# 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?

• This seems to follow from Ito's lemma but because $N$ is a finite variation process it enters similarly to $t$. – fesman Dec 17 '20 at 17:47
• @Erik Kjellgren Are you sure $\delta$ instead of $\partial$ didn't have a meaning? – fesman Dec 17 '20 at 19:33
• @fesman $\delta$ would imply it is a functional derivative. Meaning that $t$, $\lambda$, and $N$ are not variables, but functions themself. If we look at Wikipedia for Itô's lemma, en.wikipedia.org/wiki/It%C3%B4%27s_lemma. We can see that it is for partial derivatives and not functional derivatives. – Erik Kjellgren Dec 17 '20 at 19:38
• thanks guys, yeah i meant to write partial – dayum Dec 17 '20 at 19:44
• @Erik Kjellgren Ok thanks I thought it might have a different symbol if the multiplier on $dN$ is not a partial derivative but a discrete change in the function when the jump occurs, (see Ito's lemma for jump processes). – fesman Dec 17 '20 at 19:47