Two frameworks for pricing defaultable derivatives

The purpose of this paper is to present two essentially different schemes for deriving the partial differential equations (PDE) for the price of the so-called defaultable derivatives. In the first one the asset price is represented as a solution of a stochastic differential equation (SDE), stopped at a stochastic time. The second one explores the idea of adding a jump process assuming that the stopping time is the moment of its first jump. We investigate also the role of the loss rate, which represents the loss of the asset at the default moment. In both cases we examine various assumptions and dependencies between the underlying asset, the stopping time, and the loss rate. We examine separately the cases when the underlying asset price is driven by a Brownian motion or by a Levy process. We give a method to solve the PDEs for the derivative prices by the use of the so-called default premium. As an example we derive a closed form formula for the price of a contingent convertible bond. (C) 2019 Elsevier Ltd. All rights reserved.