zbMATH — the first resource for mathematics

Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels. (English) Zbl 1143.91020
The authors assume that the value of underlying assets of a firm is modelled using a general exponential spectrally negative Lévy process. It is shown that an analytical treatment of the optimal bankruptcy level is possible and that the smooth-pasting condition is not always appropriate. The analytical proof is given for the fact that, depending on the path regularity of the underlying Lévy process, a principle of either smooth pasting or continuous pasting should be applied, accordingly to unbounded or bounded variation of Lévy process, respectively. Some notions of fluctuation theory of Lévy processes, including a number of identities expressed in terms of scale functions, from which it is possible to give analytic expressions for the value and debt of a firm, are discussed. The computation of the optimal endogenous bankruptcy level is also discussed.

91G40 Credit risk
91B99 Mathematical economics
91B72 Spatial models in economics
60G51 Processes with independent increments; Lévy processes
Full Text: DOI
[1] Bertoin J. (1996) Lévy Processes. Cambridge University Press, Cambridge · Zbl 0861.60003
[2] Bertoin J. (1997) Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Probab. 7, 156–169 · Zbl 0880.60077 · doi:10.1214/aoap/1034625257
[3] Chan, T., Kyprianou, A.E.: Smoothness of scale functions for spectrally negative Lévy processes. (preprint, 2005). http://www.maths.bath.ac.uk/\(\sim\)ak257/levyc2version11.pdf
[4] Chen, N. Kou, S.: Credit spreads, optimal capital structure, and implied volatility with endogenous default and jump risk (preprint, 2005). http://www.newton.cam.ac.uk/preprints/NI05031.pdf · Zbl 1168.91379
[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, 719–740 · Zbl 0808.65140 · doi:10.1214/aoap/1177004968
[6] Duffie D., Lando D. (2001) Term structure of credit spreads with incomplete accounting information. Econometrica 69, 633–664 · Zbl 1019.91022 · doi:10.1111/1468-0262.00208
[7] Emery D.J. (1973) Exit problem for a spectrally positive process. Adv. Appl. Probab. 5, 498–520 · Zbl 0297.60035 · doi:10.2307/1425831
[8] Hilberink B., Rogers L.C.G. (2002) Optimal capital structure and endogenous default. Finance Stoch. 6, 237–263 · Zbl 1002.91019 · doi:10.1007/s007800100058
[9] Kyprianou A.E. (2006) Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin Heidelberg New York · Zbl 1104.60001
[10] Lambert A. (2000) Completely asymmetric Lévy processes confined in a finite interval. Ann. Inst. Henri Poincaré 36, 251–274 · Zbl 0970.60055 · doi:10.1016/S0246-0203(00)00126-6
[11] Leland H.E., Toft K.B. (1996) Optimal capital structure, endogeneous bankruptcy, and the term structure of credit spreads. J. Finance 51:987–1019 · doi:10.2307/2329229
[12] Leland H.E. (1994) Corporate debt value, bond covenants, and optimal capital structure with default risk. J. Finance 49:1213–1252 · doi:10.2307/2329184
[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)
[14] Surya, B.A.: Evaluating scale functions of spectrally negative Lévy processes (preprint, 2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.