Bubbles in Assets with Finite Life

We study the speculative value of a finitely lived asset when investors disagree and short sales are limited. In this case, investors are willing to pay a speculative value for the resale option they obtain when they acquire the asset. Using martingale arguments, we characterize the equilibrium speculative value as a solution to a fixed point problem for a monotone operator F. A Dynamic Programming Principle applies and is used to show that the minimal solution to the fixed-point problem is a viscosity solution of a naturally associated (non-local) obstacle problem. This obstacle type free boundary problem was the focus of Berestycki et al. (2014), who proved a comparison principle and existence and uniqueness of the viscosity solution. Combining the monotonicity of the operator F and the comparison principle we obtain several comparison of solution results. We also use the characterization of the exercise boundary of the obstacle problem to study the effect of an increase in the costs of transactions on the value of the bubble and on the volume of trade, and in particular to quantify the effect of a small transaction (Tobin) tax.