Reduced-Form Credit Bond Pricing with Stochastic Hazard Rates

Summary

The document introduces a reduced-form credit bond model in which default is represented as the first jump of a Poisson process and the hazard rate is stochastic. It writes a differential expansion for bond price as a function of time, hazard rate, and the default-count process, including a second-order term for changes in the hazard rate.

It raises questions about why both the hazard rate and jump process appear, and why the expansion includes a squared hazard-rate increment but no squared jump increment. The text does not answer these questions or provide a derivation, so it serves as a prompt about applying Itô calculus to jump processes rather than a complete pricing method. Readers would need additional assumptions about hazard-rate dynamics and the bond payoff at default to obtain a usable model.

Key ideas

• The model represents a credit event as the first jump of a Poisson process with stochastic intensity.
• Bond price is expressed as a function of time, hazard rate, and the jump process.
• The proposed differential includes a second-order term for hazard-rate variation.
• The document asks how jump-process terms should enter the expansion but does not resolve the question.

Tags

# Reduced form of credit model


# Reduced form of credit model












The price for a simple credit bond, where a credit event is modeled as the first jump of a Poisson process $N$, with stochastic hazard rate $\lambda$, is given by

$$P_t = P(t, \lambda, N)$$

such that,

$$\mathrm{d}P_t = \frac{\partial P}{\partial t}\mathrm{d}t + \frac{\partial P}{\partial \lambda }\mathrm{d}\lambda + \frac{\partial P}{\partial N}\mathrm{d}N + \frac{1}{2}\frac{\partial^2 P}{\partial \lambda^2}\mathrm{d}\lambda ^2$$

This is the general formulation of the reduced form of the model. Since $\lambda$ simply describes $N$, why do we need them both in the equation? Also, why are we keeping the $\lambda^2$ term but not the $N^2$ term? Does $N^2$ vanish somehow under Ito's lemma?

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