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

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.