股价泡沫的逆CIR模型研究

林黎, 郑海涛, 覃筱

系统工程理论与实践 ›› 2022, Vol. 42 ›› Issue (1) : 46-59.

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系统工程理论与实践 ›› 2022, Vol. 42 ›› Issue (1) : 46-59. DOI: 10.12011/SETP2021-0491
论文

股价泡沫的逆CIR模型研究

    林黎1, 郑海涛2, 覃筱3
作者信息 +

The study on inverse CIR bubble model

    LIN Li1, ZHENG Haitao2, QIN Xiao3
Author information +
文章历史 +

摘要

按照股价泡沫的定义，泡沫识别的本质是一个联合检验问题，面临着二难逻辑困境.不少学者因此尝试绕开泡沫定义，通过假设投机的行为特征来直接提出对应的泡沫价格随机过程模型，然后围绕这些模型的统计性质来创建识别方法.然而，目前的随机过程模型对泡沫形态的限制性较强，忽略了很多的泡沫形态，导致泡沫识别的灵敏度不高.另外，大部分模型缺乏对泡沫崩溃风险率的动态建模，使得对泡沫的实时预警缺乏理论支持.为了克服传统模型的这些不足，本文提出了一个新的股价泡沫模型.由于该模型的解服从倒数化的Cox-Ingersoll-Ross （CIR）过程，简称为逆CIR泡沫模型.该模型经济学意义明确且形式简约，便于估计.本文证明，该模型兼容传统泡沫模型的非线性风险溢价和价格暂态超指数膨胀特征.同时还证明，逆CIR模型能够解释传统模型所无法解释的泡沫实证效应-泡沫崩溃前的1）"暴雨前寂静"现象；2）"高位滞涨"现象.另外，此模型包含内生且具有明确经济意义的泡沫崩溃风险率，可作为实时预警指标.本文给出了崩溃风险率的计算方法-求解高斯超几何函数方程.对我国股市2015年泡沫崩溃的风险率实证估计表明，该模型具有良好的识别和预警效果.

Abstract

According to the definition of stock price bubble, the detection of bubble ought to be a test of joint hypothesis, that is somehow confronted with logical predicament. Therefore, majority of scholars choose to by pass the definition and attempt to directly model the bubbles evolution as some proper stochastic processes, which are rationalized with the assumptions on the speculative behaviors. With the the statistical properties of models of the stochastic processes, some bubble detection methods are constructed accordingly. However, the existing stochastic process models have somehow strong restrictions on the nature of bubbles, which lead much cases which truly have bubbles are overlooked by the models. As a result, the sensitivity of the methods for detecting bubbles based on those models are lower. Besides, most models suffer from the lack of dynamics of bubble burst hazard rate, and thus lead to a unconvincing status for the real-time warning for upcoming bubble crash. To remedy traditional models' deficiency mentioned above, we propose a new bubble model in this paper. This model is called inverse CIR bubble model as the solution of stock price follows a reciprocal Cox-Ingersoll-Ross stochastic process. It has crystal clear economics implication as well as parsimonious mathematical specification with only three crux parameters. Hence the model is easier to make explanation and calibration. Based on the derived analytical results, this paper rationalizes the presence of nonlinear risk premium and transient super-exponential growth of price in a bubble. Meanwhile, this model can help us in explaining for some abnormal empirical results during bubble maturation, such as the "lull before the storm" and the "stagflation in high volatile plateau" before the bubble crash. These empirical results can however not reconcile with the traditional bubble models. Further, we show that this model has an endogenous hazard rate for the crash, which is of clear economics implication and can be used as a real-time warning indicator. Meanwhile, This paper prove the hazard rate satisfies a specific equation in the form of Gaussian hypergeometric function. As shown by the empirical studies on 2015 bubble for Chinese stock market, our model not only enjoys good explanatory power but also provides timely alarms for the upcoming collapse of the bubble.

关键词

股价泡沫 / 泡沫崩溃 / 风险预警 / 超指数膨胀 / 逆CIR过程

Key words

stock bubbles / bubble crash / early-warning for risk / super-exponential growth / inverse CIR process

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林黎 , 郑海涛 , 覃筱. 股价泡沫的逆CIR模型研究. 系统工程理论与实践, 2022, 42(1): 46-59 https://doi.org/10.12011/SETP2021-0491
LIN Li , ZHENG Haitao , QIN Xiao. The study on inverse CIR bubble model. Systems Engineering - Theory & Practice, 2022, 42(1): 46-59 https://doi.org/10.12011/SETP2021-0491
中图分类号： F830.9   

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基金

国家自然科学基金（71771086，71873012，72033001）
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