissertation are the volatility characters in financial markets.
A large body of research suggests that volatility in financial market is time varying; that is to say, volatility is not constant but varying with time. In order to describe the time varying of volatility, two classes models?autoregressive conditional heteroskedasitic (ARCH) model and stochastic volatility (SV) model, were proposed. The main research objects of this dissertation are stochastic volatility modeling, estimating and their applications in Chinese stock markets.
Based on reading document widely, we summarize the development of stochastic volatility modeling from viewpoint of how to unite discrete data and continuous models. After introducing the related notation and terminology, we discuss stochastic autoregressive volatility (SARV) model, which encompasses various representations of stochastic volatility model already available in the literature, where after we research the discrete time stochastic volatility modeling in detail, extensions of SV model are also given.
The applications of the models are necessarily involved in their estimation; people have constructed many estimation methods of SV model, the simplest one is quasi-maximum likelihood (QML) estimation and it is inevitable to face with the optimization of the functions when using QML to estimate SV model. The likelihood of SV model can’t be differentiated, so we must use free derivative methods to maximize it, but the results of the classical optimization methods highly depend on the choice of initial point and many problems often appear when implementing them. Therefore, we put forward a new method?TSGA-QML (quasi-maximum likelihood based on tabu search genetic algorithm), Monte Carlo experiments show that the method performs well with respect to both parameter estimates and volatility estimates. We also illustrate the method by analyzing daily returns of index, stocks and portfolios on Shanghai Stock Exchange and find that their volatility displays high persistence while traditional portfolio can’t avoid the persistence.
Persistence in volatility can be characterized by (near) unit root and long memory. We study the persistence from the former viewpoint in chapter 4; that is to say, we test for a unit root in volatility process. After presenting the usual unit root test methods, chapter 4 discusses unit root test for SV model and analyzes persistence in volatility for the daily stock returns on Shanghai Stock Exchange; we obtain that the unit root
hypothesis are all rejected for the series we study, which suggests that they don’t satisfy the definition of persistence in volatility of Li and Zhang (2000). Meanwhile, this dissertation considers persistence in volatility from viewpoint of long memory. In chapter 5, based on introducing the concepts, theories, models with respect to long memory, we research the statistical characters, volatility proxies, the relation with ARFIMA model and temporal aggregation of long memory stochastic volatility (LMSV) model, we also analyze the finite properties of QML estimators based on TSGA of LMSV model. Finally, empirical analysis on long memory of Shanghai Stock Exchange is conducted with the related theories and methods. In order to describe the relationship on volatility among different financial markets, chapter 6 extends univariate LMSV model to the multivariate one and gives its spectral-likelihood estimator, the test procedure for fractional cointegration is also developed under multivariate LMSV model. Finally, we apply the model and method to analyze daily returns of Shanghai Composite Index and Shenzhen Component Index and find their volatilities are fractional cointegrated. Chapter 7 provides a Bayesian detecting method for structural change in volatility and construct evident test with stock returns in Shanghai stock market, based on which we introduce structural change into SV model and LMSV model. Empirical evidence and Monte Carlo simulation confirm that the persistence parameter of SV model is very sensitive to structural change in volatility while memory parameter of LMSV model is also quite sensitive to it. In the end, this dissertation provides a selective summary of the most important developments in the field of financial econometrics in the past two decades, along with a discussion of promising avenues for future research. The contents of this dissertation are the components of National Natural Science Fund Persistence in Volatility of Multivariate Time Series and Its Applications in Financial System (No: 70171001).
通过对大量金融时间序列的研究,人们发现时间序列的波动(也即不确定性)呈现出时变性,即波动不是固定不变的,而是随时间变化的。描述时变波动的模型一般有两类,即自回归条件方差(ARCH)模型和随机波动(SV)模型。本文着重对后者的建模、估计及其在中国股市的应用进行了研究,其主要内容包括:
1. 本文在广泛阅读文献的基础上,从如何把离散数据与连续模型相统一的角度,系统概括了随机波动建模问题的研究进展。首先在一些概念与术语的基础上,讨论了最为一般的所谓随机自回归随机波动(SARV)模型,这类模型包含了迄今文献上出现的所有随机波动模型;接下来,讨论了离散时间随机波动的有关建模问题,并介绍了离散时间SV 模型的各种扩展形式。
2. 模型的应用必然要涉及到参数的估计问题。对于SV 模型,学者们提出各种各样的估计方法,其中最简单、最常用的方法是伪极大似然估计(QML),这就必然涉及到函数的优化问题。由于SV 模型的(伪)似然函数的导数不能直接求得,所以其优化只能借助于不使用导数的方法,而传统方法优化结果的好坏直接依赖于初点的选择,且在实现过程中存在种种问题。为此,本文提出了一种基于禁忌遗传算法(TSGA)的QML 估计,简称为TSGA-QML。Monte Carlo试验表明TSGA-QML 方法在参数估计与波动估计上都有着很好的效果。同时,我们还用所提出的方法,对上海股市综合指数、个股、股票组合的日收益序列进行了实证分析,结果发现几乎所有的序列都呈现出较高的波动持续性,传统的投资组合理论不能消除这种持续性。
3. 对于波动持续性的描述可以有两种方式:一是用波动方程中的单位根或近单位根来描述;另一种则是在波动方程中引入分数差分算子得到长记忆随机波动(LMSV) 模型。本文第四章就是从第一个角度出发来研究SV 模型的波动持续性,即对波动过程进行单位根检验。在此部分,我们在介绍单位根检验的一般方法基础上,研究了SV 模型的单位根检验问题,并以之对上海股市收益波动的持续性进行了分析,结果发现:我们所选的所有序列的收益波动都拒绝了单位根假设,
Avoiding financial risk is always the main subject of investment theories and practice. Since Markwitz put forward mean-variance model in 1952, research on financial risk and other related issues have begun to depend on mathematical methods and all kinds of mathematical models and theories have come into being, the most famous ones of which are Efficient Market Hypothesis (EMH), Portfolio Theory, Capital Asset Pricing Model (CAPM), Arbitrage Pricing Theory (APT), Option Pricing Equation and Capital Structure Theory and all of them except EMH and Capital Structure Theory are associated with volatility, which is the theory background of this dissertation. The empirical backgrounds of this dissertation are the volatility characters in financial markets.
A large body of research suggests that volatility in financial market is time varying; that is to say, volatility is not constant but varying with time. In order to describe the time varying of volatility, two classes models?autoregressive conditional heteroskedasitic (ARCH) model and stochastic volatility (SV) model, were proposed. The main research objects of this dissertation are stochastic volatility modeling, estimating and their applications in Chinese stock markets.
Based on reading document widely, we summarize the development of stochastic volatility modeling from viewpoint of how to unite discrete data and continuous models. After introducing the related notation and terminology, we discuss stochastic autoregressive volatility (SARV) model, which encompasses various representations of stochastic volatility model already available in the literature, where after we research the discrete time stochastic volatility modeling in detail, extensions of SV model are also given.
The applications of the models are necessarily involved in their estimation; people have constructed many estimation methods of SV model, the simplest one is quasi-maximum likelihood (QML) estimation and it is inevitable to face with the optimization of the functions when using QML to estimate SV model. The likelihood of SV model can’t be differentiated, so we must use free derivative methods to maximize it, but the results of the classical optimization methods highly depend on the choice of initial point and many problems often appear when implementing them. Therefore, we put forward a new method?TSGA-QML (quasi-maximum likelihood based on tabu search genetic algorithm), Monte Carlo experiments show that the method performs well with respect to both parameter estimates and volatility estimates. We also illustrate the method by analyzing daily returns of index, stocks and portfolios on Shanghai Stock Exchange and find that their volatility displays high persistence while traditional portfolio can’t avoid the persistence.
Persistence in volatility can be characterized by (near) unit root and long memory. We study the persistence from the former viewpoint in chapter 4; that is to say, we test for a unit root in volatility process. After presenting the usual unit root test methods, chapter 4 discusses unit root test for SV model and analyzes persistence in volatility for the daily stock returns on Shanghai Stock Exchange; we obtain that the unit root
hypothesis are all rejected for the series we study, which suggests that they don’t satisfy the definition of persistence in volatility of Li and Zhang (2000). Meanwhile, this dissertation considers persistence in volatility from viewpoint of long memory. In chapter 5, based on introducing the concepts, theories, models with respect to long memory, we research the statistical characters, volatility proxies, the relation with ARFIMA model and temporal aggregation of long memory stochastic volatility (LMSV) model, we also analyze the finite properties of QML estimators based on TSGA of LMSV model. Finally, empirical analysis on long memory of Shanghai Stock Exchange is conducted with the related theories and methods. In order to describe the relationship on volatility among different financial markets, chapter 6 extends univariate LMSV model to the multivariate one and gives its spectral-likelihood estimator, the test procedure for fractional cointegration is also developed under multivariate LMSV model. Finally, we apply the model and method to analyze daily returns of Shanghai Composite Index and Shenzhen Component Index and find their volatilities are fractional cointegrated. Chapter 7 provides a Bayesian detecting method for structural change in volatility and construct evident test with stock returns in Shanghai stock market, based on which we introduce structural change into SV model and LMSV model. Empirical evidence and Monte Carlo simulation confirm that the persistence parameter of SV model is very sensitive to structural change in volatility while memory parameter of LMSV model is also quite sensitive to it. In the end, this dissertation provides a selective summary of the most important developments in the field of financial econometrics in the past two decades, along with a discussion of promising avenues for future research. The contents of this dissertation are the components of National Natural Science Fund Persistence in Volatility of Multivariate Time Series and Its Applications in Financial System (No: 70171001).
引文
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