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. Moreover, we make explicit the price processes that exclude risk-free arbitrage opportunities but allow for momentum trading strategies with time-varying horizons. The model consists finally in one specification for the non-bubble regime and a second specification for the bubble regime, with the transition from one to the other controlled by the crossing of a momentum threshold. The proposed price generating process generates the main stylized facts 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 eight empirical historical cases that exhibit large volatility bursts and are candidates for the presence of bubbles. The calibration supports the relevance of the proposed model to represent a significant component of historical bubble regimes.