Path properties of volatility: A statistical analysis

我们发展统计理论来推断随机过程波动性的路径性质。波动率是描述随机过程演化过程中不确定性的关键变量。它的路径性质决定了波动率的平稳性和相依结构，从而决定了波动率的最优估计方法、预测技术和持续性。尽管波动率的路径属性对应用和当前实证工作的主题特别感兴趣，但到目前为止还没有统计基础。最近，关于随机过程的路径性质的高级估计和检验程序从直接观测中得到了重要的贡献。关键的区别和困难在于，波动性是潜在的，因此无法直接观察到。基于涉及预估波动率的统计数据，两位申请者在第一篇侧重于变点分析的文章中开始朝着这个新方向工作。利用最具创新性的波动率估计和直接观测对路径属性的推断，我们的目标是用最优的统计方法建立一个新的研究链。目前，文献中提出了相互矛盾的波动率模型。为了建立适当的波动率模型，需要更多关于波动率路径属性的知识。我们考虑了一类丰富的随机波动过程，以协调不同的风格化事实。在这个项目中发展的理论将提供证据，关于哪些模型是合适的，而且，路径属性是持久的还是时变的。对这项工作的特别兴趣是由金融计量经济学推动的。波动率是描述价格演变中的市场风险的流行概念。因此，可靠的波动率估计是风险分析的关键要素。我们的主要关注点是在最高可用记录频率下的日内高频数据。随着大量成交量(目前近70%)归因于高频交易，对此类高频数据的建模和分析变得越来越重要。同时，需要考虑特定的市场摩擦，从而引发嘈杂的价格观察。我们设计了两种噪声规范：经典的集中市场微观结构噪声模型和适用于限价订单记录价格的不规则噪声。这是在三个互补的一揽子工作方案中进行的。申请者提出了一个工作包，提供了新的方法学基础。在一个理想化的框架中，我们确定了识别某些路径属性和构建有效的路径恢复技术所需的最小信息量。这两个博士项目的重点是针对这两个相关但不同的噪声观测模型的最佳方法。这些现实模型的结构更复杂，需要几项创新来解决波动性路径属性的识别问题。

We develop statistical theory to infer path properties of the volatility of a stochastic process. Volatility is the key quantity to describe uncertainty in the evolution of a stochastic process. Its path properties determine the smoothness and dependence structure of volatility and thus optimal estimation methods, forecasting techniques and the persistence of volatility. Even though path properties of volatility are of particular interest for applications and subject of current empirical work, there is so far no statistical groundwork. Recently, there have been important contributions to advance estimation and testing procedures on path properties of stochastic processes from direct observations. The key difference and difficulty is that volatility is latent and thus not directly observable. Based on statistics involving pre-estimated volatility, the two applicants started to work in this new direction in a first article focusing on change-point analysis. Exploiting the most innovative contributions regarding volatility estimation and inference on path properties from direct observations, we aim to establish a novel strand of research with optimal statistical approaches. Currently, conflicting models for volatility are put forward in the literature. In order to build adequate volatility models, more knowledge on the path properties of volatility is required. We consider a rich class of random volatility processes to reconcile different stylized facts. The theory developed in this project will provide evidence about which models are suitable and, moreover, if path properties are persistent or time-varying. Particular interest in this work is motivated by financial econometrics. Volatility is the prevailing concept to describe market risk in price evolutions. Reliable volatility estimates are thus key ingredients for risk analysis. Our main focus is on intra-day high-frequency data at highest available recording frequencies. Modeling and analysing such high-frequency data becomes more and more important as much volume, currently almost 70 %, is attributed to high-frequency trading. At the same time, specific market frictions need to be taken into account, inducing noisy price observations. We design approaches for two noise specifications: The classical centred market microstructure noise model and irregular noise suitable for recorded prices from limit order books. This is pursued in three complementary work packages. One work package, providing novel methodological groundwork, is addressed by the applicants. In an idealized framework, we determine the minimal amount of information necessary to identify certain path properties and construct efficient techniques for their recovery. The two doctoral projects focus on optimal methods for the two related yet different noisy observation models. The more involved structure of these realistic models ask for several innovations to address the identification of the path properties of volatility.