Shorting in Speculative Markets

We propose a continuous-time model of trading with heterogeneous beliefs. Risk-neutral agents face quadratic costs-of-carry on positions and thus their marginal valuation of the asset decreases with the size of their position, as it would be the case for risk-averse agents. In the equilibrium models of heterogeneous beliefs that followed the work by Harrison and Kreps, investors are risk-neutral, short-selling is prohibited and agents face constant marginal costs of carrying positions. The resulting resale option guarantees that the equilibrium price exceeds the price of the asset in a static buy-and-hold model where speculation is ruled out and this difference is identified as a bubble. Our model features three novelties to this literature. First, increasing marginal costs entail that the price depends on asset supply. Second, in addition to the resale option, agents also value an option to delay, and this may cause the market to equilibrate below the buy-and-hold price. Third, we introduce the possibility of short-selling. Our model shows that when shorting is very costly, price formation is dominated by optimists and pessimists’ views are hardly reflected. On the other hand, an unexpected decrease in shorting costs may lead to the collapse of a bubble; this links the financial innovations that facilitated shorting of mortgage backed securities to the subsequent collapse of prices relative to fundamentals. We use a Hamilton–Jacobi–Bellman equation of a novel form to describe the equilibrium of our model and derive comparative statics results.