# Stochastic Discount Factor of CIR bond pricing model

The CIR model states $dr=\kappa(\theta-r)dt+\sigma dW$ and the corresponding bond pricing equation can be derived from the general equilibrium approach.

The equation is: $\frac{1}{2}\sigma^2rP_{rr}+[\kappa(\theta-r)-\lambda r]P_r-rP=P_{\tau}$

where $\lambda$ is the factor of risk.

What is the derivation of the corresponding stochastic discount factor and pricing kernel of CIR model?

If you refer me to the exact text you are using I will be more specific - since your notation is not like their 1995 paper.

In any case, the derivation of the stochastic discount factor (also called the pricing kernel) in a general equilibrium model must come from the first order conditions of an investors portfolio choice. The investor will have a utility function over something he cares about (wealth, consumption, leisure, etc) and asset risk will be a function of the co-variance between the return on the asset and some function of the investors utility over something he cares about.

Consider the standard framework where an investor makes a consumption investment decision in a two period framework. Given some endowment today $e_0$ consumption is what is left after saving $c_1=e_1-s$, tomorrow you consume the uncertain return on investment $s\tilde{R}$

$$\underset{s}{max}\quad u(e_1-s) + \beta\mathbb{E}[u(s\tilde{R})]$$ The first order conditions of portfolio choice can be written as \begin{align*} u'(c_1) =& \mathbb{E}[u'(c_2)\tilde{R}]\\ 1 =& \mathbb{E}[\beta\frac{u'(c_2)}{u'(c_1)}\tilde{R}]\\ \end{align*}

Anything else you see will be a variation on the same concept.