Local martingales, bubbles and option prices

Abstract

In this article we are interested in option pricing in markets with bubbles. A bubble is defined to be a price process which, when discounted, is a local martingale under the risk-neutral measure but not a martingale. We give examples of bubbles both where volatility increases with the price level, and where the bubble is the result of a feedback mechanism. In a market with a bubble many standard results from the folklore become false. Put-call parity fails, the price of an American call exceeds that of a European call and call prices are no longer increasing in maturity (for a fixed strike). We show how these results must be modified in the presence of a bubble. It turns out that the option value depends critically on the definition of admissible strategy, and that the standard mathematical definition may not be consistent with the definitions used for trading.