  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.
