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

The aim of this paper is to define a model which allows traders to assess the value of equity and credit derivatives in a unified framework. We propose closed-form formulas which traders could use to evaluate equity, equity options and credit default swaps (CDSs) in a consistent way. The model can also be used to solve the inverse problem, that is to extract credit-risk sensitive information from market quotes of equity/credit derivatives. 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.