Motivated by the behavior of asset prices, trading volume, and price volatility during episodes of asset price bubbles, we present a continuous‐time equilibrium model in which overconfidence generates disagreements among agents regarding asset fundamentals. With short‐sale constraints, an asset buyer acquires an option to sell the asset to other agents when those agents have more optimistic beliefs. As in a paper by Harrison and Kreps, agents pay prices that exceed their own valuation of future dividends because they believe that in the future they will find a buyer willing to pay even more. This causes a significant bubble component in asset prices even when small differences of beliefs are sufficient to generate a trade. In equilibrium, bubbles are accompanied by large trading volume and high price volatility. Our analysis shows that while Tobin’s tax can substantially reduce speculative trading when transaction costs are small, it has only a limited impact on the size of the bubble or on price volatility.