摘要
对金融市场波动(Volatility)的描述是现代金融理论的核心内容之一,有关波动率大小的测度(Measurement)及其动力学机制(Dynamics)的刻画,对于资产定价理论的检验、最优资产组合的选择、衍生产品定价以及金融风险的测度和管理而言,都具有极其重要的理论和现实意义。
尽管金融波动率测度及其建模研究在过去几十年里取得了极大进展,涌现出了如GARCH类、SV、lnRV-ARFIMA等在金融理论与实务领域都得到广泛应用的众多波动模型,但这些主流的波动率测度、模型本身都存在着这样或那样的理论与方法缺陷,并由此导致了理论推论与实证结果的不一致。同时,金融物理学研究中的多分形理论(Multifractal theory)作为研究金融市场波动复杂性的一种有力工具,能够在很大程度上弥补传统方法对市场复杂波动特征描述的不足。遗憾的是,在目前绝大多数的相关研究中,这一重要方法还只停留在用于对市场多分形特征的实证检验层面。因此,本论文通过充分提炼金融价格序列多分形分析过程中所产生的对定量描述金融波动有益的间接统计信息,提出了一种新的波动率测度方法及其动力学模型,然后进一步考察了其在波动率预测、金融风险管理、衍生产品定价等领域的实际表现。具体研究内容如下：
(1)金融市场的若干复杂波动特征研究。其中包括：利用不同抽样频率的价格数据和描述性统计、收益率分布QQ图、收益率概率分布图等方法,对于金融市场收益率分布特征的研究；通过建立并估计GJRSK-M模型,实现对金融市场高阶矩波动特征的研究；基于重标极差分析(R／S)法对金融市场波动长记忆特征的研究。
(2)多分形波动率测度(Multifractal volatility, MFV)的建立及其建模。运用多分形分析中的一套基本语言,实证研究了金融市场的多分形波动特征,并探讨了这一重要典型特征与波动率测度的关系；提出了基于多分形奇异指数的全新波动率测度MFV,并对其进行了全面的统计特征检验,由此构建了用于刻画MFV动力学行为的lnMFV-ARMA模型;考察了基于多分形波动测度的条件收益率分布状况,并与基于常用GARCH模型的条件收益率分布状况进行了对比。
(3)MFV方法在金融波动率预测中的应用研究。首先,从现有的几种市场真实波动率的代理变量中,选择了平方波动率rt2作为各种波动模型对未来市场波动率预测精度的比较基准；然后,运用样本外滚动时间窗法计算了lnMFV-ARMA模型及其它多种常用波动模型(GARCH类和SV)对未来市场波动率的预测值；最后,利用现有的三种预测精度检验方法：中位数损失函数法、M-Z回归法、SPA检验法,实证检验了多分形波动模型及其它常用模型对未来市场真实波动率的预测精度。
(4)MFV方法在金融风险测度中的应用研究。以标准普尔500和上证综指两种指数为例,运用lnMFV-ARMA模型及其它常用波动模型,估计了目前在金融理论界和实务界都得到广泛应用的VaR和ES两种风险测度,然后运用似然比(Likehood ratio)检验、动态分位数回归(Dynamic quantile regression)检验、bootstrap等后验分析方法,实证比较了不同波动模型的VaR和ES测度精度。
(5)MFV方法在金融衍生品定价中的应用研究。以中国股票市场中的4只认购权证(深高CWB1、上港CWB1、马钢CWB1、武钢CWB1)为例,在B-S定价模型框架下,计算了基于多分形波动率测度参数的权证价格：然后,以权证产品的市场价格为基准,与基于其它常用波动率测度参数的定价精度进行了比较。
In modern financial theory, description of financial asset volatility act as one of the core role. The usefulness of volatility measurement and its dynamics has important significance to academic and practice.
Measurements and models to describe and predict financial asset volatility abound and many volatility models, such as GARCH models, SV and lnRV-ARFIMA, are presented to be used in financial theory and practice. However, these mainstream models are all defective, hence bring on inconsistency between theory inferrer and empirical results. As an powerful tool to describe complexity of financial volatility, multifractal theory can fetch up limitations of traditional ways. However most of these studies focus on empirical tests of multifractality in different financial data sets. So we wonder whether multifractal analysis can contribute to the measurement and forecasting accuracy of volatility in financial markets. Along with this notion, we propose a so-called multifractal volatility (MFV) measure and its dynamic model based on the multifractal spectrum of high-frequency price movements within one trading day. We further examined MFV's performance in volatility forecasting, risk measuring and financial derivatives pricing. The main research content are as follows:
(1) Research on several complex features of volatility, such as inspection to distribution characteristics of financial returns based on different frequency data and descriptive statistics, QQ plot and probability distribution plot; analysis of higher-moments volatility characteristics based on GJRSK-M model; research of long-memory feature of volatility based on R/S analysis.
(2) Construction and modeling of multifractal volatility (MFV). Through one group of multifractal language, we empirically studied multifractal feature of financial volatility and further discussed relationship between this important characteristics and volatility measuring. Based on analysis above, we constructed a new volatility measure, named MFV, and comprehensively investigated its statistical property. Consequently, a time series model, lnMFV-ARMA, is used to describe dynamic characteristics of MFV. Finally, distribution of conditional return based on MFV is investigated and compared with those based on GARCH model.
(3) Application of MFV in financial volatility forecasting. First, square return, rt2, is selected to be regarded as proxy of potential volatility from several volatility measurements such as rt2,|rt| and RV, et al. Second, rolling time windows is employed to forecast out-of-sample volatility based on some popular volatility models such as GARCH models and SV. Finally, three testing methods, which are median loss function, M-Z regression and SPA test, are employed to empirically test the difference of forecasting accuracy between MFV model and other popular ones.
(4) Application of MFV in financial risk measuring. Take S&P500 and SSEC as sample, we compute VaR and ES which are applied extensively in finance theory and practice based on lnMFV-ARMA and other popular models. Furthermore, some backtesting methods, such as LR test, DQR test and bootstrap, are employed to compare VaR and ES accuracy of different volatility models.
(5) Application of MFV in financial derivatives pricing. Take 4 call warrants of Chinese stock market as sample, model price based on MFV model is computed under Black-Scholes model framework. Then take market price as benchmark, the pricing accuracy difference between MFV model and other volatility models is compared.
引文
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