[Paper Review] Super-exponential endogenous bubbles in an equilibrium model of rational and noise traders

Taisei Kaizoji, Matthias Leiss+2 more|arXiv (Cornell University)|Sep 22, 2011

Complex Systems and Time Series Analysis36 references9 citations

TL;DR

This paper proposes a dynamic equilibrium model with rational and noise traders to explain super-exponential financial bubbles, where noise traders' herding behavior via momentum trading drives faster-than-exponential price growth. The model analytically explains log-periodic bubble patterns and accurately replicates the dotcom bubble (1995–2000), showing transient momentum profitability and endogenous bubble formation without agent switching or fundamental shifts.

ABSTRACT

We introduce a model of super-exponential financial bubbles with two assets (risky and risk-free), in which rational investors and noise traders co-exist. Rational investors form expectations on the return and risk of a risky asset and maximize their constant relative risk aversion expected utility with respect to their allocation on the risky asset versus the risk-free asset. Noise traders are subjected to social imitation and follow momentum trading. Allowing for random time-varying herding propensity, we are able to reproduce several well-known stylized facts of financial markets such as a fat-tail distribution of returns and volatility clustering. In particular, we observe transient faster-than-exponential bubble growth with approximate log-periodic behavior and give analytical arguments why this follows from our framework. The model accounts well for the behavior of traders and for the price dynamics that developed during the dotcom bubble in 1995-2000. Momentum strategies are shown to be transiently profitable, supporting these strategies as enhancing herding behavior.

Motivation & Objective

• To develop a micro-founded equilibrium model that explains endogenous financial bubbles without relying on exogenous shocks or fundamental shifts.
• To analyze how coexistence of rational and noise traders leads to transient, faster-than-exponential bubble growth.
• To provide an analytical explanation for log-periodic behavior and super-exponential dynamics observed in real market bubbles.
• To test the model’s ability to reproduce stylized facts such as fat-tailed returns, volatility clustering, and momentum profitability during the dotcom bubble.
• To assess whether rational investors can stabilize markets when noise traders exhibit herding behavior.

Proposed method

• Introduces a two-asset economy with a risky and risk-free asset, where rational investors maximize constant relative risk aversion (CRRA) utility over portfolio allocation.
• Models noise traders as following momentum strategies driven by social imitation and time-varying herding propensity.
• Incorporates random time-varying herding intensity to generate endogenous bubble dynamics without agent strategy switching.
• Uses stochastic differential equations to model wealth dynamics and price formation under heterogeneous expectations.
• Employs analytical techniques to derive conditions under which price growth exceeds exponential rates, leading to super-exponential bubbles.
• Calibrates the model to empirical data from the dotcom bubble (1995–2000), matching cumulative returns and volatility patterns.

Experimental results

Research questions

• RQ1 Can a model with rational and noise traders generate endogenous, super-exponential bubble growth without fundamental shifts?
• RQ2 What mechanism explains the observed faster-than-exponential price acceleration in financial bubbles?
• RQ3 How does momentum trading by noise traders contribute to bubble formation and volatility clustering?
• RQ4 To what extent do noise traders dominate market dynamics even when they are a minority?
• RQ5 Can the model reproduce the log-periodic behavior and transient profitability of momentum strategies seen in the dotcom bubble?

Key findings

• The model produces super-exponential bubble growth with approximate log-periodic behavior, providing a first analytical explanation for this stylized fact in the literature.
• During the dotcom bubble (1998–2000), the Internet stock index returned 932.5% over 24 months, matching empirical data and demonstrating transient momentum profitability.
• After the bubble burst, the Internet stock index lost 54.2% over five years (1998–2002), while the non-Internet index gained 8.6%, confirming momentum reversal.
• Noise traders, despite being a minority, dominate market dynamics during bubbles due to superior performance, leading to endogenous bubble formation.
• The model reproduces fat-tailed return distributions and volatility clustering through time-varying herding behavior.
• Rational investors fail to stabilize markets when noise traders exhibit persistent herding, as their wealth dynamics are overwhelmed by momentum-driven speculation.