Multi-dimensional Rational Bubbles and fat tails: application of stochastic regression equations to financial speculation

Abstract

We extend the model of rational bubbles of Blanchard and of Blanchard and Watson to arbitrary dimensions d: a number d of market time series are made linearly interdependent via d times d stochastic coupling coefficients. We first show that the no-arbitrage condition imposes that the non-diagonal impacts of any asset i on any other asset j different from i has to vanish on average, i.e., must exhibit random alternative regimes of reinforcement and contrarian feedbacks. In contrast, the diagonal terms must be positive and equal on average to the inverse of the discount factor. Applying the results of renewal theory for products of random matrices to stochastic recurrence equations (SRE), we extend the theorem of Lux and Sornette (cond-mat/9910141) and demonstrate that the tails of the unconditional distributions associated with such d-dimensional bubble processes follow power laws (i.e., exhibit hyperbolic decline), with the same asymptotic tail exponent mu<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. This small value mu<1 of the tail exponent has far-reaching consequences in the non-existence of the means and variances. Although power-law tails are a pervasive feature of empirical data, the numerical value mu<1 is in disagreement with the usual empirical estimates mu approximately equal to 3. It, therefore, appears that generalizing the model of rational bubbles to arbitrary dimensions does not allow us to reconcile the model with these stylized facts of financial data. The non-stationary growth rational bubble model seems at present the only viable solution (see cond-mat/0010112).