长记忆波动率模型的结构性质、统计推断及应用研究

长记忆现象广泛存在于资产回报率的波动率序列。普通的GARCH类模型并不能捕捉波动率的长记忆性。本项目基于双曲GARCH类模型对长记忆波动率进行研究。我们的主要任务包括：（1）通过研究记忆参数在FIGARCH模型和ARFIMA模型中的不同作用，重新定义波动率的长记忆性，并发展有效的检验方法来检验这种长记忆性是否存在；（2）针对长记忆FIGARCH模型方差不存在的缺陷，构造一类新的协方差平稳的长记忆GARCH模型，得到宽平稳框架下模型估计和检验的渐近性质；（3）通过考虑长记忆现象产生的内在机制以及长记忆与非平稳，结构突变的关系，构造一类具有区制转移性质的长记忆GARCH模型。我们对提出的每一类模型进行结构性质，统计推断方面的研究，完善长记忆波动率模型的理论结果，同时将新提出的波动率模型应用于实际市场波动率预测，为金融预警和衍生品定价提供更精确的量化工具。

Long memory volatility exists widely in many asset returns. The common GARCH model in which autocorrelations of squares decay very quickly can not capture the property of long memory in volatility. In this proposal, we study long memory in volatility based on a family of hyperbolic GARCH models. Our main tasks include: (1) by investigating the distinction of memory parameter in the FIGARCH model and the ARFIMA model, we re-define the concept of memory in volatility series and construct effective tests to test whether long memory in volatility exist or not; (2) to overcome the shortcoming of the FIGARCH whose variance is always infinite, we propose a new long memory GARCH model with covariance stationarity and derive its asymptotics under weak stationary conditions; (3) by considering the generating mechanism of long memory and the relationship among long memory, non-stationarity and structural breaks, we construct a family of regime switching long memory GARCH models. For each kind of models mentioned above, we study its structural property and statistical inference to fulfill the theoretical consequences of long memory time series models. The new volatility models are applied to forecast the real market volatility. They provide new measurement tools for financial risk management and derivative pricing.