Vulnerable Options, Risky Corporate Bond and Credit Spread¤

In the current literature, the focus of credit risk analysis has been either on the valuation of risky corporate bond and credit spread, or on the valuation of vulnerable options, but never both in the same context. There are two main concerns with existing studies. First, corporate bonds and credit spreads are generally analyzed in a context where corporate debt is the only liability of the ̄rm and ̄rm's value follows a continuous stochastic process. This set-up implies a zero short term spread, which is strongly rejected by empirical observations. The failure of generating non-zero short term credit spreads may be attributed to the simpli ̄ed assumption on corporate liabilities. Since a corporation generally has more than one type of liabilities, modelling multiple liabilities may help to incorporate discontinuity in ̄rm's value and hence lead to realistic credit term structures. Second, vulnerable options are generally valued under the assumption that a ̄rm can fully payo® the option if the ̄rm's value is above the default barrier at option's maturity. Such an assumption is not realistic since a corporation can ̄nd itself in a solvent position at option's maturity but with assets insu±cient to payo® the option. The main contribution of this study is to address the above concerns. The proposed framework extends the existing equity-bond capital structure to an equity-bond-derivative setting, and encompasses many existing models as special cases. The ̄rm under study has two types of liabilities: a corporate bond and a short position in a call option. The risky corporate bond, credit spreads and vulnerable options are analyzed, and are compared with their counterparts from previous models. Numerical results show that adding a derivative type of liability can lead to positive short term credit spreads and various shapes of credit spread term structures, which are not possible in previous models. In addition, as a surprising result we ̄nd that vulnerable options need not always be worth less than their default free counterparts. ¤ We are grateful to the Social Sciences and Humanities Research Council of Canada for ̄nancial support. We also acknowledge, respectively, the Edith Whyte Research Grant at the Queen's and the University of Toronto Connaught Fund. The analysis and conclusions of this paper are those of the authors and do not indicate concurrence by the Chicago Mercantile Exchange. We thank participants at the Queen's University's workshop, the 11th annual PACAP/FMA conference, and the 1999 Northern Finance Association conference for their comments. We especially thank Andrew Chen and two anonomous referees for their comments and suggestions.