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

We revisit the previous work of Leland [13], Leland and Toft [12] and Hilberink and Rogers [8] on optimal capital structure and show that the issue of an optimal endogenous default 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 phenomena, namely that, depending on the nature of the small jumps, the optimal default 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 default level according to the appropriate choice of fit.