The study on inverse CIR bubble model

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.