  A bubble is characterized by the presence of an underlying asset whose
discounted price process is a strict local martingale under the pricing
measure. In such markets, many standard results from option pricing theory do
not hold, and in this paper we address some of these issues. In particular, we
derive existence and uniqueness results for the Black--Scholes equation, and we
provide convexity theory for option pricing and derive related ordering results
with respect to volatility. We show that American options are convexity
preserving, whereas European options preserve concavity for general payoffs and
convexity only for bounded contracts.
