Equity Options, Credit Default Swaps and Leverage: A Simple Stochastic-Volatility Model for Equity and Credit Derivatives

The aim of this paper is to extract credit-risk sensitive information from the quotes of equity options and CDSs. In particular, we wish to estimate the firm’s leverage, as it is perceived by traders. This goal is achieved within a model à la Leland (1994), where stockholders have a perpetual American option to default. After making the case for modeling debt in terms of a single perpetual-bond equivalent issue, we define leverage, show the stochastic nature of equity volatility and derive the term structures of default probabilities and credit spreads by making use of the first-passage time distribution function. Then, we give new formulas for call and put options written on stockholders’ equity. The formulas, which depend on the leverage parameter L and make use of the univariate normal distribution function, are consistent with the volatility skew observed in the equity options market and converge to the Black-Scholes-Merton (BSM) equations for L → 1. All the Greeks are simple functions of the standard corresponding letters of the BSM model. The paper concludes with an application of the model to the case of Lehman Brothers and General Motors