[Paper Review] A simple model for asset price bubble formation and collapse

Alexander Kiselev, Lenya Ryzhik|arXiv (Cornell University)|Sep 1, 2010

Complex Systems and Time Series Analysis37 references7 citations

TL;DR

This paper proposes a simple stochastic differential equation modeling asset price bubbles driven by mean reversion, trend-following speculation, and random fluctuations. It demonstrates that parameter regimes lead to distinct behaviors, including bubble ignition via sudden fundamentals shifts, with rigorous analysis in the weakly random limit and numerical evidence of regime transitions.

ABSTRACT

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.

Motivation & Objective

• To develop a minimal, interpretable model of asset price bubbles with clear psychological and economic drivers.
• To study how mean reversion, speculative trend-following, and random noise interact to produce bubble formation and collapse.
• To rigorously analyze the weakly random regime and assess the role of fundamentals in triggering bubbles.
• To explore the statistical and dynamic behavior of the model, including regime shifts and bubble persistence.
• To lay groundwork for future extensions to spatially distributed systems and statistical properties of returns.

Proposed method

• A stochastic differential equation (SDE) is formulated with three components: mean reversion to a stable price, speculative trend-following response, and additive Brownian motion noise.
• The model incorporates a nonlinear feedback term representing speculative behavior, proportional to price deviation from a moving benchmark.
• Numerical simulations are used to explore parameter regimes and visualize bubble dynamics under varying conditions.
• Rigorous analysis is performed in the weakly random limit, focusing on the stability and transition behavior of the system.
• The model is extended to spatially distributed systems using a graph-based framework with inter-city price diffusion.
• Theoretical scaling limits are conjectured to connect the continuous SDE to discrete bubble models like Blanchard-Watson.

Experimental results

Research questions

• RQ1 Under what parameter conditions does the model exhibit bubble formation and collapse?
• RQ2 How does a sudden jump in fundamentals trigger bubble ignition?
• RQ3 What is the role of random fluctuations in enabling or suppressing bubble dynamics?
• RQ4 Can the model's behavior be rigorously analyzed in the weakly random regime?
• RQ5 How do spatial interactions affect bubble propagation and localization in real estate or market networks?

Key findings

• The model exhibits distinct dynamical regimes depending on parameter values, including stable, oscillatory, and explosive (bubble) phases.
• A large enough jump in fundamentals can trigger bubble ignition, with probability increasing as the jump size approaches the noise amplitude ν.
• Numerical simulations show that bubbles can form even with small noise when speculative feedback is strong.
• In the weakly random regime, the system's behavior can be rigorously analyzed, supporting the existence of regime transitions.
• The model suggests that bubbles may arise not from fundamentals alone, but from the interplay of psychology and noise.
• Spatial extensions suggest possible mechanisms for bubble contagion through price diffusion across interconnected markets.