# Pricing kernel representation

I am reading this paper https://mpra.ub.uni-muenchen.de/4969/1/MPRA_paper_4969.pdf pp.6-7 on discrete-time bond pricing. The model adopted is a a common affine model,

the short rate follows $$$$r_t = a + b' X_t$$$$

the state variables $$X_t$$, under $$Q$$, follows $$$$X_t = \bar{\mu} + \bar{\rho} X_{t-1} + \Sigma \eta_t$$$$ so that the price at time $$t$$ of a bond paying a unitary amount at time $$t+n$$ equals $$$$p_t^n = E_t^Q(\exp{(-r_t) p_{t+1}^{n-1})}$$$$ According to the authors "The link between the risk-neutral distribution Q and the historical distribution P is given by the prices of risk, denoted by $$\lambda_0 = \Sigma^{-1} (\mu-\bar{\mu})$$ and $$\lambda_1 = \Sigma^{-1} (\rho-\bar{\rho})$$:

$$$$\frac{dQ}{dP}\vert_t = \frac{\xi_{t+1}}{E(\xi_{t+1})}$$$$ with $$$$\xi_{t+1} = \prod_{j=1}^{\infty} \exp{((-\lambda_0 + \lambda_1 X_{t+j-1}) \epsilon_{t+j})}.$$$$

However, I do not understand the representation of the pricing kernel: where this last product comes from?

Usually, for instance in https://web.stanford.edu/~piazzesi/AP.pdf, the pricing kernel is $$$$M_{t,t+1}= \exp{(-r_t -\frac{1}{2} \lambda_t' \lambda_t - \lambda_t' \epsilon_t)}$$$$ so that

$$$$E_t[M_{t,t+1}]= \exp{(-r_t)}$$$$

And defining

$$$$\xi_{t,t+1}=\exp{(r_t M_{t,t+1}})=\exp{(-\frac{1}{2} \lambda_t' \lambda_t - \lambda_t'\epsilon_{t+1})}$$$$

we can define a new measure $$Q$$ equivalent to $$P$$ since $$E[\xi_{t,t+1}]=1$$ and $$\xi_{t,t+1}>0$$, and clearly,$$\frac{dQ}{dP}\vert_t = \xi_{t,t+1}$$ is the Radon-Nikodym derivative

Are these two representation equivalent?

thank you very much

• I think the notion is confusing though I have seen both forms before. Apparently the Radon-Nikodym derivative is defined over a random variable with arbitrary time horizons versus the latter definition concerns time $t+1$ variables. This is similar to arbitrary horizon vs period by period stochastic discount factors.
– fes
Apr 18, 2021 at 21:30