ABSTRACT
We propose two rational expectation models of transient financial bubbles with heterogeneous arbitrageurs and positive feedbacks leading to self-reinforcing transient stochastic faster-than-exponential price dynamics. As a result of the nonlinear feedbacks, the termination of a bubble is found to be characterized by a finite-time singularity in the bubble price formation process ending at some potential critical time t̃ c, which follows a mean-reverting stationary dynamics. Because of the heterogeneity of the rational agents’ expectations, there is a synchronization problem for the optimal exit times determined by these arbitrageurs, which leads to the survival of the bubble almost all the way to its theoretical end time. The explicit exact analytical solutions of the two models provide nonlinear transformations which allow us to develop novel tests for the presence of bubbles in financial time series. Avoiding the difficult problem of parameter estimation of the stochastic differential equation describing the price dynamics, the derived operational procedures allow us to diagnose bubbles that are in the making and to forecast their termination time. The tests have been performed on four financial markets, the US S&P500 index from 1 February 1980 to 31 October 2008, the US NASDAQ Composite index from 1 January 1980 to 31 July 2008, the Hong Kong Hang Seng index from 1 December 1986 to 30 November 2008 and the US Dow Jones Industrial Average Index from 3 January 1920 to 31 December 1931. Our results suggest the feasibility of advance bubble warning using stochastic models that embody the mechanism of positive feedback.
