Finance and Stochastics

, Volume 11, Issue 1, pp 131–152 | Cite as

Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels

  • A. E. KyprianouEmail author
  • B. A. Surya


We revisit the previous work of Leland [J Finance 49:1213–1252, 1994], Leland and Toft [J Finance 51:987–1019, 1996] and Hilberink and Rogers [Finance Stoch 6:237–263, 2002] on optimal capital structure and show that the issue of determining an optimal endogenous bankruptcy level can be dealt with analytically and numerically when the underlying source of randomness is replaced by that of a general spectrally negative Lévy process. By working with the latter class of processes we bring to light a new phenomenon, namely that, depending on the nature of the small jumps, the optimal bankruptcy level may be determined by a principle of continuous fit as opposed to the usual smooth fit. Moreover, we are able to prove the optimality of the bankruptcy level according to the appropriate choice of fit.


Credit risk Endogenous bankruptcy Scale functions Fluctuation identity Continuous and smooth pasting principles Wiener–Hopf factorization 

JEL Classification


Mathematics Subject Classification (2000)

91B28 91B99 91B72 


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  1. 1.
    Bertoin J. (1996) Lévy Processes. Cambridge University Press, CambridgezbMATHGoogle Scholar
  2. 2.
    Bertoin J. (1997) Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Probab. 7, 156–169CrossRefMathSciNetGoogle Scholar
  3. 3.
    Chan, T., Kyprianou, A.E.: Smoothness of scale functions for spectrally negative Lévy processes. (preprint, 2005). Scholar
  4. 4.
    Chen, N. Kou, S.: Credit spreads, optimal capital structure, and implied volatility with endogenous default and jump risk (preprint, 2005). Scholar
  5. 5.
    Choudhury G.L., Lucantoni D.M., Whitt W. (1994) Multidimensional transform inversion with applications to the transient M/G/1 queue. Ann. Appl. Probab. 4, 719–740MathSciNetGoogle Scholar
  6. 6.
    Duffie D., Lando D. (2001) Term structure of credit spreads with incomplete accounting information. Econometrica 69, 633–664CrossRefMathSciNetGoogle Scholar
  7. 7.
    Emery D.J. (1973) Exit problem for a spectrally positive process. Adv. Appl. Probab. 5, 498–520CrossRefMathSciNetGoogle Scholar
  8. 8.
    Hilberink B., Rogers L.C.G. (2002) Optimal capital structure and endogenous default. Finance Stoch. 6, 237–263CrossRefMathSciNetGoogle Scholar
  9. 9.
    Kyprianou A.E. (2006) Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin Heidelberg New YorkzbMATHGoogle Scholar
  10. 10.
    Lambert A. (2000) Completely asymmetric Lévy processes confined in a finite interval. Ann. Inst. Henri Poincaré 36, 251–274CrossRefMathSciNetGoogle Scholar
  11. 11.
    Leland H.E., Toft K.B. (1996) Optimal capital structure, endogeneous bankruptcy, and the term structure of credit spreads. J. Finance 51:987–1019CrossRefGoogle Scholar
  12. 12.
    Leland H.E. (1994) Corporate debt value, bond covenants, and optimal capital structure with default risk. J. Finance 49:1213–1252CrossRefGoogle Scholar
  13. 13.
    Pistorius, M.R.: An excursion theoretical approach to some boundary crossing problems and the Skorokhod embedding for reflected Lévy processes. In: Séminaire de Probabilités. Springer, Berlin Heidelberg New York (to appear, 2006)Google Scholar
  14. 14.
    Surya, B.A.: Evaluating scale functions of spectrally negative Lévy processes (preprint, 2006)Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of Mathematical SciencesThe University of BathBathUK
  2. 2.Mathematical InstituteUniversity of UtrechtUtrechtThe Netherlands

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