Debt–equity swap with finite time horizon—variational inequality approach

This paper concerns the finite-horizon optimal reorganization problem under debt–equity swap. The model of equity is formulated as a parabolic variational inequality, or equivalently, a free boundary problem, where the free boundary corresponds to the optimal reorganization boundary. The existence and uniqueness of the solution are proven and the behavior of the free boundary, such as smoothness, monotonicity and boundedness, is studied. To the best of our knowledge, this is the first complete set of results on debt–equity swap for finite maturity obtained using PDE techniques.