The natural place to start the exposition is with the Black and Scholes (1973) and Merton (1974) milestones. The development of option-pricing techniques and the application to the study of corporate liabilities is where the modeling of credit risk has its foundations. While there was of course research out before this, the optionpricing literature, which views the bonds and stocks issued by a firm as contingent claims on the assets of the firm, is the first to give us a strong link between a statistical model describing default and an economic-pricing model. Obtaining such a link is a key problem of credit risk modeling. We make models describing the distribution of the default events and we try to deduce prices from these models. With pricing models in place we can then reverse the question and ask, given the market prices, what is the market’s perception of the default probabilities. To answer this we must understand the full description of the variables governing default and we must understand risk premiums. All of this is possible, at least theoretically, in the option-pricing framework.
Chapter 2 starts by introducing the Merton model and discusses its implications for the risk structure of interest rates—an object which is not to be mistaken for a term structure of interest rates in the sense of the word known from modeling government bonds. We present an immediate application of the Merton model to bonds with different seniority. There are several natural ways of generalizing this, and to begin with we focus on extensions which allow for closed-form solutions. One direction is to work with different asset dynamics, and we present both a case with stochastic interest rates and one with jumps in asset value. A second direction is to introduce a default boundary which exists at all time points, representing some sort of safety covenant or perhaps liquidity shortfall. The Black-Cox model is the classic model in this respect. As we will see, its derivation has been greatly facilitated by the development of option-pricing techniques. Moreover, for a clever choice of default boundary, the model can be generalized to a case with stochastic interest rates. A third direction is to include coupons, and we discuss the extension both to discretetime, lumpy dividends and to continuous flows of dividends and continuous coupon payments. Explicit solutions are only available if the time horizon is made infinite.