This paper studies asset price bubbles in a continuous time model using the local martingale framework. Providing careful definitions of the asset's market and fundamental price, we characterize all possible price bubbles in an incomplete market satisfying the “no free lunch with vanishing risk (NFLVR)” and “no dominance” assumptions. We show that the two leading models for bubbles as either charges or as strict local martingales, respectively, are equivalent. We propose a new theory for bubble birth that involves a nontrivial modification of the classical martingale pricing framework. This modification involves the market exhibiting different local martingale measures across time—a possibility not previously explored within the classical theory. Finally, we investigate the pricing of derivative securities in the presence of asset price bubbles, and we show that: (i) European put options can have no bubbles; (ii) European call options and discounted forward prices have bubbles whose magnitudes are related to the asset's price bubble; (iii) with no dividends, American call options are not exercised early; (iv) European put-call parity in market prices must always hold, regardless of bubbles; and (v) futures price bubbles can exist and they are independent of the underlying asset's price bubble. Many of these results stand in contrast to those of the classical theory. We propose, but do not implement, some new tests for the existence of asset price bubbles using derivative securities.