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

Abstract

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.

Keywords:

• Financial bubbles
• Momentum
• Positive feedback
• Time-horizon
• Quasi-likelihood
• Finite-time-singularity

JEL Classification:

• C52
• G01
• G17

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

† Below we will allow for processes with explosion and thus deal with solutions S and coefficient functions f,g being defined only on stochastic intervals $[0,\zeta )$ for a random time $\zeta :\mathrm{\Omega }\to [0,\mathrm{\infty })$.

† This description reflects information provided by professionals through many private communications.

‡ Up to a random time, at which the processes may explode. For (pathwise) uniqueness and (strong) existence of solutions see, e.g. Theorem 3.1 and 3.2 in Ikeda and Watanabe (1981). For details on explosion see comments below.

† To avoid confusion with the term transient as used in economics for temporary phenomena, we explicitly add diffusive.

Additional information

Funding

LL acknowledges financial support from the National Natural Science Foundation of China (Grant No. 71771086) and (Grant No. 71301051). MS was supported by the Swiss National Foundation with grant No. 2-77156-16 entitled Financial mathematics of positive-feedback bubbles and crashes.