Equity Options and Bond Options in the Leland Model

We reviewed (extended and made more compact / self-explained) some formulas, previously appeared in the literature, for the evaluation of equity options and bond options in the Leland model (1994), where stockholders have a perpetual American option to default. Our formulas have been expressed in terms of binary barrier options, using the results obtained by Rubinstein and Reiner (1991). We also used the results obtained by Barone (2010), where it has been proved that perpetual American options follow a geometric Brownian motion, under the standard Black-Scholes-Merton assumptions. We generalized the formulas for bond options and extended the formulas for both equity and bond options to the case of premature expiry of the contracts at the time of default. Besides, we derived a simple put-call parity for both equity options and bond options. These relationships allow to check our formulas.