Decisions in Economics and Finance

, Volume 31, Issue 1, pp 51–72 | Cite as

The optimal capital structure of the firm with stable Lévy assets returns

  • Olivier Le CourtoisEmail author
  • François Quittard-Pinon


This article builds a new structural default model under the assumption that a firm’s assets return follows a dynamics displaying jumps of both signs. In essence, we expand the work of Hilberink and Rogers (itself an extension of the Leland and Toft framework), which deals only with negative jumps. In contrast, we make use of stable Lévy processes, and we compute the values of the firm, debt and equity under this assumption. Theoretical credit spreads can also be obtained in our framework. They prove to be consistent with the empirical credit spreads observed in financial markets.


Optimal capital structure Default risk Stable processes Credit spreads 

JEL Classification

C60 G32 

Mathematics Subject Classification (2000)

60G52 91B28 


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  1. Abate J. and Whitt W. (1995). Numerical Inversion of Laplace Tranforms of probability distributions. ORSA J. Comput. 7: 36–43 Google Scholar
  2. Bernard C., Le Courtois O. and Quittard-Pinon F. (2005). A new procedure for pricing parisian options. J. Deriv. 12(4): 45–53 CrossRefGoogle Scholar
  3. Bertoin J. (1998). Lévy Processes. Cambridge University Press, Cambridge Google Scholar
  4. Bingham N.H. (1975). Fluctuation theory in continuous time. Adv. Appl. Prob. 7: 705–766 CrossRefGoogle Scholar
  5. Black F. and Cox J.C. (1976). Valuing corporate securities: some effects of bond indenture provisions. J. Financ. 31(2): 351–367 CrossRefGoogle Scholar
  6. Carr P., Geman H. and MadanD.B. Yor M. (2002). The fine structure of asset returns: an empirical investigation. J. Bus. 75(2): 305–332 CrossRefGoogle Scholar
  7. Chen, N., Kou, S.G.: Credit spreads, optimal capital structure, and implied volatility with endogenous default and jump risk. Working Paper (2005)Google Scholar
  8. Dao, T.B., Jeanblanc, M.: Double exponential jump-diffusion process: a structural model of endogenous default barrier with roll-over debt structure. Working Paper (2005)Google Scholar
  9. Doney R.A. (1987). On Wiener-Hopf factorisation and the distribution of extrema for certain stable processes. Ann. Prob. 15(4): 1352–1362 CrossRefGoogle Scholar
  10. Geman H. and Yor M. (1993). Bessel processes, asian options and perpetuities. Math. Financ. 3: 349–375 CrossRefGoogle Scholar
  11. Hilberink B. and Rogers L.C.G. (2002). Optimal capital structure and endogenous default. Financ. Stoch. 6: 237–263 CrossRefGoogle Scholar
  12. Hull J.C., Predescu M. and White A. (2005). Bond prices, default probabilities, and risk premiums. J. Credit Risk 1(2): 53–60 Google Scholar
  13. Jarrow R.A. and Turnbull S.M. (1995). Pricing derivatives on financial securities subject to credit risk. J. Financ. 50: 53–85 CrossRefGoogle Scholar
  14. Jensen M.C. and Meckling W.H. (1976). Theory of the firm: managerial behavior, agency costs and ownership structure. J. Financ. Econ. 3: 305–360 CrossRefGoogle Scholar
  15. Le Courtois O. and Quittard-Pinon F. (2006). Risk-neutral and actual default probabilities with an endogenous bankruptcy jump-diffusion model. Asia-Pacific Financ. Markets 13: 11–39 CrossRefGoogle Scholar
  16. Leland H.E. (1994a). Corporate debt value, bond covenants and optimal capital structure. J. Financ. 49(4): 1213–1252 CrossRefGoogle Scholar
  17. Leland, H.E.: Bond prices, yield spreads, and optimal capital structure with default risk. Working Paper No. 240, IBER, University of California, Berkeley (1994b)Google Scholar
  18. Leland H.E. (2004). Predictions of default probabilities in structural models of debt. J. Invest. Manage. 2(2): 5–20 Google Scholar
  19. Leland H.E. and Toft K. (1996). Optimal capital structure, endogenous bankruptcy and the term structure of credit spreads. J. Finance 51(3): 987–1019 CrossRefGoogle Scholar
  20. Longstaff F.A. and Schwartz E.S. (1995). A simple approach to valuing risky fixed and floating rate debt. J. Financ. 50(3): 789–819 CrossRefGoogle Scholar
  21. Madan D. and Unal H. (1998). Pricing the risks of default. Rev. Deriv. Res. 2: 121–160 Google Scholar
  22. McGill P. (1989). Computing the overshoot of a Lévy process. Stoch. Anal. Path Integ. Dyn. 200: 165–196 Google Scholar
  23. McGill, P.: A Plug for Wiener-Hopf methods. Preprint (2003)Google Scholar
  24. Merton R.C. (1974). On the pricing of corporate debt: the risk structure of interest rates. J. Financ 29(2): 449–470 CrossRefGoogle Scholar
  25. Modigliani F. and Miller M. (1958). The cost of capital, corporation finance and the theory of investment. Am. Econ. Rev. 48(3): 261–297 Google Scholar
  26. Rogers L.C.G. (2000). Evaluating first-passage probabilities for spectrally one-sided Lévy processes. J. Appl. Prob. 37: 1173–1180 CrossRefGoogle Scholar
  27. Samorodnisky G. and Taqqu M.S. (1994). Stable non-Gaussian Random Processes. Chapman-Hall, London Google Scholar
  28. Sarig O. and Warga A. (1989). Some empirical estimates of the risk structure of interest rates. J. Finan. 44(5): 1351–1360 CrossRefGoogle Scholar
  29. Satō K. (1987). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, CambridgeGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Olivier Le Courtois
    • 1
    Email author
  • François Quittard-Pinon
    • 1
    • 2
  1. 1.EM Lyon Business SchoolEcully CedexFrance
  2. 2.ISFA Graduate School of Actuarial StudiesUniversity of Lyon 1Lyon 1France

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