Hysteresis phenomenon and expected shortfall of financial returns

Crouhy和Rockinger（1997）发现，坏消息会增加股票收益的波动性，而好消息对波动性的影响非常小，除非它们在几天内聚集在一起，在这种情况下会降低波动性。这表明波动性对坏消息和好消息的反应不同，即使有坏消息或好消息发布，它也可能保持不变，除非其他一些条件也得到满足。我们称这种现象为滞后现象。为了对金融过程中的滞后现象进行建模，Li等人（2015）提出了一种滞后AR模型。他们表明，滞后现象可以用滞后区来模拟，滞后区将底层过程的分布范围分为三个区域。过程的演化将保持在滞回区内，直到满足某些其他条件。Zhu et al.（2017）将Li et al.（2015）的工作扩展到滞后AR-Gestival模型，使得基础金融过程的条件均值和波动率都受到滞后现象的影响。金融过程中的滞后现象反映了滞后区与市场心理和交易者处理金融经济信息的方式之间的关系。换句话说，市场参与者可能会留在原地，直到他们获得更多信息，使他们能够评估他们可能采取的行动可能导致的金融风险。然而，这些滞后模型只关注基础金融过程的水平和波动性。因此，它们可以用来模拟滞后现象，但它们可能不适合金融风险管理。这是因为更合适的金融风险管理指标是风险价值（VaR）和预期缺口（ES），而不是水平或波动性。在金融学中，风险价值（VaR）给出了在一个预先定义的置信水平下，在给定时间范围内预期损失的最大金额，ES则度量了损失超出风险价值水平的条件预期损失。参见例如Artzner等人（1997，1999）、Acerbi和Tasche（2002）以及Yamai和Yoshiba（2002）及其参考文献。在这个项目中，我们将重点关注财务回报的ES。Nadarajah等人（2014）对ES的估计方法进行了出色的回顾，其中讨论了150多篇参考文献。他们表明，ES估计的常用方法之一是使用由参数统计模型定义的条件分布。然而，现有的滞后模型定义的分布可能无法捕获ES估计急需的尾部相关信息。这也是现有滞后模型不适合金融风险管理的另一个原因。注意，与尾部相关的信息的示例是分布尾部形状。根据ES的定义，尾部形状在ES估计中起着重要的作用，因此，了解如何对金融过程中的滞后现象进行建模，以及如何利用金融过程尾部形状的信息来估计过程的ES是很重要的。目前尚不清楚如何使用单一统计模型解决这两个问题。这是统计和金融文献中的一个空白。因此，本计画将借由发展一种新颖的基于分位数函数的统计模型来填补这一差距，使所发展的模型能够捕捉到金融过程中的滞后现象，并且也特别适合于ES估计。

Crouhy and Rockinger (1997) found that bad news increases volatility of stock returns, while good news has a very small impact on volatility except when they are clustered over a few days, which in this case reduces volatility. This suggests that volatility responds to bad and good news differently, and it may remain unchanged even when there is a release of bad or good news unless some other conditions are satisfied as well. We call this phenomenon the hysteresis phenomenon. To model the hysteresis phenomenon in financial process, Li et al. (2015) proposed a hysteretic AR model. They showed that the hysteresis phenomenon can be modelled by a hysteretic zone, which divides the distribution range of the underlying process into three regimes. The evolution of the process will remain in the hysteretic zone until some other conditions have been satisfied. Zhu et al. (2017) extended the work of Li et al. (2015) to hysteretic AR-GARCH model so that both conditional mean and volatility of the underlying financial process are affected by the hysteresis phenomenon. It is seen that the hysteresis phenomenon in financial process reflects the relation between a hysteresis zone and the psychology of the markets and the way that traders process financial and economic information. In other words, market participants may stay where they are until they obtain more information that enables them to assess the financial risks that may be caused by the actions they may take. However, these hysteretic models only focus on the level and volatility of the underlying financial process. Therefore, they can be used for modelling the hysteresis phenomenon, but they may not be suitable for financial risk management. This is because the more appropriate metrics for financial risk management are value-at-risk (VaR) and expected shortfall (ES), rather than level or volatility. In finance, VaR gives the maximum amount expected to be lost over a given time horizon, at a pre-defined confidence level, and ES measures the conditional expectation of loss given that the loss is beyond the VaR level. See e.g. Artzner et al. (1997,1999), Acerbi and Tasche (2002) and Yamai and Yoshiba (2002) and references therein. In this project, we will focus on the ES of financial returns. Nadarajah et al. (2014) gave an excellent review on the estimation methods for ES, in which more than 150 references were discussed. They showed that one of the common methods for ES estimation is to use the conditional distribution defined by a parametric statistical model. However, the distribution defined by the existing hysteretic models may not capture the much-needed tail-related information for ES estimation. This gives another reason why the existing hysteretic models are not suitable for financial risk management. Note that an example of the information related to the tail is the distributed tail shape. According to the definition of ES, tail shape plays an important role in ES estimation.Therefore, it is important to know how to model the hysteresis phenomenon in a financial process and how to use the information about the tail shape of the financial process to estimate the ES of the process. Currently, it is not clear how to address these two issues using a single statistical model. This is a gap in the statistical and financial literature. Therefore, this project will fill the gap by developing a novel statistical model based on quantile function, so that the developed model can capture the hysteresis phenomenon in the financial process, and it is also particularly suitable for ES estimation.