From rational bubbles to crashes

We study and generalize in various ways the model of rational expectation (RE) bubbles introduced by Blanchard and Watson in the economic literature. Bubbles are argued to be the equivalent of Goldstone modes of the fundamental rational pricing equation, associated with the symmetry-breaking introduced by non-vanishing dividends. Generalizing bubbles in terms of multiplicative stochastic maps, we summarize the result of Lux and Sornette that the no-arbitrage condition imposes that the tail of the return distribution is hyperbolic with an exponent μ<1. We then outline the main results of Malevergne and Sornette, who extend the RE bubble model to arbitrary dimensions d: a number d of market time series are made linearly interdependent via d×d stochastic coupling coefficients. We derive the no-arbitrage condition in this context and, with the renewal theory for products of random matrices applied to stochastic recurrence equations, we extend the theorem of Lux and Sornette to demonstrate that the tails of the unconditional distributions associated with such d-dimensional bubble processes follow power laws, with the same asymptotic tail exponent μ<1 for all assets. The distribution of price differences and of returns is dominated by the same power-law over an extended range of large returns. Although power-law tails are a pervasive feature of empirical data, the numerical value μ<1 is in disagreement with the usual empirical estimates μ≈3. We then discuss two extensions (the crash hazard rate model and the non-stationary growth rate model) of the RE bubble model that provide two ways of reconciliation with the stylized facts of financial data.