基于高频时间序列的波动率预测新模型的统计推断

In high frequency data analysis, there are many breakthroughs for the estimation problems. But for the forecasting problem, which is more important in practice, there are a lot to be improved. Especially, in the forecasting problem, there is no model derived theoretically from the underlying dynamics of price process. There is also no guidance on how to choose the predictors based on the model implied by the price dynamics. This project will study the forecasting problem starting from the forecasting of stochastic volatility. It will provide a new theoretical model of forecasting implied by the price dynamics directly and reveal all the predictors undiscovered (e.g., leverage effect) in the existing forecasting models. The new model will also present the correct form of each predictor and the contribution from each predictor to the forecasting. For those predictors which cannot be observed or directly estimated, they will be converted to estimable parameters and the statistical properties of the converted estimable parameters will be studied too. With a clear view on the forecasting of volatility, the forecasting methods will be generalized to forecast a unified form of parameters. In addition, for the asymptotic variance emerged in the aforementioned estimation problems, a unified approach of estimation will be studied, especially, for the case with jumps.

在高频数据分析中，波动率的估计问题取得了很多突破性进展，但在实际中更受关注的预测问题却仍有诸多难点亟待突破。例如考虑存在杠杆效应、波动率的波动率等影响因素的波动率预测就是其中的一大难点，尤其是缺少从数据产生的理论机制推导出的正确理论预测模型。首先，本项目拟从理论模型出发推导出新的波动率预测模型，研究新模型在存在杠杆效应和波动率的波动率等因素时的正确表达形式以及这些因素对预测的影响，建立新提出模型的统计性质。其次，研究新的波动率预测模型中各影响因素估计的渐近分布，构造渐近方差的估计方法。进一步给出当存在跳跃点时的这些估计的渐近方差的估计方法。最后，将上述新提出的模型以及估计方法推广到具有更一般形式的参数预测模型，得到模型中各影响因素的估计以及相应的渐近方差的估计。

本项目在三年研究期间，按照项目研究内容开展研究，在国内外高水平学术期刊上和会议上发表论文7篇，圆满完成了研究任务，实现项目的研究目标。波动率是金融风险管理和度量的重要指标，尤其在高频交易和海量数据出现后，对波动率的研究亟需新的统计方法，推断理论和实际应用。本项目的主要研究包括在离散时间模型和连续时间模型下的波动率估计和预测的新方法，建立相应的优良统计大样本性质，并将波动率的估计方法结合当下的统计机器学习应用于金融资产选择等实际问题当中。