基于α-VG分布的多元时间序列模型及其在金融建模中的应用研究

As the foundation of option pricing, risk management, portfolio selection and other financial applications, building stochastic models for asset price has been being an active research area for many years. The key is to seek models that can reproduce some well documented stylized facts of asset return series, such as heavy tails, gain/loss asymmetry and volatility clustering. In the last decades, two main types of models, the stochastic volatility (SV) models and the Generalized Autoregressive conditional heteroscedasticity (GARCH) type models, have been broadly used in various financial modeling. However, in seeking what really caused the 2008 financial crisis, people realized that it is urgent to improve our techniques to measure the risks in various financial markets, particularly in derivative market, and the crucial modification is that we must take one more stylized feature into account when constructing asset price models. This important feature, termed as the extreme dependence, basically telling about the dependencies between (extreme) financial asset-returns, though not new in empirical studies, has been increasingly observed during recent time periods in almost all international markets. And some researchers believe it is a direct consequence of globalization and relaxed market regulation in finance and insurance industry.The increasing extremal dependencies strongly impact the companies' profit contributions and may weaken the financial stability of entire industrial sectors. Recently, a lot of copula type models have been invented to study this phenomenon, using a concept called tail dependence. Though this type of model has been used in practice, to make it applicable in derivative market definitely need more efforts. To work out an option pricing method to work well with the new models, we may also need to invent some new techniques. One main problem is that all the classical option price approaches can not be directly applied in these copula based models. Recently, we find we may complete this task in a quite different way. We may construct a new multivariate time series in the spirit of GARCH, but with new quantity and a brand new probability distribution we have just developed recently. Using this new model, we can not only reproduce those well known stylized facts of asset returns, including extreme dependence, but also apply the famous Esscher transformation for option pricing. In this project, we will comprehensively study the probability and statistical issues on building the option pricing system to work with the new model. Various financial applications, risk management and portfolio optimization will also be considered.

建立能准确刻画收益率序列特征，又便于应用的模型是金融数学多年的热点，也是进一步建立期权定价模型、风险管理模型及最优投资组合模型的基础。对以上序列厚尾、尖峰、向左倾斜，及Volatility Clustering等特征的研究产生了若干有影响的模型。然而，对当前经济危机的反思让人们意识到，在全球化的大环境之下，亟待发展新的模型来刻画极值相依特征，以提升各类金融市场的风险度量和管理的水平。近年来，沿着这一方向涌现了大量Corpula类模型。这些模型虽各具优点，但均尚未发展起与之成功配套的期权定价理论。由于构造的特点，经典期权定价方法要在此类模型中直接应用远非易事。最近，申请者及合作者发现，基于他们建立的α-VG分布族，有望构造出一类新多元时间序列模型，既能刻画上述实证特征，又能借助经典方法完成期权定价。本项目拟对新模型相关的概率统计理论问题及期权定价，风险度量，投资组合等应用问题展开深入研究。

项目首先创立了一类新的分布族α-VG，该分布族具有如下特征：1.通过其中参数α的变化 能够细致的刻画程度不同的尖峰、厚尾，同时又能够以参数β的变化反应α-VG分布向左或向右的倾斜程度，这就为量化刻画金融收益率序列众所周知的尖峰、厚尾不对称现象提供了一个基础的概率模型。此外，对于金融市场风险管理相关的一类特征现象，即多个时间序列之间的相关性常常因某些极值事件的发生而发生突变，新的分布族也为构造能够刻画这类现象的时间序列模型提供了一个有吸引力的基础模块。 2. α-VG是一类条件正态的分布族，因此依其构造的投资组合模型能够较好的利用经典的基于正态分布的相应模型建立的成果。总而言之，α-VG分布族展示出的概率特征使其成为构造能够刻画金融时间序列中最重要的一些特征，特别是极值动态相关性，同时新分布族较好的数学解析性，又使以其为基础模块构造的金融时间序列模型具备较好的可应用性。作为α-VG分布的一个重要的应用，项目组利用该分布的上述特性，构造了一个一类能够刻画多元资产序列其相关性能够依时间变化的时间序列模型，利用这类模型深入研究了金融时间序列中的comovement现象，并建立了一个能够刻画金融市场系统风险的发生发展的一个数学模型，该模型同时为度量相关关系动态变化的金融市场的总体风险提供了一个基础的新的工具。作为新模型一个应用特例，项目研究了中美两个证券市场之间在3次大的金融危机之间的相互影响。作为将新模型推向应用领域的重要一步，项目对模型中相关的统计分析展开了研究。. 作为新模型的应用与拓展，项目还对计量金融学和医学统计中的若干问题展开了研究，包括：股灾预警指标构建，期指信息挖掘与分析，金融市场投资者情绪的度量，商业银行的头寸管理，含治愈个体的复发事件的半参数回归模型等。. 项目的成果和提出的新方法将对以下问题的研究发挥重要作用： 金融市场的系统性风险的度量与预警，医院流行病的管理，环境监测。.. 关键词：动态相关，Comovement，系统风险度量，投资组合管理，α-VG 分布