A simple mechanism for financial bubbles: time-varying momentum horizon

Building on the notion that bubbles are transient self-fulfilling prophecies created by positive feedback mechanisms, we construct the simplest continuous price process whose expected returns and volatility are functions of momentum only. The momentum itself is measured by a simple continuous moving average of past prices over a given time horizon. We introduce a simple dynamics of the time horizon used by the representative investor, which is motivated by the race of trend following agents to forerun their competitors. We provide the full set of solutions, which includes an explosive regime where the price and momentum explodes stochastically in finite time to infinity, transient price dynamics escaping to infinity and recurrent behaviors, where the momentum remains either strictly positive or undergoes instantaneous reflections at the origin. The proposed price generating process produces price dynamics that are in agreement with the main qualitative properties of empirical financial time series. Moreover, it produces realistic regime shifts between non bubble and bubble regimes. We construct a quasi-likelihood methodology to calibrate the model to empirical financial time series, which is applied to an Internet index and a ‘brick and mortar’ index, over the period of the dotcom bubble and its subsequent crash, from Jan. 1998 to Dec. 2002. The Wilks test of nested hypotheses shows a very strong skill in diagnosing the bubble of the Internet index and in disqualifying a bubble in the ‘brick and mortar’ index.