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. Preprint submitted to Elsevier Preprint 1 February 2008 1 The model of rational bubbles Blanchard [5] and Blanchard and Watson [6] originally introduced the model of rational expectations (RE) bubbles to account for the possibility, often discussed in the empirical literature and by practitioners, that observed prices may deviate significantly and over extended time intervals from fundamental prices. While allowing for deviations from fundamental prices, rational bubbles keep a fundamental anchor point of economic modelling, namely that bubbles must obey the condition of rational expectations. In contrast, recent works stress that investors are not fully rational, or have at most bound rationality, and that behavioral and psychological mechanisms, such as herding, may be important in the shaping of market prices [41,33,34]. However, for fluid assets, dynamic investment strategies rarely perform over simple buy-and-hold strategies [30], in other words, the market is not far from being efficient and little arbitrage opportunities exist as a result of the constant search for gains by sophisticated investors. Here, we shall work within the conditions of rational expectations and of no-arbitrage condition, taken as useful approximations. Indeed, the rationality of both expectations and behavior often does not imply that the price of an asset be equal to its fundamental value. In other words, there can be rational deviations of the price from this value, called rational bubbles. A rational bubble can arise when the actual market price depends positively on its own expected rate of change, as sometimes occurs in asset markets, which is the mechanism underlying the models of [5] and [6]. In order to avoid the unrealistic picture of ever-increasing deviations from fundamental values, Blanchard [6] proposed a model with periodically collapsing bubbles in which the bubble component of the price follows an exponential explosive path (the price being multiplied by at = ā > 1) with probability π and collapses to zero (the price being multiplied by at = 0) with probability 1− π. It is clear that, in this model, a bubble has an exponential distribution of lifetimes with a finite average lifetime π/(1 − π). Bubbles are thus transient phenomena. The condition of rational expectations imposes that ā = 1/δ, where δ is the discount factor. In order to allow for the start of new bubbles after the collapse, a stochastic zero mean normally distributed component bt is added to the systematic part of Xt. This leads to the following dynamical equation Xt+1 = atXt + bt, (1) where, as we said, at = ā with probability π and at = 0 with probability 1 − π. Both variables at and bt do not depend on the process Xt. There is a huge literature on theoretical refinements of this model and on the empirical detectability of RE bubbles in financial data (see [11] and [1], for surveys of this literature). Model (1) has also been explored in a large variety of contexts, for instance in ARCH processes in econometry [14], 1D random-field Ising models [13] using Mellin transforms, and more recently using extremal properties of the G−harmonic func-