Modeling and assessing stochastic discount factors
Asset pricing theory tells us that there is no arbitrage in the economy if and only if there exists a pricing kernel (also called a stochastic discount factor). In an equilibrium setting, the pricing kernel is given by the marginal utility of optimal consumption.
Once a pricing kernel is given, zero coupon bond prices are determined. We show that pricing kernels that give rise to meaningful term structure models are uniquely determined by their drifts. This result enables us to develop precise connections among the pricing kernel approach, the Heath-Jarrow-Morton framework and the Flesaker-Hughston formulation of positive interests. As applications of these connections, we demonstrate that any Heath-Jarrow-Morton model can be supported by a Cox-Ingersoll-Ross production economy, and we provide a way of verifying positivity of interest rates directly from the primitive data of the Heath-Jarrow-Morton framework. Our result also yields a new class of positive interest rate models that can fit any type of initial yield curve, match the variances and covariances of bond returns, and yet have tractable expressions for bond prices and forward rates.
Hansen and Jagannathan use maximum pricing errors to assess the misspecification of pricing kernels. We propose an alternative way based on large deviation theory. The key idea is to use relative entropy to measure the distance between the empirical measure and the set of measures that lead to correct pricing under a given pricing kernel. This relative entropy measure is shown to represent the rate at which the probability of correct pricing decreases.
The application of our approach to the existing observable pricing kernels shows that the degree of misspecification for different pricing kernels is more varied using the relative entropy measure than using Hansen and Jagannathan's maximum pricing error criterion, which does not provide a lot of variety in specification errors across all the tested models. Also, our ranking of pricing kernels is slightly different from Hansen and Jagannathan's ranking in that the pricing kernel in the power utility model is less misspecified than the one in the logarithmic utility model.