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Convertible Bonds in a Defaultable Diffusion Model

  • Tomasz R. Bielecki
  • Stéphane Crépey
  • Monique Jeanblanc
  • Marek Rutkowski
Conference paper
Part of the Progress in Probability book series (PRPR, volume 65)

Abstract

In this paper, we study convertible securities (CS) in a primary market model consisting of: a savings account, a stock underlying a 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 associated with a CS has a solution, meaning that super-hedging of the arbitrage value of a convertible security is feasible in the present setup for both issuer and holder at the same initial cost, and we provide the related (super-)hedging strategies. Moreover, we characterize the price of a 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 game exchange option components, and provide numerical results.

Keywords

BSDE Superhedging Markov models Variational inequalities 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Tomasz R. Bielecki
    • 1
  • Stéphane Crépey
    • 2
  • Monique Jeanblanc
    • 2
    • 3
  • Marek Rutkowski
    • 4
  1. 1.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA
  2. 2.Laboratoire Analyse et probabilitésUniversité d’Évry Val d’EssonneÉvry CedexFrance
  3. 3.Europlace Institute of FinanceParisFrance
  4. 4.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

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