An Economic Bubble Model and Its First Passage Time

We introduce a new diffusion process Xt to describe asset prices within an economic bubble cycle. The main feature of the process, which differs from existing models, is the drift term where a mean-reversion is taken based on an exponential decay of the scaled price. Our study shows the scaling factor on Xt is crucial for modelling economic bubbles as it mitigates the dependence structure between the price and parameters in the model. We prove both the process and its first passage time are well-defined. An efficient calibration scheme, together with the probability density function for the process are given. Moreover, by employing the perturbation technique, we deduce the closed-form density for the downward first passage time, which therefore can be used in estimating the burst time of an economic bubble. The object of this study is to understand the asset price dynamics when a financial bubble is believed to form, and correspondingly provide estimates to the bubble crash time. Calibration examples on the US dot-com bubble and the 2007 Chinese stock market crash verify the effectiveness of the model itself. The example on BitCoin prediction confirms that we can provide meaningful estimate on the downward probability for asset prices.