Convertible Bonds in a Defaultable Diffusion Model

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


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.


BSDE Superhedging Markov models Variational inequalities 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andersen, L. and Buffum, D.: Calibration and implementation of convertible bond models. Journal of Computational Finance 7 (2004), 1–34.Google Scholar
  2. 2.
    Ayache, E., Forsyth, P. and Vetzal, K.: Valuation of convertible bonds with credit risk. Journal of Derivatives 11 (2003), 9–29.CrossRefGoogle Scholar
  3. 3.
    Barles, G. and Souganidis, P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Analysis 4 (1991), 271–283.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bielecki, T.R., Crépey, S., Jeanblanc, M. and Rutkowski, M.: Arbitrage pricing of defaultable game options with applications to convertible securities. Quantitative Finance 8 (2008), 795–810.Google Scholar
  5. 5.
    Bielecki, T.R., Crépey, S., Jeanblanc, M. and Rutkowski, M.: Valuation and hedging of defaultable game options in a hazard process model. Journal of Applied Mathematics and Stochastic Analysis, Article ID 695798, 2009.Google Scholar
  6. 6.
    Bielecki, T.R., Crépey, S., Jeanblanc, M. and Rutkowski, M.: Defaultable options in a Markovian intensity model of credit risk. Mathematical Finance 18 (2008), 493–518.Google Scholar
  7. 7.
    Bielecki, T.R., Jeanblanc, M. and Rutkowski, M.: Pricing and trading credit default swaps in a hazard process model. Annals of Applied Probability 18 (2008), 2495–2529.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bielecki, T.R. and Rutkowski, M.: Credit Risk: Modeling, Valuation and Hedging. Springer-Verlag, Berlin, 2002.Google Scholar
  9. 9.
    Björk, T.: Arbitrage Theory in Continuous Time. Oxford University Press, Oxford, 1998.Google Scholar
  10. 10.
    Bouchard, B. and Chassagneux, J.-F.: Discrete-time approximation for continuously and discretely reflected BSDEs (2008). Stochastic Processes and their Applications, 118(12), 2269–2293.Google Scholar
  11. 11.
    Bouchard, B. and Touzi, N.: Discrete-time approximation and Monte Carlo simulation of backward stochastic differential equations. Stochastic Processes and their Applications 111(2) (2004), 175–206.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Brennan, M.J. and Schwartz, E.S.: Convertible bonds: valuation and optimal strategies for call and conversion. Journal of Finance 32 (1977), 1699–1715.CrossRefGoogle Scholar
  13. 13.
    Brennan, M.J. and Schwartz, E.S.: Analyzing convertible bonds. Journal of Financial and Quantitative Analysis 15 (1980), 907–929.CrossRefGoogle Scholar
  14. 14.
    Carr, P. and Linetsky, V.: A jump to default extended CEV model: an application of Bessel processes. Finance and Stochastics 10 (2006), 303–330.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Chassagneux, J.-F.: A discrete-time approximation for doubly reflected BSDE (2009). Advances in Applied Probability, 41.Google Scholar
  16. 16.
    Chassagneux, J.F., Crépey, S., and Rahal, A.: Pricing convertible bonds with call protection. In preparation.Google Scholar
  17. 17.
    Crandall, M., Ishii, H., and Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bulletin of the American Mathematical Society 27(1) (1992), 1–67.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Crépey, S.: About the pricing equations in finance (2011). Paris-Princeton Lectures in Mathematical Finance 2010, Lecture Notes in Mathematics, Springer, 63–203, 2011.Google Scholar
  19. 19.
    Crépey, S. and Matoussi, A.: Reflected and doubly reflected BSDEs with jumps: A priori estimates and comparison principle. Annals of Applied Probability 18 (2008), 2041–2069.Google Scholar
  20. 20.
    Cvitanić, J. and Karatzas, I.: Backward stochastic differential equations with reflection and Dynkin games. Annals of Probability 24 (1996), 2024–2056.Google Scholar
  21. 21.
    Darling, R. and Pardoux, E.: Backward SDE with random terminal time and applications to semilinear elliptic PDE. Annals of Probability 25 (1997), 1135–1159.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Davis, M. and Lischka, F.R.: Convertible bonds with market risk and credit risk. In: Applied Probability, Studies in Advanced Mathematics 26, edited by Chan R., Kwok Y.K., Yao D. and Zhang Q., American Mathematical Society and International Press, 2002, pp. 45–58.Google Scholar
  23. 23.
    Delbaen, F. and Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Mathematische Annalen 312 (1997), 215–250.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Duffie, D. and Singleton, K.J.: Modeling term structures of defaultable bonds. Review of Financial Studies 12 (1999), 687–720.CrossRefGoogle Scholar
  25. 25.
    El Karoui, N., Hamad`ene, S. and Matoussi, A.: Backward stochastic differential equations and applications. Chapter 8 of “Indifference Pricing: Theory and Applications”, René Carmona, ed., Springer-Verlag, 267–320 (2008).Google Scholar
  26. 26.
    El Karoui, N., Kapoudjian, E., Pardoux, C., Peng, S., and Quenez, M.-C.: Reflected solutions of backward SDEs, and related obstacle problems for PDEs. Annals of Probability 25 (1997), 702–737.Google Scholar
  27. 27.
    El Karoui, N., Peng, S., and Quenez, M.-C.: Backward stochastic differential equations in finance. Mathematical Finance 7 (1997), 1–71.Google Scholar
  28. 28.
    Fleming, W.H. and Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, Second Edition. Springer-Verlag, Berlin, 2006.Google Scholar
  29. 29.
    Jacod, J. and Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin, 2003.Google Scholar
  30. 30.
    Jeanblanc, M. and Rutkowski, M.: Default risk and hazard processes. In: Mathematical Finance – Bachelier Congress 2000, edited by Geman, H., Madan, D., Pliska, S.R. and Vorst, T., Springer-Verlag, Berlin, 2002, pp. 281–312.Google Scholar
  31. 31.
    Kallsen, J. and Kühn, C.: Convertible bonds: financial derivatives of game type. In: Exotic Option Pricing and Advanced Lévy Models, edited by Kyprianou, A., Schoutens, W. and Wilmott, P., Wiley, 2005, pp. 277–288.Google Scholar
  32. 32.
    Kifer, Y.: Game options. Finance and Stochastics 4 (2000), 443–463.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Kusuoka, S.: A remark on default risk models. Advances in Mathematical Economics 1 (1999), 69–82.MathSciNetGoogle Scholar
  34. 34.
    Linetsky, V.: Pricing equity derivatives subject to bankruptcy. Mathematical Finance 16(2) (2006), 255–282.MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Kwok, Y. and Lau, K.: Anatomy of option features in convertible bonds. Journal of Futures Markets 24 (2004), 513–532.CrossRefGoogle Scholar
  36. 36.
    Longstaff, F.A. and Schwartz, E.S.: Valuing American options by simulations: a simple least-squares approach. Review of Financial Studies 14(1) (2001), 113–147.CrossRefGoogle Scholar
  37. 37.
    Lvov, D., Yigitbasioglu, A.B. and El Bachir, N.:. Pricing convertible bonds by simulation. Working paper, 2004.Google Scholar
  38. 38.
    Ma, J. and Zhang, J.: Representations and regularities for solutions to BSDEs with reflections. Stochastic Processes and their Applications 115(4) (2005), 539–569.MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Morton, K. and Mayers, D.: Numerical Solution of Partial Differential Equations. Cambridge University Press, Cambridge, 1994.Google Scholar
  40. 40.
    Pardoux, E. and Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations, In: Stochastic Partial Differential Equations and Their Applications. Lecture Notes in Control Inf. Sci. 176, Springer-Verlag, Berlin, 1992, pp. 200–217.Google Scholar
  41. 41.
    Romano, M. and Touzi, N.: Contingent claims and market completeness in a stochastic volatility model. Mathematical Finance 7 (1997), 399–412.MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Sˆırbu, M., Pikovsky, I. and Shreve, S.: Perpetual convertible bonds. SIAM J. Control Optim. 43 (2004), 58–85.Google Scholar
  43. 43.
    Takahashi, A., Kobayashi, T. and Nakagawa, N.: Pricing convertible bonds with default risk. Journal of Fixed Income 11 (2001), 20–29.CrossRefGoogle Scholar
  44. 44.
    Tsitsiklis, J.N. and Van Roy, B.: Regression methods for pricing complex American-style options. IEEE Transactions on Neural Networks 12(4) (2001), 694–703.CrossRefGoogle Scholar
  45. 45.
    Tsiveriotis, K. and Fernandes, C.: Valuing convertible bonds with credit risk. Journal of Fixed Income 8 (1998), 95–102.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Tomasz R. Bielecki
    • 1
    Email author
  • 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

Personalised recommendations