Pricing Kernels and Risk-Neutral Measures in Affine Bond Models
Summary
The document asks how a paper’s infinite-product representation of a pricing kernel relates to the familiar one-period stochastic discount factor and the change of measure from historical probabilities to risk-neutral probabilities. It outlines an affine term-structure setup in which the short rate depends on state variables, whose dynamics differ under the two measures, and identifies the associated prices of risk.
The discussion raises a useful conceptual question about linking those measure changes across time, but it does not provide an answer or establish that the displayed formulas are equivalent. The product is presented without a derivation, and the questioner’s alternative expression also appears to contain notation or indexing ambiguities. Readers should therefore treat this as a question about reconciling conventions, timing, and normalization, rather than as a worked pricing-kernel derivation.
Key ideas
- Affine bond models can specify different state-variable dynamics under historical and risk-neutral measures.
- Prices of risk describe how those dynamics are related.
- A one-period stochastic discount factor can define a change of measure when properly normalized and positive.
- The document poses, but does not resolve, how an infinite product over future shocks relates to a one-period kernel.
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# Pricing kernel representation
# 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 \begin{equation} r_t = a + b' X_t \end{equation}
the state variables $X_t$, under $Q$, follows \begin{equation} X_t = \bar{\mu} + \bar{\rho} X_{t-1} + \Sigma \eta_t \end{equation} so that the price at time $t$ of a bond paying a unitary amount at time $t+n$ equals \begin{equation} p_t^n = E_t^Q(\exp{(-r_t) p_{t+1}^{n-1})} \end{equation} 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 $\lambda_1 = \Sigma^{-1} (\rho-\bar{\rho})$:
\begin{equation} \frac{dQ}{dP}\vert_t = \frac{\xi_{t+1}}{E(\xi_{t+1})} \end{equation} with \begin{equation} \xi_{t+1} = \prod_{j=1}^{\infty} \exp{((-\lambda_0 + \lambda_1 X_{t+j-1}) \epsilon_{t+j})}. \end{equation}
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 \begin{equation} M_{t,t+1}= \exp{(-r_t -\frac{1}{2} \lambda_t' \lambda_t - \lambda_t' \epsilon_t)} \end{equation} so that
\begin{equation} E_t[M_{t,t+1}]= \exp{(-r_t)} \end{equation}
And defining
\begin{equation} \xi_{t,t+1}=\exp{(r_t M_{t,t+1}})=\exp{(-\frac{1}{2} \lambda_t' \lambda_t - \lambda_t'\epsilon_{t+1})} \end{equation}
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 muchShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.