Two frameworks for pricing defaultable derivatives. (June 2019)

• Record Type: Journal Article
• Authors: Zaevski, Tsvetelin S.; Kounchev, Ognyan; Savov, Mladen
• Is Part Of: Chaos, solitons and fractals. Volume 123(2019)
• Page Start: 309
• Page End: 319
• Publication Date: 2019-06
• Subjects: Stopping times -- Default -- Risk-neutral measure -- Asset pricing -- Derivative pricing -- Convertible bonds; C22 -- C25 -- G12 -- G13; 35Q91 -- 60G44 -- 91G20 -- 91G40; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2019.04.025
• Languages: English
• ISSNs: 0960-0779

Abstract

Highlights: We generalize the classical Merton–Black–Scholes model by introducing a stochastic time. We use various assumptions for the underlying asset (Gaussian or Lévy) and for the stopping time. We derive the risk-neutral condition and therefore the set of the risk-neutral measures. We derive the PDEs for the derivative prices and solve them by the default premium. As an example, we derive the prices of a convertible bond and a European call. Abstract: 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 Lévy 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.