An Equity-Based Credit Risk Model

Abstract

This chapter presents a structural model a la Leland (1994) that is, at the same time, novel, simple, and able to explain the quotes of credit default swaps (CDS), equity, and equity options. The model gives a closed-form formula for the term structure of default probabilities and can be calibrated to fit the CDS spreads. It also offers closed-form formulas for equity, equity volatility, and equity options. Differently from other structural models, debt has been modeled as a perpetual fixed-rate bond, instead of a zero-coupon bond with finite maturity. Therefore, default can happen at any time, and not only at the bond's maturity. The model (which belongs to the class of first-passage models) specifies default as the first time the firm's asset value hits a lower barrier. The barrier is endogenously determined as a solution of an optimal stopping problem (stockholders’ equity maximization). Equity is seen as a portfolio that contains a perpetual American option to default and can be valuated by using the results of Rubinstein-Reiner (1991) for barrier options. Equity options are valued by a closed-form formula that requires only an extra parameter (leverage) with respect to the standard input list of Black–Scholes–Merton equation. The formula is consistent with the volatility skew that is generally observed in the equity options markets and can be used to estimate the firms’ implied leverage, as it is perceived by traders. The chapter concludes with an application of the model to the case of Goldman Sachs.