A Stochastic Model of Asset Bubble Formation and Collapse

Summary

This work presents a simple stochastic differential equation for explaining how bubbles can form and collapse in asset prices. The model combines three forces: mean reversion toward a stable value, speculative social response associated with trend following, and random fluctuations. Their interaction can produce bubble-like behavior, connecting market dynamics with psychological and social processes that may also appear in nonfinancial settings.

The authors use numerical simulations to show that the model has distinct regimes as its parameters change. They also provide rigorous analysis for the weakly random regime and examine how shifts in fundamentals can ignite a bubble. The description identifies mechanisms and analytical approaches, but gives no parameter estimates, asset-specific calibration, or empirical validation against market data. The model is presented as a deliberately simple account of possible dynamics, so its results do not by themselves establish that a particular observed price rise is a bubble or predict when a real market will collapse.

Key ideas

• The model combines mean reversion, trend-following social response, and random fluctuations.
• Interactions among these forces can produce bubble formation and collapse.
• Numerical simulations identify distinct behaviors under different parameter settings.
• The analysis treats the weakly random regime rigorously and studies changes in fundamentals as a possible trigger.
• The description does not report empirical calibration to specific assets or markets.

Tags

# A simple model for asset price bubble formation and collapse


# A simple model for asset price bubble formation and collapse









We consider a simple stochastic differential equation for modeling bubbles in social context. A prime example is bubbles in asset pricing, but similar mechanisms may control a range of social phenomena driven by psychological factors (for example, popularity of rock groups, or a number of students pursuing a given major). Our goal is to study the simplest possible model in which every term has a clear meaning and which demonstrates several key behaviors. The main factors that enter are tendency of mean reversion to a stable value, speculative social response triggered by trend following and random fluctuations. The interplay of these three forces may lead to bubble formation and collapse. Numerical simulations show that the equation has distinct regimes depending on the values of the parameters. We perform rigorous analysis of the weakly random regime, and study the role of change in fundamentals in igniting the bubble.

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