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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.

MSC:
91G40 Credit risk
91B99 Mathematical economics
91B72 Spatial models in economics
60G51 Processes with independent increments; Lévy processes
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[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)
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