A Simple Model for Pricing Securities with Equity, Interest-Rate, and Default Risk∗

We develop a model for pricing derivative and hybrid securities whose value may depend on different sources of risk, namely, equity, interestrate, and default risks. In addition to valuing such securities the framework is also useful for extracting probabilities of default (PD) functions from market data. Our model is not based on the stochastic process for the value of the firm, which is unobservable, but on the stochastic processes for interest rates and the equity price, which are observable. The model comprises a risk-neutral setting in which the joint process of interest rates and equity are modeled together with the default conditions for security payoffs. The model is embedded on a recombining lattice which makes implementation feasible with polynomial complexity. We demonstrate the simplicity of calibration of the model to market observable data. The framework is shown to nest many familiar models as special cases. The model is able to uncover not just default probabilities, but also default functions. The framework is extensible to handling correlated default risk and may be used to value distressed convertible bonds, debt-equity swaps, and credit portfolio products such as CDOs. We present numerical and calibration examples to demonstrate the applicability and implementation of our approach. Equity, Interest-rate, Default Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3