Stochastic Discount Factor of CIR bond pricing model

Question

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?

Answer 1

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 $s\tilde{R}$

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