This paper reviews and extends the mathematical finance literature on bubbles in complete markets. We provide a new characterization theorem for bubbles under the standard no arbitrage (NFLVR) framework, showing that bubbles can be of three types. Type 1 bubbles are uniformly integrable martingales, and these can exist with an infinite lifetime. Type 2 bubbles are non-uniformly integrable martingales, and these can exist for a finite, but unbounded, lifetime. Last, type 3 bubbles are strict local martingales, and these can exist for a finite lifetime only. When one adds a no dominance assumption (from Merton [24]), only type 1 bubbles remain. In addition, under Merton’s no dominance hypothesis, put-call parity holds and there are no bubbles in standard call and put options. Our analysis implies that if one believes asset price bubbles exist and are an important economic phenomena, then asset markets must be incomplete.

Consider the postage stamp. As title to a future good (or, in this case, service) with monetary value, this humble object is essentially the same as a security. Its value, 37 cents, can be identiÞed with the present value of the service (delivery of a letter) to which its owner is entitled.

We show pathological behavior of asset price processes modeled by continuous strict local martingales under a risk-neutral measure. The inspiration comes from recent results on financial bubbles. We analyze, in particular, the effect of the strict nature of the local martingale on the usual formula for the price of a European call option, especially a strong anomaly when call prices decay monotonically with maturity. A complete and detailed analysis for the archetypical strict local martingale, the reciprocal of a three dimensional Bessel process, has been provided. Our main tool is based on a general h-transform technique (due to Delbaen and Schachermayer) to generate positive strict local martingales. This gives the basis for a statistical test to verify a suspected bubble is indeed one (or not).

In inÞnite-date models the received deÞnitions of the payoffs ofÞnite portfolio strategies imply discontinuous valuation. Accordingly, in the absence of trading restrictions, arbitrage results when inÞnite trading strategies are admitted. We propose an alternative that is free of these problems. The alternative produces a cleaner, if more abstract, treatment of equilibrium in Þnancial models in inÞnite-date settings. We consider the bearing of the revised treatment on the theory of overlapping generations models and equivalent martingale measures. 1

We show that Arrow-Debreu equilibria with countably additive prices in infinite-time economy under uncertainty can be implemented by trading infinitely-lived securities in complete sequential markets under two different portfolio feasibility constraints: wealth constraint, and essentially bounded portfolios. Sequential equilibria with no price bubbles implement Arrow-Debreu equilibria, while those with price bubbles implement Arrow-Debreu equilibria with transfers. Transfers are equal to price bubbles on initial portfolio holdings. Price bubbles arise in sequential equilibrium under the wealth constraint if some securities are in zero supply or negative prices are permitted, but cannot arise with essentially bounded portfolios. JEL Classification Codes: D50, G12, E44. Equilibrium models of dynamic competitive economies extending over infinite time play an important role in contemporary economic theory. The basic solution concept for such models is the Arrow-Debreu (or Walrasian) equilibrium. In Arrow-Debreu equilibrium it is assumed that agents simultaneously trade arbitrary

Analysis of continuous strict local martingales via h-transforms