Convertible Bonds in a Defaultable Diffusion Model

by Tomasz R. Bielecki of the Illinois Institute of Technology,
Stéphane Crépey of the Université d'Évry Val d'Essonne,
Monique Jeanblanc of the Université d'Évry Val d'Essonne & Institut Europlace de Finance, and
Marek Rutkowski of the University of New South Wales & Warsaw University of Technology

December 23, 2007

Abstract: In this paper, we study Convertible Securities (CS), in particular Convertible Bonds, in a primary market consisting of: a savings account, a stock underlying the CS, and an associated CDS contract (or, alternatively to the latter, a rolling CDS more realistically used as an hedging instrument). We model the dynamics of these three securities in terms of Markovian diffusion set-up with default. In this model, we show that a doubly reflected Backward Stochastic Differential Equation related to the CS has a solution, and we provide the related (super-)hedging strategy. Moreover, we characterize the price of the CS in terms of viscosity solutions of associated variational inequalities, and we prove the convergence of suitable approximation schemes. We finally specify these results to Convertible Bonds and their Straight Bond and Option components.

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