基于价格极差的波动率模型

Volatility of asset prices is one of the most investigated subjects in fiancial research because of its crucial role in different financial applications, such as asset pricing, portfolio construction and risk management. Usually volatility is defined as the variance or the standard deviation of the return over a period of time, therefore, different estimators or forecasting models (such as GARCH and stochastic volatility model) for volatility are basically based on the second moment of the return series, which means that only the closing prices of the time intervals have been used. Even if high frequency data is available, the most popular volatility estimator is the so-called realized volatility (RV), which is still constructed by summing up the squared returns over much finer intervals such as five or thirty minutes. . However, in the early 1980s, some authors had already pointed out that utilizing public daily price information adequately, such as the daily high, low and the opening price, rather than using the closing price only, can derive different volatility estimators which are much more efficient (with smaller variance) than that based on the squared returns. These estimators, such as the Parkinson estimator, the Garman-Klass estiamtor and the Rogers-Satchell estimator, etc., are called extreme value estimators in the literature. For instance, the efficiency of the Parkinson estimator, based on the range (the difference between the high and low prices) of the daily price, is about 4.9 in comparison with the standard sample variance estimator.. Over the last few years, we have made some tentative progresses on both the dynamic volatility modeling with extreme value estimators for the lower frequency data and the construction of realized extreme value based volatility in the case of high frequency data. This project is a continuous work of our present research. More sepecifically, the following eight problems will be addressed: 1) How to specify an extreme value based dynamic model to account for the possible long memory of volatility? 2) Can we estimate an optimal power transformation simultaneously as we construct the dynamic models using these estimators, just as the case of APGARCH models? 3) How to compare their forecasting performance with those based on SV models? 4)Can we achieve more precise forecasts by averaging these dynamic models which utilizes different aspect of the public daily price information? Morever, as high frequency data is available and the realized extreme value based volatilities are considered, 5) how to rectify the possible biases originated by the discontinuity of the trading over a whole day? 6)How to make statistical inference for these estimators especially under the case of finite samples? 7) Does power tranformation work for these estimator as well? 8)How to accunt for the effect of possible jump component in the observed prices to these estimators

金融资产价格的波动率对于资产定价和风险管理等各类金融决策至关重要。对波动率的估计和预测方法通常是基于收益率的平方，但是理论和实证的结果均表明利用价格极差（即一段时间内最高价和最低价之差）能够更加有效地估计波动率。本项目是关于利用极差进行波动率估计和预测的系统性的研究。具体包括两个方面内容：一是利用公开得到的低频价格信息（比如每天的最高、最低以及开、收盘价等）来给出波动率的预测，包括基于极值估计的GRACH-R模型的形式设定、理论性质以及与基于极差的随机波动率（SV）模型的比较、模型平均方法的应用等；二是利用日内高频交易数据构造已实现极差波动率（RRV）来进行波动率的估计，具体问题包括在考虑无交易时间情况下已实现极差波动率的构造、基于Bootstrap的统计推断、已实现极差幂次变分统计量的构造及性质、带有跳跃过程中利用已实现极差幂次变分统计量进行波动率估计和跳跃检验等。

波动率和流动性是金融市场微观结构里面两个核心的变量，对于资产定价、投资组合以及风险管理等各类金融决策都具有非常重要的意义。由于波动率和流动性不是可以直接观测到的，利用价格的信息对其进行估计和建模一直是金融计量领域里面重要的研究问题。本课题从价格极差的角度考虑对波动率和交易成本的静态估计和动态建模，分别在能够获取低频交易数据和高频交易数据的情形下，提出了一些新的估计和建模方法，并研究比较了与以往文献中的不同方法之间的区别和性质。. 获得的主要成果包括：1）在低频数据下，考察了各种波动率的极值估计对波动率动态预测的作用，改进了传统的GARCH模型；2）利用波动率的低频极值估计构造了一类异质波动率预测模型HGARCH-X，不仅能够刻画出波动率中的长记忆性特征，而且能够接近文献中常用的基于高频数据给出的Corsi预测方法；3）基于不同采样频率下的价格极差，给出了一类对于跳跃具有稳健性的已实现信息波动率（realized information volatility，RIV）估计，提出了已实现信息幂次变差（realized information power variation, RIPV）等新的统计量，并由此构造一类新的跳跃检验统计量。4）研究了基于价格极差的有效价差的Corwin-Schultz估计的渐近性质，并与Roll的协方差估计进行了比较；5）基于价格极差提出了几种新的有效价差的矩估计方法并研究了其性质；6）基于价格极差给出了有效价差的一种拟极大似然估计，并研究了其性质；7）对于流动性的LOT度量的估计方法进行了研究，指出了文献中常用的Mixed估计方法存在的错误，分析了Y-split方法的合理性；8）基于上述结果，实证地考察了适合中国证券市场的流动性度量。. 上述结果属于基础性和理论性的研究，不仅完善了波动率和流动性估计和建模的理论基础，而且对于各类这个领域的实证研究具有极其重要的参考意义，对于实际的金融决策和金融监管也具有一定的指导意义。